ALBERT

Description: 〈p〉Publication date: Available online 24 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Mikhail V. Korobkov, Konstantin Pileckas, Remigio Russo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the boundary value problem for the stationary Navier–Stokes system in two dimensional exterior domain. We prove that any solution of this problem with finite Dirichlet integral is uniformly bounded. Also we prove the existence theorem under zero total flux assumption.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 24 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Eduard Feireisl, Christian Klingenberg, Ondřej Kreml, Simon Markfelder〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The Euler system in fluid dynamics is a model of a compressible inviscid fluid incorporating the three basic physical principles: Conservation of mass, momentum, and energy. We show that the Cauchy problem is basically ill-posed for the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈/math〉-initial data in the class of weak entropy solutions. As a consequence, there are infinitely many measure-valued solutions for a vast set of initial data. Finally, using the concept of relative energy, we discuss a singular limit problem for the measure-valued solutions, where the Mach and Froude number are proportional to a small parameter.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 23 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): M.C. Bortolan, C.A.E.N. Cardoso, A.N. Carvalho, L. Pires〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we deal with Lipschitz continuous perturbations of gradient Morse-Smale semigroups (all critical elements are equilibria). We study the permanence of connections between equilibrium points (structural stability) when subjected to Lipschitz perturbations. To this end we extend the notions of 〈em〉hyperbolicity〈/em〉 and 〈em〉transversality〈/em〉 to the cases in which differentiability is no longer available.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 13 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): P. Braz e Silva, F.W. Cruz, M. Loayza, M.A. Rojas-Medar〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show global existence and uniqueness of solutions for the 3D nonhomogeneous asymmetric fluids equations through a Lagrangian approach. In particular, uniqueness of the solution is proved under quite soft assumptions about its regularity, which brings the knowledge about nonhomogeneous asymmetric fluids to the same level as for the variable density Navier-Stokes equations.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 13 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Yannick Holle〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions. First, we construct approximate solutions by a vector-valued BGK model with a kinetic coupling function. Introducing so-called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations. We will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution. These inequalities define the macroscopic coupling condition. Our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles. We give several examples for coupling functions and prove corresponding entropy flux inequalities. We prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross-sectional area.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Guoqiang Ren, Bin Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work we consider the two-species chemotaxis system with logistic source. We present the global existence of generalized solutions under appropriate regularity assumptions on the initial data. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 16 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Hantaek Bae〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we establish analyticity of solutions to the barotropic compressible Navier-Stokes equations describing the motion of the density 〈em〉ρ〈/em〉 and the velocity field 〈em〉u〈/em〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉. We assume that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is a small perturbation of 1 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 are analytic in Besov spaces with analyticity radius 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉ω〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉. We show that the corresponding solutions are analytic globally in time when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 are sufficiently small. To do this, we introduce the exponential operator 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msup〉〈mrow〉〈mi〉e〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉ω〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉θ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mi mathvariant="script"〉D〈/mi〉〈/mrow〉〈/msup〉〈/math〉 acting on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈mi〉ρ〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈mi mathvariant="script"〉D〈/mi〉〈/math〉 is the differential operator whose Fourier symbol is given by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉ξ〈/mi〉〈msub〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈msub〉〈mrow〉〈mi〉ξ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉|〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈msub〉〈mrow〉〈mi〉ξ〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉|〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈msub〉〈mrow〉〈mi〉ξ〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉|〈/mo〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"〉〈mi〉θ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is chosen to satisfy 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉θ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉ω〈/mi〉〈/math〉 globally in time.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 16 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Ziyatkhan S. Aliyev, Gunay T. Mamedova〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider a spectral problem for ordinary differential equations of fourth order with the spectral parameter contained in three of the boundary conditions. We study the oscillatory properties of the eigenfunctions and, using these properties, we obtain sufficient conditions for the system of eigenfunctions of the problem in question to form a basis in the space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mo〉∞〈/mo〉〈/math〉, after removing three functions.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Julián López-Gómez, Pierpaolo Omari〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉/〈/mo〉〈msqrt〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/msqrt〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mspace width="0.25em"〉〈/mspace〉〈mtext〉in 〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉λ〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉 is a parameter, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉a〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 changes sign once in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 at the point 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉z〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi〉f〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉∩〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="script"〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is positive and increasing in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 with a potential, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msubsup〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉t〈/mi〉〈/math〉, superlinear at +∞. In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"〉〈mi〉λ〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈msup〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈/math〉, we can characterize the existence of singular bounded variation solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈msup〉〈mrow〉〈mo stretchy="true"〉(〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈munderover〉〈mo movablelimits="false"〉∫〈/mo〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉z〈/mi〉〈/mrow〉〈/munderover〉〈mi〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉t〈/mi〉〈mo stretchy="true"〉)〈/mo〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉z〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mtext〉and〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mo stretchy="true"〉(〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈munderover〉〈mo movablelimits="false"〉∫〈/mo〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉z〈/mi〉〈/mrow〉〈/munderover〉〈mi〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉t〈/mi〉〈mo stretchy="true"〉)〈/mo〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉z〈/mi〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo〉.〈/mo〉〈/math〉〈/span〉 No previous result of this nature is known in the context of the theory of superlinear indefinite problems.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Qian Zhang, Peiguang Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The four-component chemotaxis-Navier-Stokes system〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mrow〉〈mrow〉〈mo stretchy="true"〉{〈/mo〉〈mtable displaystyle="true" columnspacing="0.2em"〉〈mtr〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈mi〉u〈/mi〉〈mo〉⋅〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉n〈/mi〉〈mo〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉⋅〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉c〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉−〈/mo〉〈mi〉n〈/mi〉〈mi〉m〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈mi〉u〈/mi〉〈mo〉⋅〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉c〈/mi〉〈mo〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉c〈/mi〉〈mo〉−〈/mo〉〈mi〉c〈/mi〉〈mo〉+〈/mo〉〈mi〉m〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈mi〉u〈/mi〉〈mo〉⋅〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉m〈/mi〉〈mo〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉m〈/mi〉〈mo〉−〈/mo〉〈mi〉n〈/mi〉〈mi〉m〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉⋅〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉P〈/mi〉〈mo〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mo〉+〈/mo〉〈mi〉m〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉ϕ〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉⋅〈/mo〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mrow〉〈/math〉〈/span〉 is considered in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉. By using Fourier localization technique and the structure of equations, we obtain the existence and uniqueness of weak solutions for the above system for a large class of initial data.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Aram L. Karakhanyan, Ahmad Sabra〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study the near-field refractor problem with point source at the origin and prescribed target on the given receiver surface Σ. This nonvariational problem can be studied in the framework of prescribed Jacobian equations. We construct the corresponding generating function and show that the Aleksandrov and the Brenier type solutions are equivalent. Our main result establishes local smoothness of Aleksandrov's solutions when the data is smooth and when the medium containing the source has smaller refractive index than the medium containing the target. This is done by deriving the Monge-Ampère type equation that smooth solutions satisfy and establishing the validity of the MTW condition for a large class of receiver surfaces, which in turn implies the local 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 regularity of the refractor.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Suting Wei, Jun Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the Fife-Greenlee problem〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mo stretchy="true" maxsize="2.4ex" minsize="2.4ex"〉(〈/mo〉〈mi〉u〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="true" maxsize="2.4ex" minsize="2.4ex"〉)〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in 〈/mtext〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="2em"〉〈/mspace〉〈mfrac〉〈mrow〉〈mo〉∂〈/mo〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉∂〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/mfrac〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈mtext〉on〈/mtext〉〈mspace width="0.25em"〉〈/mspace〉〈mo〉∂〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈/math〉〈/span〉 where Ω is a bounded domain in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 with smooth boundary, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉ε〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 is a small parameter, 〈em〉ν〈/em〉 denotes the unit outward normal of ∂Ω. Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi mathvariant="normal"〉Γ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉y〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉:〈/mo〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 be a simple smooth curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two disjoint nonempty components. We assume that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈mspace width="0.2em"〉〈/mspace〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/math〉 on Ω and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi mathvariant="bold"〉a〈/mi〉〈mo〉≠〈/mo〉〈mn〉0〈/mn〉〈/math〉 on Γ, and also some admissibility conditions hold for 〈strong〉a〈/strong〉, Γ and ∂Ω. For any fixed integer 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"〉〈mi〉N〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉2〈/mn〉〈mi〉m〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉1〈/mn〉〈mo〉≥〈/mo〉〈mn〉3〈/mn〉〈/math〉, we will show the existence of a clustered solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈/math〉 with 〈em〉N〈/em〉-transition layers near Γ with mutual distance 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"〉〈mi〉O〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉ε〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mi mathvariant="normal"〉log〈/mi〉〈mo〉⁡〈/mo〉〈mi〉ε〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, provided that 〈em〉ε〈/em〉 stays away from a discrete set of values at which resonance occurs.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Xinyue Evelyn Zhao, Bei Hu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Large number of papers have been devoted to the study of tumor models. Wellpostedness as well as properties of solutions are systematically studied [1], [2], [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. The properties include asymptotic behavior, stability, bifurcation analysis, etc. In this paper we study a tumor model with a time-delay, and establish a bifurcation result for all even mode 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉. The time delay represents the time taken for cells to undergo replication (approximately 24 hours). In contrast to some results in the literature where bifurcation is established for sufficiently large bifurcation parameter, our result includes the smallest bifurcation point, which is crucial as this is the point of stability changes under certain conditions as we have shown in [40]. The inclusion of a time delay, although biologically very reasonable, introduces two significant mathematical challenges: (a) the explicit solution utilized to verify the bifurcation theorem is no longer available; (b) the system becomes non-local because of the time-delay, which in term produces technical difficulties for the PDE estimates. To our knowledge, this is the first paper on the study of bifurcation for tumor models with a time delay.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Paolo Maremonti, Senjo Shimizu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the Navier-Stokes initial boundary value problem in 3D-exterior domains with non-decaying initial data. We investigate the existence of weak solutions defined for all 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉t〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 for large non-decaying initial data. To achieve the result, we extend the half-space technique developed by Maremonti and Shimizu (2018) [26] to the case of an exterior domain.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Kaiqiang Li, Weike Wang, Xiongfeng Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we consider the asymptotic stability of rarefaction wave to the equilibrium diffusion limit equations without viscosity from radiation hydrodynamic. The present pressure includes a fourth order term about the absolute temperature from radiation effect as well as the ideal polytropic part, which brings the main difficulty to prove the asymptotic stability of the rarefaction wave. To overcome it, we impose an additional restriction condition on the density and the temperature at the far field, see (1.14). This condition is sufficient to achieve the a priori estimates of the solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): De Tang, Yuming Chen〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we mainly study a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates. One interesting feature of the model is that the boundary condition at the downstream end represents a net loss of individuals, which is tuned by a parameter 〈em〉b〈/em〉 to measure the magnitude of the loss. When the upstream end has the no-flux condition, Lou and Zhou (2015) [11] have confirmed that large diffusion rate is more favorable when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mn〉0〈/mn〉〈mo〉≤〈/mo〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/math〉. Here we consider the case where the upstream end has the free-flow condition, which means that the upstream end is linked to a lake. We firstly investigate the corresponding single species model. Here we establish the existence and uniqueness of positive steady states. Then for the two-species model, we find that the parameter 〈em〉b〈/em〉 can be regarded as a bifurcation parameter. Precisely, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mn〉0〈/mn〉〈mo〉≤〈/mo〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/math〉, large diffusion rate is more favorable while when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉, small diffusion rate is selected (if exists). When 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈/math〉, the system is degenerate in the sense that there is a compact global attractor consisting of a continuum of steady states.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Alberto Bressan, Sondre T. Galtung, Audun Reigstad, Johanna Ridder〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The models introduced in this paper describe a uniform distribution of plant stems competing for sunlight. The shape of each stem, and the density of leaves, are designed in order to maximize the captured sunlight, subject to a cost for transporting water and nutrients from the root to all the leaves. Given the intensity of light, depending on the height above ground, we first solve the optimization problem determining the best possible shape for a single stem. We then study a competitive equilibrium among a large number of similar plants, where the shape of each stem is optimal given the shade produced by all others. Uniqueness of equilibria is proved by analyzing the two-point boundary value problem for a system of ODEs derived from the necessary conditions for optimality.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Jann-Long Chern, Masato Hashizume, Gyeongha Hwang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the following nonlinear Neumann problem〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mrow〉〈mo stretchy="true"〉{〈/mo〉〈mtable columnspacing="0em"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉−〈/mo〉〈mi〉γ〈/mi〉〈mfrac〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo〉+〈/mo〉〈mi〉μ〈/mi〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mfrac〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈msubsup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉 in 〈/mtext〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈/msub〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈mn〉3〈/mn〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mfrac〉〈mrow〉〈mo〉∂〈/mo〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉∂〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉 on 〈/mtext〉〈mo〉∂〈/mo〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈/msub〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉γ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mover accent="true"〉〈mrow〉〈mi〉γ〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mo〉:〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉2〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msubsup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mn〉2〈/mn〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is the ball centered at the origin with radius 〈em〉R〈/em〉. Firstly, we establish the existence of infinitely many positive radial solutions which are singular at the origin. Secondly, we investigate the existence and regularity of a least-energy solution. Lastly, we study the symmetric properties of a regular least-energy solution.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 268, Issue 7〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: Available online 20 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Matthias Hieber, Christian Stinner〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉It is shown that the classical as well as quasilinear Keller-Segel systems with non-degenerate diffusion possess for given 〈em〉T〈/em〉-periodic and sufficiently small forcing functions a unique, strong 〈em〉T〈/em〉-time periodic solution. The proof given relies on the existence of strong 〈em〉T〈/em〉-periodic solutions for the linearized system, its characterization in terms of maximal 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-regularity of the underlying operator and a quasilinear version of the Arendt-Bu Theorem. The latter is of independent interest and yields the existence of strong 〈em〉T〈/em〉-periodic solutions to general quasilinear evolution equations under suitable conditions on the operators and the forcing terms.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 16 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Lin He, Feimin Huang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In the present paper, it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic Navier-Stokes equations, in which the pressure could be general and includes 〈em〉γ〈/em〉-law, and the viscosity coefficient is a smooth function of density. The strength of shock wave could be arbitrarily large. The proof is given by introducing a new variable, which can formulate the original system into a new one, and the elementary energy method introduced in [21].〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Evgeny Yu. Panov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Yunfei Su, Lei Yao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study the hydrodynamic limit for the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations in a two or three dimensional bounded domain when the initial density is bounded away from zero. The proof relies on the relative entropy argument to obtain the strong convergence of macroscopic density of the particles 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉;〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, which extends the works of Goudon-Jabin-Vasseur [15] and Mellt-Vasseur [26] to inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations. Precisely, the relative entropy estimates in [15] and [26] give the strong convergence of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si220.svg"〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉, respectively. However, we only obtain the strong convergence of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si220.svg"〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 from the relative entropy estimate, and we use another way to obtain the strong convergence of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 via the convergence of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si220.svg"〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉. Furthermore, when the initial density may vanish, taking advantage of compactness result 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉↪〈/mo〉〈mo stretchy="false"〉↪〈/mo〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 of Orlicz spaces in 2D, we obtain the convergence of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si373.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉;〈/mo〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, which is used to obtain the relative entropy estimate, thus we also show the hydrodynamic limit for 2D inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations when there is initial vacuum.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 8 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Hong-Mei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 8 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): S. Giuffrè〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The aim of the paper is to study a gradient constrained problem associated with a linear operator. Two types of problems are investigated. The first one is the equivalence between a non-constant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a Hamilton-Jacobi equation in the viscosity sense. The equivalence result is obtained under a condition on the gradient constraint. The second problem is the existence of Lagrange multipliers. We prove that the non-constant gradient constrained problem admits a Lagrange multiplier, which is a Radon measure if the free term of the equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉f〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉. If 〈em〉f〈/em〉 is a positive constant, we regularize the result, namely we prove that the Lagrange multipliers belong to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 8 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Shuxing Chen, Dening Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉For the 3-dimensional steady irrotational isentropic supersonic flow against a general conic projectile, an improved explicit condition is obtained for the construction of approximate solution for the conical shock waves, and thereupon the existence of such conical shock solution is established under assumptions much weaker than previous ones.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 7 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Caidi Zhao, Yanjiao Li, Tomás Caraballo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this article, we first prove, from the viewpoint of infinite dynamical system, sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations. Then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a Liouville type equation. Moreover, we reveal that the equation describing the invariant property of the trajectory statistical solutions is a particular situation of the Liouville type equation. Finally, we study the equations of three-dimensional incompressible magneto-micropolar fluids in detail and illustrate how to apply our abstract results to some concrete autonomous evolution equations.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 8 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Fuke Wu, George Yin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Itô formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Itô formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random “noise”, under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 27 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Yuxi Hu, Reinhard Racke〈/p〉

Description: 〈p〉Publication date: Available online 20 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Amru Hussein〈/p〉

Description: 〈p〉Publication date: Available online 19 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Yang Liu, Takeshi Wada〈/p〉

Description: 〈p〉Publication date: Available online 19 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Hyeonbae Kang, KiHyun Yun〈/p〉

Description: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 268, Issue 9〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: Available online 13 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Sergei A. Nazarov, Jari Taskinen〈/p〉

Description: 〈p〉Publication date: Available online 7 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Dan Dai, Shuai-Xia Xu, Lun Zhang〈/p〉

Description: 〈p〉Publication date: Available online 7 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Tomáš Dohnal, Lisa Wahlers〈/p〉

Description: 〈p〉Publication date: Available online 7 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Huyuan Chen, Laurent Véron〈/p〉

Description: 〈p〉Publication date: Available online 6 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Jaume Llibre, Rafael Ramírez, Valentín Ramírez〈/p〉

Description: 〈p〉Publication date: Available online 6 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Tian-Yi Wang, Jiaojiao Zhang〈/p〉

Description: 〈p〉Publication date: 5 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 268, Issue 8〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: Available online 31 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Shengliang Pan, Yanlong Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The behavior of admissible polygons moving under a crystalline curvature flow is discussed. It is proved that the perimeter of the evolving polygon is preserved, its area is increasing, its crystalline curvature is uniformly bounded under this flow, and the final shape of the solution polygon is the boundary of a Wulff shape.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 16 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Tadahiro Oh, Yuzhao Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉In this paper, we first introduce a new function space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mtext mathvariant="italic"〉M〈/mtext〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉 whose norm is given by the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉ℓ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-sum of modulated 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈/msup〉〈/math〉-norms of a given function. In particular, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.svg"〉〈mi〉θ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉, we show that the space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mtext mathvariant="italic"〉M〈/mtext〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉 agrees with the modulation space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"〉〈msup〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 on the real line and the Fourier-Lebesgue space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"〉〈mi mathvariant="script"〉F〈/mi〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.svg"〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉 is globally well-posed in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"〉〈msup〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 for any 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mo〉∞〈/mo〉〈/math〉, while the renormalized cubic NLS on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/math〉 is globally well-posed in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"〉〈mi mathvariant="script"〉F〈/mi〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 for any 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mo〉∞〈/mo〉〈/math〉.〈/p〉 〈p〉In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 15 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): V. Clark, J.C. Meyer〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉η〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mo〉∞〈/mo〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 15 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Yuanyuan Lian, Kai Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we obtain the boundary pointwise 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 regularity for viscosity solutions of fully nonlinear elliptic equations. That is, if ∂Ω is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 (or 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉) at 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈mo〉∂〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/math〉, the solution is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 (or 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉) at 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉. Our results are new even for the Laplace equation. Moreover, our proofs are simple.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 10 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Shibo Liu, Sunra Mosconi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider a class of stationary Schrödinger-Poisson systems with a general nonlinearity 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and coercive sign-changing potential 〈em〉V〈/em〉 so that the Schrödinger operator 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉V〈/mi〉〈/math〉 is indefinite. Previous results in this framework required 〈em〉f〈/em〉 to be strictly 3-superlinear, thus missing the paramount case of the Gross-Pitaevskii-Poisson system, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi〉t〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉t〈/mi〉〈/math〉; in this paper we fill this gap, obtaining non-trivial solutions when 〈em〉f〈/em〉 is not necessarily 3-superlinear.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 10 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Víctor Ayala, Adriano Da Silva, Philippe Jouan, Guilherme Zsigmond〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study the main properties of control sets with nonempty interior of linear systems on semisimple Lie groups. We show that, unlike the solvable case, linear systems on semisimple Lie groups may have more than one control set with nonempty interior and that they are contained in right translations of the one around the identity.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 10 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Qiudong Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we go beyond what was proposed in theory by Melnikov ([15]) to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time-periodic equations. Not only we derive integral formula for splitting distances of all orders, but also we develop an analytic theory to evaluate the acquired multiple integrals. We reveal that the dominance of the exponentially small Poincaré/Melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order Melnikov integrals. This symmetry is beheld by many non-Hamiltonian equations.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 10 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Ting Chen, Shimin Li, Jaume Llibre〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the global dynamical behavior of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉Z〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉-equivariant cubic Hamiltonian vector fields with a linear type bi-center at 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mo〉±〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. By using a series of symbolic computation tools, we obtain all possible phase portraits of these 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉Z〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉-equivariant Hamiltonian systems.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 10 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Xiao Ma〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉For 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉r〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 random dynamical systems 〈em〉F〈/em〉 on a compact smooth Riemannian manifold 〈em〉M〈/em〉, the fiber topological entropy is bounded above by an integral formula. Particularly, for the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈/math〉 random dynamical systems, the integral formula coincides with the fiber topological entropy.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 268, Issue 7〈/p〉 〈p〉Author(s): Santiago Barbieri, Laurent Niederman〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We construct a local Nekhoroshev-like result of stability with sharp constants for the planar three-body problem, both in the planetary and in the restricted circular case, by using the periodic averaging technique. Our constructions can be generalized to any near-integrable hamiltonian system whose unperturbed hamiltonian is quasi-convex. The dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account. This allows for a deep analytical understanding of the limits of such techniques in insuring Nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases. Finally, two examples with concrete values are considered, one for the planetary case and one for the restricted case.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 10 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): The Anh Bui〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈/msub〉〈/math〉 be a Schrödinger operator with inverse square potential 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉a〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mn〉3〈/mn〉〈/math〉. The main aim of this paper is to develop the theory of new Besov and Triebel–Lizorkin spaces associated to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈/msub〉〈/math〉 based on the new space of distributions. As applications, we apply the theory to study some problems on the parabolic equation associated to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈/msub〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 10 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Veronica Felli, Roberto Ognibene〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider a class of singularly perturbed domains, obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero. We use an Almgren-type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues, via Courant-Fischer Min-Max characterization and blow-up analysis for scaled eigenfunctions.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 10 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Wen Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be a complete Riemannian manifold. We prove that for any 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mfrac〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉k〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉p〈/mi〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is small enough, some parabolic type gradient bounds hold for the positive solutions of a nonlinear parabolic equation〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉a〈/mi〉〈mi〉u〈/mi〉〈mi mathvariant="normal"〉log〈/mi〉〈mo〉⁡〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈/math〉〈/span〉 on geodesic balls 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉B〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉O〈/mi〉〈mo〉,〈/mo〉〈mi〉r〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 in 〈strong〉M〈/strong〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si122.svg"〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉r〈/mi〉〈mo〉≤〈/mo〉〈mn〉1〈/mn〉〈/math〉. We can also derive the gradient estimates for any solutions to the above nonlinear parabolic equation along the Ricci flow on a closed manifold without any curvature conditions. As its application, we derive some parabolic type gradient estimates for a positive solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of the heat equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈/math〉. Moreover, our estimate is stronger than Zhang and Zhang's estimate (see Remark 2 in Section 4). Those results are generalizations of Li-Yau, Hamilton, Li-Xu type gradient estimates under the integral Ricci curvature bounds.〈/p〉 〈p〉By utilizing the gradient estimates of the heat equation, we obtain Harnack inequalities, the upper bound and the lower bound for the heat kernel, eigenvalue estimate and the lower bound of Green's function on Riemannian manifolds under the integral Ricci curvature bounds.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 9 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): József J. Kolumbán〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid with Navier slip-with-friction conditions at the solid boundary. The fluid-solid system occupies the whole plane. We prove the small-time global exact controllability of the position and velocity of the solid when the control takes the form of a distributed force supported in a compact subset (with nonvoid interior) of the fluid domain, away from the body. The strategy relies on the introduction of a small parameter: we consider fast and strong amplitude controls for which the “Navier-Stokes+rigid body” system behaves like a perturbation of the “Euler+rigid body” system. By the means of a multi-scale asymptotic expansion we construct a controlled solution to the “Navier-Stokes+rigid body” system thanks to some controlled solutions to “Euler+rigid body”-type systems and by using that the influence of the boundary layer on the solid motion turns out to be sufficiently small.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Zhengzheng Chen, Huijiang Zhao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We are concerned with the time-asymptotic behavior toward rarefaction waves for strong non-vacuum solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes equations with degenerate density-dependent viscosity. The case when the pressure 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉p〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉ρ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msup〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈mrow〉〈mi〉γ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and the viscosity coefficient 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉μ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉ρ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msup〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 for some parameters 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉α〈/mi〉〈mo〉,〈/mo〉〈mi〉γ〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉 is considered. For 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉α〈/mi〉〈mo〉≥〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉γ〈/mi〉〈mo〉≥〈/mo〉〈mi mathvariant="normal"〉max〈/mi〉〈mo〉⁡〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉, if the initial data is assumed to be sufficiently regular, without vacuum and mass concentrations, we show that the Cauchy problem of the one-dimensional compressible Navier-Stokes equations admits a unique global strong non-vacuum solution, which tends to the rarefaction waves as time goes to infinity. Here both the initial perturbation and the strength of the rarefaction waves can be arbitrarily large. The proof is established via a delicate energy method and the key ingredient in our analysis is to derive the uniform-in-time positive lower and upper bounds on the specific volume.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Paolo Tilli, Davide Zucco〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the optimal partitioning of a (possibly unbounded) interval of the real line into 〈em〉n〈/em〉 subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as 〈em〉n〈/em〉 tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 7 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Aifang Qu, Hairong Yuan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We formulate a mathematical problem on hypersonic-limit of three-dimensional steady uniform non-isentropic compressible Euler flows of polytropic gases passing a straight cone with arbitrary cross-section and attacking angle, which is to study Radon measure solutions of a nonlinear hyperbolic system of conservation laws on the unit 2-sphere. The construction of a measure solution with density containing Dirac measures supported on the surface of the cone is reduced to find a regular periodic solution of highly nonlinear and singular ordinary differential equations (ODE). For a circular cone with zero attacking angle, we then proved the Newton's sine-squared law by obtaining such a measure solution. This provides a mathematical foundation for the Newton's theory of pressure distribution on three-dimensional bodies in hypersonic flows.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 March 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 268, Issue 6〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: 5 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 268, Issue 10〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: Available online 24 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Daoyin He, Jie Liu, Keyan Wang〈/p〉

Description: 〈p〉Publication date: Available online 24 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Manassés de Souza〈/p〉

Description: 〈p〉Publication date: Available online 20 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Jun Geng, Bojing Shi〈/p〉

Description: 〈p〉Publication date: Available online 20 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Vladimir Bobkov, Pavel Drábek, Yavdat Ilyasov〈/p〉

Description: 〈p〉Publication date: Available online 19 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Tianwen Luo, Peng Qu〈/p〉

Description: 〈p〉Publication date: Available online 17 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Han-Su Zhang, Tiexiang Li, Tsung-fang Wu〈/p〉

Description: 〈p〉Publication date: Available online 17 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): D. De Silva, O. Savin〈/p〉

Description: 〈p〉Publication date: 5 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 268, Issue 10〈/p〉 〈p〉Author(s): Seonghak Kim, Hoang-Hung Vo〈/p〉

Description: 〈p〉Publication date: Available online 10 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Irina Kmit, Lutz Recke, Viktor Tkachenko〈/p〉

Description: 〈p〉Publication date: Available online 28 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Hildebrando M. Rodrigues, J. Solà-Morales〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The purpose of this paper is to present an example of a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="script"〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 (in the Fréchet sense) discrete dynamical system in a infinite-dimensional separable Hilbert space for which the origin is an exponentially asymptotically stable fixed point, but such that its derivative at the origin has spectral radius larger than unity, and this means that the origin is unstable in the sense of Lyapunov for the linearized system. The possible existence or not of an example of this kind has been an open question until now, to our knowledge. The construction is based on a classical example in Operator Theory due to Kakutani.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Yu Tian, Zhaoyin Xiang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we investigate a 3D chemotaxis-Stokes system with general sensitivity, nonlinear cell diffusion and Robin signal boundary condition. After introducing a Lions-Magenes type transformation, we transform the Robin signal boundary condition into the usual homogeneous Neumann boundary value, and then use a two-step iterative method to establish the global existence of bounded weak solutions. Here, the key is to deal with the extra terms coming from the transformation. In the case of homogeneous Robin boundary value, we will adapt the method of Winkler (Calc. Var., 2015 [35]) to investigate the large time behavior and the eventual smoothness of the above weak solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Daniele Bartolucci, Aleks Jevnikar, Youngae Lee, Wen Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We are concerned with the mean field equation with singular data on bounded domains. By assuming a singular point to be a critical point of the 1-vortex Kirchhoff-Routh function, we prove local uniqueness and non-degeneracy of bubbling solutions blowing up at a singular point. The proof is based on sharp estimates for bubbling solutions of singular mean field equations and a suitably defined Pohozaev-type identity.〈/p〉〈/div〉