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  • Articles  (901,864)
  • Mathematics  (901,864)
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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 24-61 
    ISSN: 1422-6952
    Keywords: Keywords. Water waves, bifurcations, spectral theory, dynamical systems.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. The mathematical study of 2D travelling waves in the potential flow of two superposed layers of perfect fluid, with free surface and interfaces (with or without surface tensions) and with the bottom layer of infinite depth, is set as an ill-posed reversible evolution problem, where the horizontal space variable plays the role of a “time”. We give the structure of the spectrum of the linearized operator near equilibrium. This spectrum contains a set of isolated eigenvalues of finite multiplicities, a small number of which lie near or on the imaginary axis, and the entire real axis constitutes the essential spectrum, where there is no eigenvalue, except 0 in some cases. We give a general constructive proof of bifurcating periodic waves, adapting the Lyapunov-Schmidt method to the present (reversible) case where 0 (which is “resonant”) belongs to the continuous spectrum. In particular we give the results for the generic case and for the 1 : 1 resonance case.
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 168-186 
    ISSN: 1422-6952
    Keywords: Keywords. Navier-Stokes equations, heat-conducting fluids, steady states, asymptotic behaviour.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. We prove that any solution to the full Navier-Stokes system of equations of heat-conducting compressible fluid stabilizes to an equilibrium when time tends to infinity.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 282-308 
    ISSN: 1422-6952
    Keywords: Keywords. Viscous, compressible, heat conducting fluid, liquid—solid phase transition, free boundary, classical solution.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. A new model for liquid—solid phase transitions within the frame of complete Navier—Stokes equations in a liquid phase is proposed. It takes into account such properties of liquid as compressibility, viscosity, and heat conductivity. The local existence and uniqueness of a smooth solution to the related initial-boundary value problem is proved.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 356-387 
    ISSN: 1422-6952
    Keywords: Keywords. Navier—Stokes equations, initial-boundary value problems, partial regularity, Hausdorff's dimension.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. We prove a criterion of local Hölder continuity for suitable weak solutions to the Navier—Stokes equations. One of the main part of the proof, based on a blow-up procedure, has quite general nature and can be applied to other problems in spaces of solenoidal vector fields.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 2 (2000), S. 151-184 
    ISSN: 1422-6952
    Keywords: Keywords. The stationary Navier—Stokes system, homogeneous harmonic polynomials, power series expansion and isolated singularity.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in $ B_R \setminus \{0\} $ , $ n \geq 3 $ and $ |u(x)| = o\,(|x|^{-(n - 1)/2}) $ as $ |x| \to 0 $ or $ u \in L^{2n/(n - 1)}(B_R) $ , then (u, p) is a distribution solution and if in addition, $ u \in L^{\beta}(B_R) $ for some $ \beta 〉 n $ then ( u, p) is smooth in B R .
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 2 (2000), S. 99-106 
    ISSN: 1422-6952
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. It is well known that a weak solution (v, p) to the Navier-Stokes equations is regular if v satisfies some suitable extra conditions (see (1.2), (1.3)). However, with the exception of the recent papers [BV4], [BV5] (see also [K], [Be]) not so much attention has been payed to “alternative natural assumptions” that p may fulfill, in order that (v, p) be regular. By “alternative natural assumptions”, we mean assumptions that formally follow from the Poisson equation relating pressure and velocity (see (1.4)). The objective of this paper is to prove that (v, p) is regular if $ |p|/(1 + |v|) $ obeys some conditions that are in formal agreement with this relation.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 2 (2000), S. 315-323 
    ISSN: 1422-6952
    Keywords: Keywords. Navier—Stokes equations, regularity, projection.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. We improve regularity criteria for weak solutions to the Navier-Stokes equations stated in references [1], [3] and [12], by using in the proof given in [3], a new idea introduced by H. O. Bae and H. J. Choe in [1]. This idea allows us, in one of the main hypothesis (see eq. (1.7)), to replace the velocity u by its projection $ \bar u $ into an arbitrary hyperplane of $ {\Bbb R}^n $ ; see Theorem A. For simplicity, we state our results for space dimension $ n \le 4 $ , since if $ n \ge 5 $ the proofs become more technical and additional hypotheses are needed. However, for the interested reader, we will present the formal calculations for arbitrary dimension n.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 2 (2000), S. 365-380 
    ISSN: 1422-6952
    Keywords: Keywords. Euler equation, inviscid flow, instability, method of averaging.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. An example is presented of a class of periodic, two-dimensional, inviscid fluid flows where the stability spectrum contains both discrete unstable eigenvalues and an unstable essential spectrum. The method of averaging is used to demonstrate the existence of unstable eigenvalues. For such flows spectral instability implies nonlinear instability.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 235-281 
    ISSN: 1422-6952
    Keywords: Keywords. The modified Navier—Stokes equations, initial-boundary value problems, interior regularity, Hausdorff's dimension.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. We discuss interior regularity of solutions to the three-dimensional modified Navier—Stokes equations. In particular, we formulate sufficient conditions that guarantee the local Hölder continuity of the velocity gradient.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 388-408 
    ISSN: 1422-6952
    Keywords: Keywords. Lagrange functional, stationary points, C2 solutions of the Euler equation.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. We show in detail in which sense the following two properties of a time dependent, C 2-smooth, divergence-free vector field v are equivalent:¶a) v satisfies the Euler equation of hydrodynamics (with some pressure function p)¶b) v is a stationary point of a suitable Lagrange functional.¶Important steps are the study of surjectivity properties of the derivative of the action functional, and the identification of vector fields orthogonal to the divergence-free fields as gradients, in the sense of classical differentiability. Thus, a foundation of the Euler equation from a variational principle is provided in a form which, to the author's knowledge, was not available so far.
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