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  • Articles  (85,144)
  • 1995-1999  (85,144)
  • Mathematics  (85,144)
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  • 1
    Electronic Resource
    Electronic Resource
    Gravity and Capillary-Gravity Periodic Travelling Waves for Two Superposed Fluid Layers, One Being of Infinite Depth (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 24-61 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050003
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  • 2
    Electronic Resource
    Electronic Resource
    Unconditional Stability of Stationary Flows of Compressible Heat-Conducting Fluids Driven by Large External Forces (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 168-186 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050008
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  • 3
    Electronic Resource
    Electronic Resource
    Well-Posedness of a Gas—Solid Phase Transition Problem (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 282-308 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050012
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  • 4
    Electronic Resource
    Electronic Resource
    On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier—Stokes equations (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 356-387 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050015
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  • 5
    Electronic Resource
    Electronic Resource
    Interior Regularity for Solutions to the Modified Navier—Stokes Equations (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 235-281 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050011
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  • 6
    Electronic Resource
    Electronic Resource
    Equivalence of the Euler Equation with a Variational Problem (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 388-408 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050016
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  • 7
    Electronic Resource
    Electronic Resource
    On the Solvability of the Stokes and Navier-Stokes Problems in the Domains That Are Layer-Like at Infinity (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 78-116 
    Details
    ISSN: 1422-6952
    Keywords: Keywords. Stokes and Navier-Stokes equations, weak solutions, layer-like domains.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. Weak solutions to the Stokes and Navier-Stokes problems are proved to exist in domains which, outside a ball, coincide with the three-dimensional layer $\Bbb R^2\times (0,1)$ . Apart from solutions with the finite Dirichlet integral, solutions to the linear problem are constructed with a prescribed behavior at infinity such that the plane-parallel Poiseuille and Couette flows, the rotational flow. A solution to the nonlinear problem is found that drives a nonzero flux to infinity and becomes unique under the data smallness assumption. Estimates for weighted norms of the pressure are derived as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050005
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  • 8
    Electronic Resource
    Electronic Resource
    On the Existence of the Pressure for Solutions of the Variational Navier—Stokes Equations (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 225-234 
    Details
    ISSN: 1422-6952
    Keywords: Keywords. Navier—Stokes, pressure, strong solutions, variational solutions.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. We prove that in general it is not possible to associate a pressure field to the velocity field of a weak solution of the Navier—Stokes equations, if the body force has values in the space V'.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050010
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  • 9
    Electronic Resource
    Electronic Resource
    Partial Regularity of Weak Solutions to the Navier—Stokes Equations in the Class $ L^{\infty}(0,T;\, L^3(\Omega)^3) $ (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 309-325 
    Details
    ISSN: 1422-6952
    Keywords: Keywords. Navier—Stokes equations, weak solutions, regularity.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. We show that if v is a weak solution to the Navier—Stokes equations in the class $ L^{\infty}(0,T;\, L^3(\Omega)^3) $ then the set of all possible singular points of v in $ \Omega $ , at every time $ t_0\in(0,T) $ , is at most finite and we also give the estimate of the number of the singular points.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050013
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  • 10
    Electronic Resource
    Electronic Resource
    Soliton Collisions in the Ion Acoustic Plasma Equations (1999)
    Springer
    Journal of mathematical fluid mechanics 1 (1999), S. 117-130 
    Details
    ISSN: 1422-6952
    Keywords: Keywords. KdV equation, soliton interactions, plasma equations.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. Numerical experiments involving the interaction of two solitary waves of the ion acoustic plasma equations are described. An exact 2-soliton solution of the relevant KdV equation was fitted to the initial data, and good agreement was maintained throughout the entire interaction. The data demonstrates that the soliton interactions are virtually elastic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000210050006
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