Variational methods in nonlinear problems
Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations
Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they capture successfully the nonlinear features of the flows, such as shocks and rarefaction waves for...
Vertex centred Discretization of Two-Phase Darcy flows on General Meshes
This paper concerns the discretization of multiphase Darcy flows, in the case of heterogeneous anisotropic porous media and general 3D meshes used in practice to represent reservoir and basin geometries. An unconditionally coercive and symmetric vertex centred approach is introduced in this paper. This scheme extends the Vertex Approximate Gradient scheme (VAG), already introduced for single phase diffusive problems in [9], to multiphase Darcy flows....
Vibrations of a beam between obstacles. Convergence of a fully discretized approximation
We consider mathematical models describing dynamics of an elastic beam which is clamped at its left end to a vibrating support and which can move freely at its right end between two rigid obstacles. We model the contact with Signorini's complementary conditions between the displacement and the shear stress. For this infinite dimensional contact problem, we propose a family of fully discretized approximations and their convergence is proved. Moreover some examples of implementation are presented....
Viscosity solutions and standard Riemann semigroup for conservation laws with boundary
Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients
We introduce small viscosity solutions of hyperbolic systems with discontinuous coefficients accross the fixed noncharacteristic hypersurface Under a geometric stability assumption, our first result is obtained, in the multi-D framework, for piecewise smooth coefficients. For our second result, the considered operator is with (expansive case not included in our first result), thus resulting in an infinity of weak solutions. Proving that this problem is uniformly Evans-stable, we show that...
Viscous profiles for traveling waves of scalar balance laws: The uniformly hyperbolic case.
Vitesse de convergence vers le réel des résonances
Wave equation and multiplier estimates on ax + b groups
Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive Sobolev...
Wave equation with a concentrated moving source
A tempered distribution which is an exact solution of the wave equation with a concentrated moving source on the right-hand side, is obtained in the paper by means of the Cagniard - de Hoop method.
Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators
In this paper, we consider one-dimensional wave equation with real-valued square-summable potential. We establish the long-time asymptotics of solutions by, first, studying the stationary problem and, second, using the spectral representation for the evolution equation. In particular, we prove that part of the wave travels ballistically if q ∈ L2(ℝ+) and this result is sharp.
Wave equations with low regularity coefficients.
Wave front tracking in systems of conservation laws
This paper contains several recent results about nonlinear systems of hyperbolic conservation laws obtained through the technique of Wave Front Tracking.
Weak entropic solution to a scalar hyperbolic-parabolic law.
In this paper we are interested in the Dirichlet problem of a hyperbolic-parabolic degenerate equation. Thanks to a global entropic formulation in the sense of F. Otto, we propose a result of existence and uniqueness of the entropic measure valued solution and of the entropic weak solution in the space DM2.
Weak formulation for nonlinear hyperbolic Stefan problems.
Weak Limits of Solutions to Semilinear Hyperbolic Sytems.
Weak solutions for a hyperbolic system with unilateral constraint and mass loss
Weak solutions for a simple hyperbolic system.
Weak solutions for a viscous -Laplacian equation.
Weak solutions to a nonlinear variational wave equation and some related problems
In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the Young measure theory and related compactness results, in the first section. Then we use the Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section....