Existence of Compressible Current-Vortex Sheets: Variable Coefficients Linear Analysis

We study the initial-boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the Rankine-Hugoniot relations about an unsteady piecewise smooth solution. This solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity (current-vortex sheet). Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev space W₂^1,σ. Despite the fact that the constant coefficients linearized problem does not meet the uniform Kreiss-Lopatinskii condition, the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current-vortex sheets. The result of this paper is a necessary step in proving the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of magnetohydrodynamics.     

An Elastic-viscoplastic Quasistatic Contact Problem: Existence and Uniqueness of a Weak Solution

In this paper, the contact between an elastic-viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, is also considered. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is established by using approximate problems, the pseudomonotone operators theory, Schauder fixed-point theorem and a comparison result.     

Uniqueness of Weak Solutions to Systems of Conservation Laws

Consider a strictly hyperbolic system of conservation laws in one space dimension: Relying on the existence of the Standard Riemann Semigroup generated by , we establish the uniqueness of entropy-admissible weak solutions to the Cauchy problem, under a mild assumption on the variation of along space-like segments.     

Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L ^∞ in space and time.     

Existence of Weak Solutions for a Two-dimensional Fluid-rigid Body System

We consider the problem of a rigid body immersed in an inviscid incompressible fluid in two dimensional space. The motion of the fluid is described by the incompressible Euler equations and the motion of the rigid body is governed by the balance of linear and angular momentum. A global weak solution is obtained, without any assumption on the weighted norm of the initial vorticity.     

The FENE Dumbbell Polymer Model: Existence and Uniqueness of Solutions for the Momentum Balance Equation

We consider the FENE dumbbell polymer model which is the coupling of the incompressible Navier-Stokes equations with the corresponding Fokker–Planck–Smoluchowski diffusion equation. We show global well-posedness in the case of a 2D bounded domain. We assume in the general case that the initial velocity is sufficiently small and the initial probability density is sufficiently close to the equilibrium solution; moreover an additional condition on the coefficients is imposed. In the corotational case, we only assume that the initial probability density is sufficiently close to the equilibrium solution.     

Existence and Integral Representations of Weak Solutions for Elastic Plates with Cracks

The existence and continuous dependence on the data are investigated in Sobolev spaces for the problem of bending of a Reissner-Mindlin-type plate weakened by a crack when the displacements or the moments and force are prescribed along the two sides of the crack. The cases of both an infinite and a finite plate are considered, and representations are sought for the solutions in terms of single layer and double layer potentials with distributional densities. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation with a velocity which is a non-local quantity depending on the whole shape of the dislocation line. We study the special case where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.     

Existence of Weak Solutions for the Motion of Rigid Bodies in a Viscous Fluid

. We study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of with Dirichlet boundary conditions. By introducing an appropriate weak formulation for the complete problem, we prove existence of solutions for initial velocities in . In the absence of collisions, solutions exist for all time in dimension 2, whereas in dimension 3 the lifespan of solutions is infinite only for small enough data. (Accepted June 10, 1998)     

The Bending-Gradient Theory for Thick Plates: Existence and Uniqueness Results

This paper is devoted to the mathematical justification of the Bending-Gradient theory which is considered as the extension of the Reissner-Mindlin theory (or the First Order Shear Deformation Theory) to heterogeneous plates. In order to rigorously assess the well-posedness of the Bending-Gradient problems, we first assume that the compliance tensor related to the generalized shear force is positive definite. We define the functional spaces to which the variables of the theory belong, then state and prove the existence and uniqueness theorems of solutions of the Bending-Gradient problems for clamped and free plates, as well as for simply supported plates. The obtained results are afterward extended to the general case, i.e., when the compliance tensor related to generalized shear forces is not definite.     

Existence of Weak Solutions for the Three-Dimensional Motion of an Elastic Structure in an Incompressible Fluid

We study here the three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid. The structure and the fluid are contained in a fixed bounded connected set Ω. We show the existence of a weak solution for regularized elastic deformations as long as elastic deformations are not too important (in order to avoid interpenetration and preserve orientation on the structure) and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural (and it corresponds to many physical applications) to consider that its rigid motion (translation and rotation) may be large. The existence result presented here has been announced in [4]. Some improvements have been provided on the model: the model considered in [4] is a simplified model where the structure motion is modelled by decoupled and linear equations for the translation, the rotation and the purely elastic displacement. In what follows, we consider on the structure a model which represents the motion of a structure with large rigid displacements and small elastic perturbations. This model, introduced by [15] for a structure alone, leads to coupled and nonlinear equations for the translation, the rotation and the elastic displacement.     

Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

The purpose of this work is to study the existence of solutions for an unsteady fluid-structure interaction problem. We consider a three-dimensional viscous incompressible fluid governed by the Navier–Stokes equations, interacting with a flexible elastic plate located on one part of the fluid boundary. The fluid domain evolves according to the structure’s displacement, itself resulting from the fluid force. We prove the existence of at least one weak solution as long as the structure does not touch the fixed part of the fluid boundary. The same result holds also for a two-dimensional fluid interacting with a one-dimensional membrane.     

Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics

Compressible vortex sheets are fundamental waves, along with shocks and rarefaction waves, in entropy solutions to multidimensional hyperbolic systems of conservation laws. Understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number \(M > \sqrt{2}\) and instability when \(M < \sqrt{2}\) ; and Artola and Majda’s analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when \(M > \sqrt{2}\) . For the Euler equations in three dimensions, every compressible vortex sheet is violently unstable and this instability is the analogue of the Kelvin–Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free-boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free-boundary problems. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss–Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients. Then we exploit these results to develop a suitable iteration scheme of the Nash–Moser–Hörmander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in three-dimensional MHD.     

The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics

We prove the local-in-time existence of solutions with a surface of current-vortex sheet (tangential discontinuity) of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity. This paper is a natural completion of our previous analysis (Trakhinin in Arch Ration Mech Anal 177:331–366, 2005) where a sufficient condition for the weak stability of planar current-vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities. The original nonlinear problem is a free boundary hyperbolic problem. Since the free boundary is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces . The fact that the Kreiss–Lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates. Therefore, we prove our existence theorem by a suitable Nash–Moser-type iteration scheme.     

Global Existence of Weak Solutions for Viscous Incompressible Flows around a Moving Rigid Body in Three Dimensions

We study the motion of a rigid body of arbitrary shape immersed in a viscous incompressible fluid in a bounded, three-dimensional domain. The motion of the rigid body is caused by the action of given forces exerted on the fluid and on the rigid body. For this problem, we prove the global existence of weak solutions.     

Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity

In the present paper we prove the existence of weak solutions to the equations of non-stationary motion of an incompressible fluid with shear rate dependent viscosity in a cylinder Q = Ω × (0,T), where denotes an open set. For the power-low model with we are able to construct a weak solution with ∇ · u = 0.