Finite Energy Weak Solutions Research Articles

Existence of finite energy weak solutions for the equations MHD of compressible fluids

This article concerns the existence of global weak solutions for a compressible Magnetohydrodynamic model. We assume the viscosity and the resistivity to be constant and we prove that Feireisl and Lions's strategies dedicated to the usual barotropic compressible flows may be extended to our system. The only difficulty to be taken into account is the magnetic field dependency. The case with density-dependent viscosity and resistivity coefficients will be treated in a forthcoming paper following Bresch and Desjardins's strategy.

On the existence of finite energy weak solutions to the Navier–Stokes equations in irregular domains

AbstractThis paper studies the existence of weak solutions of the Navier–Stokes system defined on a certain class of domains in ℝ3 that may contain cusps. The concept of such a domain and weak energy solution for the system is defined and its existence is proved. However, thinness of cusps must be related to the adiabatic constant appearing in the pressure law. Copyright © 2008 John Wiley & Sons, Ltd.

On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics

In this paper we consider the global existence of weak solutions to a class of Quantum Hydrodynamics (QHD) systems with initial data, arbitrarily large in the energy norm. These type of models, initially proposed by Madelung, have been extensively used in Physics to investigate Supefluidity and Superconductivity phenomena and more recently in the modeling of semiconductor devices . Our approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method, which iterates a Schrodinger Madelung picture with a suitable wave function updating mechanism. Therefore several \emph{a priori} bounds of energy, dispersive and local smoothing type allow us to prove the compactness of the approximating sequences. No uniqueness result is provided.

Propagation of density-oscillations in solutions to the compressible Navier-Stokes-Poisson system

Concerning a bounded sequence of finite energy weak solutions to the compressible Navier-Stokes-Poisson system (denoted by CNSP), which converges up to extraction of a subsequence, the limit system may not be the same system. By introducing Young measures as in [6, 15], the authors deduce the system (HCNSP) which the limit functions must satisfy. Then they solve this system in a subclass where Young measures are convex combinations of Dirac measures, to give the information on the propagation of density-oscillations. The results for strong solutions to (CNSP) (see Corollary 6.1) are also obtained.

Smoothness of weak solutions to a nonlinear fluid-structure interaction model

The nonlinear fluid-structure interaction coupling the Navier-Stokes equations with a dynamic system of elasticity is consid- ered. The coupling takes place on the boundary (interface) via the con- tinuity of the normal component of the Cauchy stress tensor. Due to a mismatch of parabolic and hyperbolic regularity, previous results in the literature dealt with either a regularized version of the model, or with very smooth initial conditions leading to local existence only. In con- trast, in the case of small but rapid oscillations of the interface, in (3 )t he authors established existence of finite energy weak solutions that are de- fined globally. This is achieved by exploiting new hyperbolic trace regu- larity results which provide a way to deal with the mismatch of parabolic and hyperbolic regularity. The goal of this paper is to establish regularity of weak solutions, for initial data satisfying the appropriate regularity and compatibility conditions imposed on the interface. It is shown that weak solutions equipped with smooth initial data become classical.

On the existence of solutions to the Navier–Stokes–Poisson equations of a two‐dimensional compressible flow

AbstractIn this paper, we consider the Navier–Stokes–Poisson equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(ϱ)=aϱlogd (ϱ) for large ϱ. Here d>1 and a>0. Copyright © 2006 John Wiley & Sons, Ltd.

Superconductors surrounded by normal materials

We consider a generalized Ginzburg–Landau energy functional modelling a superconductor surrounded by a material in the normal state. In this model, the order parameter is defined in the whole space. We derive existence of a global minimizer in weighted Sobolev spaces for both square-integrable and constant-applied magnetic fields. We then prove boundedness and classical elliptic estimates for the order parameter, in order to study the loss of superconductivity for high applied magnetic fields. In two dimensions for the general case and in three dimensions for the case of constant permeability, we show the existence of an upper critical field above which the only finite-energy weak solutions are the normal states. For the three-dimensional case, we show that as the applied field tends to infinity, finite-energy weak solutions tend to the normal state

Propagation of oscillations, complete trajectories and attractors for compressible flows

We study the qualitative properties and the long time behaviour of the finite energy weak solutions to the Navier-Stokes equations of compressible flows. Under certain restrictions imposed on the adiabatic constant $ \gamma $ , the existence of global compact attractors as well as other results on stabilization are obtained.

On the existence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow

We consider the Navier–Stokes equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(ϱ) = aϱlogd(ϱ) for large ϱ. Here d>1 and a > 0. Copyright © 2003 John Wiley & Sons, Ltd.

Existence globale pour un fluide barotrope autogravitant

We consider the global existence of a solution for the Navier–Stokes system of compressible barotropic self-gravitating fluids in three dimensions. We prove that this system admits at least one finite energy weak solution on any time interval [0,T], provided that the adiabatic constant γ satisfies γ>3/2.

Asymptotic behaviour of solutions to nonlinear shells in a supersonic flow 1

Asymptotic behaviour of a nonlinear PDE system describing nonlinear vibrations of a shell immersed in a supersonic gas is considered. The model under consideration is a prototype of hyperbolic-like, nondissipative PDE dynamics with nonlinear damping. The main result is the existence of a compact attractor which “attracts” all finite energy (weak) solutions. It is also shown that the Hausdorf dimension of this attractor is finite. The results obtained previously in the literature on related dynamics deal with weakly compact attractors and regular solutions. Thus, a novel contribution of this paper is to obtain compact in the finite energy space attractors for all finite energy solutions (rather than regular).

The Klein-Gordon equation with light-cone data

It is shown that the characteristic Cauchy problem\(\left( {\frac{{\partial ^2 }}{{\partial t^2 }} - \Delta + 1} \right)\)·u(x,t)=0,u(x,−|x|)=f(x),x∈ℝn,n≧1 has a unique finite energy weak solution for allf such that ∫dx(|∇f|2+|f|2)<∞ and all finite energy weak solutions of the equation are obtained in this way.

Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model

A compressible Oldroyd--B type model with stress diffusion is derived from a compressible Navier--Stokes--Fokker--Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic, compressible, isothermal, viscous Newtonian solvent, are idealized as pairs of massless beads connected with Hookean springs. We develop a-priori bounds for the model, including a logarithmic bound, which guarantee the nonnegativity of the elastic extra stress tensor, and we prove the existence of large data global-in-time finite-energy weak solutions in two space dimensions.