Analysis of the adiabatic piston problem via methods of continuum mechanics

We consider a system modelling the motion of a piston in a cylinder filled by a viscous heat conducting gas. The piston is moving longitudinally without friction under the influence of the forces exerted by the gas. In addition, the piston is supposed to be thermally insulating (adiabatic piston). This fact raises several challenges which received a considerable attention, essentially in the statistical physics literature. We study the problem via the methods of continuum mechanics, specifically, the motion of the gas is described by means of the Navier–Stokes–Fourier system in one space dimension, coupled with Newton's second law governing the motion of the piston. We establish global in time existence of strong solutions and show that the system stabilizes to an equilibrium state for $t\to \infty$.     

Local strong solutions to the stochastic compressible Navier–Stokes system

We study the Navier–Stokes system describing the motion of a compressible viscous fluid driven by a nonlinear multiplicative stochastic force. We establish local in time existence (up to a positive stopping time) of a unique solution, which is strong in both PDE and probabilistic sense. Our approach relies on rewriting the problem as a symmetric hyperbolic system augmented by partial diffusion, which is solved via a suitable approximation procedure. We use the stochastic compactness method and the Yamada–Watanabe type argument based on the Gyöngy–Krylov characterization of convergence in probability. This leads to the existence of a strong (in the PDE sense) pathwise solution. Finally, we use various stopping time arguments to establish the local existence of a unique strong solution to the original problem.     

On the Long-Time Behaviour of Compressible Fluid Flows Subjected to Highly Oscillating External Forces

We show that the global-in-time solutions to the compressible Navier-Stokes equations driven by highly oscillating external forces stabilize to globally defined (on the whole real line) solutions of the same system with the driving force given by the integral mean of oscillations. Several stability results will be obtained.     

On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries

We study the asymptotic behavior of solutions to the incompressible Navier-Stokes system considered on a sequence of spatial domains, whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter. Imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are non-degenerate, meaning not oriented in a single direction.     

Shape Optimization in Viscous Compressible Fluids

Abstract. We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier—Stokes system for a general viscous barotropic fluid.     

Long‐time convergence of solutions to a phase‐field system

We prove that any global bounded solution of a phase field model tends to a single equilibrium state for large times though the set of equilibria may contain a nontrivial continuum of stationary states. The problem has a partial variational structure, specifically, only the elliptic part of the first equation represents an Euler–Lagrange equation while the second does not. This requires some modifications in comparison with standard methods used to attack this kind of problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Long-Time Stabilization of Solutions to the Ginzburg–Landau Equations of Superconductivity

 The long-time dynamical properties of solutions (φ,A) to the time-dependent Ginzburg–Landau (TDGL) equations of superconductivity are investigated. The applied magnetic field varies with time, but it is assumed to approach a long-time asymptotic limit. Sufficient conditions (in terms of the time rate of change of the applied magnetic field) are given which guarantee that the dynamical process defined by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamical system as time goes to infinity. Analyticity of an energy functional is used to show that every solution of the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg–Landau equations. The standard “φ = − ∇ · A” gauge is chosen.     

Long-Time Stabilization of Solutions to the Ginzburg–Landau Equations of Superconductivity

 The long-time dynamical properties of solutions (φ,A) to the time-dependent Ginzburg–Landau (TDGL) equations of superconductivity are investigated. The applied magnetic field varies with time, but it is assumed to approach a long-time asymptotic limit. Sufficient conditions (in terms of the time rate of change of the applied magnetic field) are given which guarantee that the dynamical process defined by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamical system as time goes to infinity. Analyticity of an energy functional is used to show that every solution of the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg–Landau equations. The standard “φ = − ∇ · A” gauge is chosen. (Received 30 June 2000; in revised form 30 December 2000)