Continuity of Drag and Domain Stability in the Low Mach Number Limits

We consider a mathematical model of a rigid body immersed in a viscous, compressible fluid moving with a velocity prescribed on the boundary of a large channel containing the body. We assume that the Mach number is proportional to a small parameter ε and that the general boundary of the body contains small asperities of amplitude proportional to ε ^α for a certain α?>?0 and suppose the Navier’s slip condition on this rough boundary. We show that time averages of the drag functional converge, as ε → 0, to the corresponding time averages of the drag for the limit system, whereas the limit system is turning out to be the incompressible Navier–Stokes system with no-slip condition on the smooth limit body.     

Long-time behavior and convergence for semilinear wave equations on ℝ N

We prove that any bounded solution (u, u ₁) ofu _u +du _t –u+f(u)=0,u=u(x, t), x^N,N3, converges to a fixed stationary state provided its initial energy is appropriately small. The theory of concentrated compactness is used in combination with some recent results concerning the uniqueness of the so-called ground-state solution of the corresponding stationary problem.     

Dimension Reduction for Compressible Viscous Fluids

We consider the barotropic Navier-Stokes system describing the motion of a compressible viscous fluid confined to a cavity shaped as a thin rod Ω _ε =εQ×(0,1), Q?R ². We show that the weak solutions in the 3D domain converge to (strong) solutions of the limit 1D Navier-Stokes system as ε→0.     

On the motion of rigid bodies in a viscous fluid

We consider the problem of motion of several rigid bodies in a viscous fluid. Both compressible and incompressible fluids are studied. In both cases, the existence of globally defined weak solutions is established regardless possible collisions of two or more rigid objects.     

On weak solutions to the 2D Savage–Hutter model of the motion of a gravity-driven avalanche flow

We consider the Savage–Hutter system consisting of two-dimensional depth-integrated shallow water equations for the incompressible fluid with the Coulomb-type friction term. Using the method of convex integration we show that the associated initial value problem possesses infinitely many weak solutions for any finite energy initial data. On the other hand, the problem enjoys the weak-strong uniqueness property provided the system of equations is supplemented with the energy inequality.     

A Comparison Theorem for a Nonlinear system Arising in Porous Medium Combustion

Comparison theorems for the initial value finite domain one dimensional heat equation with a discontinuous forcing term are extended to a coupled system of a heat equation and an ordinary differential equation in space, rather than the usual ordinary differential equation in time, that arises in combustion theory.     

The Oberbeck–Boussinesq Approximation as a Singular Limit of the Full Navier–Stokes–Fourier System

We consider the asymptotic limit for the complete Navier–Stokes–Fourier system as both Mach and Froude numbers tend to zero. The limit is investigated in the context of weak variational solutions on an arbitrary large time interval and for the ill-prepared initial data. The convergence to the Oberbeck–Boussinesq system is shown.      

Relative Entropies, Suitable Weak Solutions, and Weak-Strong Uniqueness for the Compressible Navier–Stokes System

We introduce the notion of relative entropy for the weak solutions to the compressible Navier–Stokes system. In particular, we show that any finite energy weak solution satisfies a relative entropy inequality with respect to any couple of smooth functions satisfying relevant boundary conditions. As a corollary, we establish the weak-strong uniqueness property in the class of finite energy weak solutions, extending thus the classical result of Prodi and Serrin to the class of compressible fluid flows.