Ondřej Kreml

Weak solutions to the barotropic Navier–Stokes system with slip boundary conditions in time dependent domains

Abstract We consider the compressible (barotropic) Navier–Stokes system on time dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained.

STABILITY OF THE ISENTROPIC RIEMANN SOLUTIONS OF THE FULL MULTIDIMENSIONAL EULER SYSTEM

We consider the complete Euler system describing the time evolution of an inviscid nonisothermal gas. We show that the rarefaction wave solutions of the one-dimensional (1-D) Riemann problem are stable, in particular unique, in the class of all bounded weak solutions to the associated multidimensional problem. This may be seen as a counterpart of the nonuniqueness results of physically admissible solutions emanating from 1-D shock waves constructed recently by the method of convex integration.

Uniqueness of rarefaction waves in multidimensional compressible Euler system

We show that 1-D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.

INCOMPRESSIBLE LIMITS OF FLUIDS EXCITED BY MOVING BOUNDARIES

We consider the motion of a viscous compressible fluid confined to a physical space with a time dependent kinematic boundary. We suppose that the characteristic speed of the fluid is dominated by the speed of sound and perform the low Mach number limit in the framework of weak solutions. The standard incompressible Navier--Stokes system is identified as the target problem.

Non-uniqueness of admissible weak solutions to the Riemann problem for isentropic Euler equations

We study the Riemann problem for multidimensional compressible isentropic Euler equations. Using the framework developed in Chiodaroli et al (2015 Commun. Pure Appl. Math. 68 1157–90), and based on the techniques of De Lellis and Szekelyhidi (2010 Arch. Ration. Mech. Anal. 195 225–60), we extend the results of Chiodaroli and Kreml (2014 Arch. Ration. Mech. Anal. 214 1019–49) and prove that it is possible to characterize a set of Riemann data, giving rise to a self-similar solution consisting of one admissible shock and one rarefaction wave, for which the problem also admits infinitely many admissible weak solutions.

On the low Mach number limit of compressible flows in exterior moving domains

We study the incompressible limit of solutions to the compressible barotropic Navier–Stokes system in the exterior of a bounded domain undergoing a simple translation. The problem is reformulated using a change of coordinates to fixed exterior domain. Using the spectral analysis of the wave propagator, the dispersion of acoustic waves is proved by means of the RAGE theorem. The solution to the incompressible Navier–Stokes equations is identified as a limit.