Antonín Novotný

On the Existence of Globally Defined Weak Solutions to the Navier—Stokes Equations

Abstract. We prove the existence of globally defined weak solutions to the Navier—Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant satisfies

Singular limits in thermodynamics of viscous fluids

\gamma > 3/2

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

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Weak-strong uniqueness property for the full Navier-Stokes-Fourier system

In this series of lectures we discuss various aspects of the problems arising in scale analysis of the full Navier-Stokes-Fourier system. In particular, several simplied systems used in meteorology, geophysics, or astrophysics will be identied as singular limits of the complete system. The theory is based on a concept of weak solutions and thermodynamic stability. In the rst part of the talk, we briey discuss the available existence theory for the complete Navier-Stokes-Fourier system in the framework of weak solutions. We also shortly address an alternative approach to singular limits based on strong solutions dened on a possibly very short time interval. Next, we examine the role of thermodynamic stability hypotheses in obtaining uniform bounds independent of the scaling parameter. We introduce the ballistic free energy and deduce the so-called total dissipation balance. Then we associate to each variable its essential and residual parts according to the respective scaling. We discuss compactness of the family of solutions to the scaled Navier-Stokes-Fourier system resulting from the uniform bounds. Propagation of acoustic waves plays a crucial role in the analysis of the incompressible limits. We discuss this topic in detail. In particular, we introduce Lighthill’s acoustic analogy and other form of the acoustic equation. We introduce several methods how to handle convergence problem in the convective term. In particular, we develop a method based on dispersion of acoustic waves on large domains. We introduce an abstract formulation of the acoustic equation based on spectral analysis of the Neumann Laplacean. We examine the associated spectral measures and introduce two basic tools for solving the problem: the celebrated RAGE theorem and a result of Tosio Kato. Final part of the lecture is devoted to applications of the abstract method to various specic problems, including those dened on physical domains that may change their

Lp-approach to steady flows of viscous compressible fluids in exterior domains

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.

Weak and Variational Solutions to Steady Equations for Compressible Heat Conducting Fluids

The Navier–Stokes–Fourier system describing the motion of a compressible, viscous and heat conducting fluid is known to possess global-in-time weak solutions for any initial data of finite energy. We show that a weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists. In particular, strong solutions are unique within the class of weak solutions.

Inviscid Incompressible Limits of the Full Navier-Stokes-Fourier System

We investigate steady compressible flows in three-dimensional exterior domains for small data and for both zero and nonzero (but constant) velocity at infinity. We prove existence and uniqueness of solutions in Lp-spaces, p>3, and study their regularity as well as their decay at infinity.

On a simple model of reacting compressible flows arising in astrophysics

We study a steady compressible Navier–Stokes–Fourier system in a bounded three-dimensional domain. We consider a general pressure law of the form

Global solution to the compressible isothermal multipolar fluid

p=(\gamma-1)\varrho e

Anelastic Approximation as a Singular Limit of the Compressible Navier–Stokes System

which includes in particular the case