Bum Ja Jin

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.

Temporal and spatial decays for the Navier–Stokes equations

We obtain spatial and temporal decay rates of weak solutions of the Navier–Stokes equations, and for strong solutions. For the spatial decay rate of the weak solutions, the power of the weight given by He and Xin in 2001 does not exceed 3/2;. However, we show the power can be extended up to 5/2;.

TEMPORAL AND SPATIAL DECAY RATES OF NAVIER-STOKES SOLUTIONS IN EXTERIOR DOMAINS

We obtain spatial-temporal decay rates of weak solutions of incompressible flows in exterior domains. When a domain has a boundary, the pressure term yields difficulties since we do not have enough information on the pressure term near the boundary. For our calculations we provide an idea which does not require any pressure information. We also estimated the spatial and temporal asymptotic behavior for strong solutions.

EXISTENCE OF STRONG MILD SOLUTION OF THE NAVIER-STOKES EQUATIONS IN THE HALF SPACE WITH NONDECAYING INITIAL DATA

We construct a mild solutions of the Navier-Stokes equations in half spaces for nondecaying initial velocities. We also obtain the uni- form bound of the velocityeld and its derivatives.

Navier--Stokes--Fourier System on Unbounded Domains: Weak Solutions, Relative Entropies, Weak-Strong Uniqueness

We investigate the Navier--Stokes--Fourier system describing the motion of a compressible, viscous, and heat conducting fluid on large class of unbounded domains with no slip and slip boundary conditions. We propose a definition of weak solutions that is particularly convenient for the treatment of the Navier--Stokes--Fourier system on unbounded domains. We introduce suitable weak solutions as weak solutions that satisfy the relative entropy inequality. We prove existence of weak solutions and of suitable weak solutions for arbitrary large initial data for potential forces with an arbitrary growth at large distances. Finally we prove the weak-strong uniqueness principle, meaning that the suitable weak solutions coincide with strong solutions emanating from the same initial data (as long as the latter exist), at least when the potential force vanishes at large distances.

Regularity for the Navier–Stokes equations with slip boundary condition

For the Navier-Stokes equations with slip boundary conditions, we obtain the pressure in terms of the velocity. Based on the representation, we consider the relationship in the sense of regularity between the Navier-Stokes equations in the whole space and those in the half space with slip boundary data.

Inviscid incompressible limits of strongly stratified fluids

We consider the motion of a compressible viscous fluid in the asymptotic regime of low Mach and high Reynolds numbers under strong stratification imposed by a conservative external force. Assuming a bi-dimensional character of the flow, we identify the limit system represented by the so-called lake equation the Euler system supplemented by an anelastic type constraint imposed by the limit density profile.

NAVIER-STOKES EQUATIONS IN BESOV SPACE B -s ∞ , ∞ (ℝ n + )

In this paper we consider the Navier-Stokes equations in the half space. Our aim is to construct a mild solution for initial data in B −� 1 ,1 (R n ), 0 < � < 1. To do this, we derive the estimate of the Stokes flow with singular initial data in B −� 1 ,q(R n), 0 < � < 1, 1 < q � 1.