A  system of the Hamilton--Jacobi  and the continuity equations in the vanishing viscosity limit

Thomas Strömberg

doi:10.3934/cpaa.2011.10.479

Article Contents

2011, Volume 10, Issue 2: 479-506. Doi: 10.3934/cpaa.2011.10.479

This issue Previous Article Free boundary problem for compressible flows with density--dependent viscosity coefficients Next Article Bifurcations of some elliptic problems with a singular nonlinearity via Morse index

A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit

Thomas Strömberg^1,

1.
Department of Mathematics, Luleå University of Technology, SE-971 87 Luleå

Received: January 2010

Revised: July 2010

Published: March 2011

Abstract / Introduction Related Papers Cited by

Abstract

We study the following system of the viscous Hamilton--Jacobi and the continuity equations in the limit as $\varepsilon \downarrow 0$:

$ S^\varepsilon_t+\frac{1}{2}|D S^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon =0$ in $Q_T$, $S^\varepsilon(0,x)=S_0(x)$ in $R^n;$

$ \rho^\varepsilon_t+$ Div$(\rho^\varepsilon D S^\varepsilon)=0$ in $Q_T$, $\rho^\varepsilon(0,x)=\rho_0(x)$ in $R^n$.

Here $Q_T=(0,T]\times R^n$. The potential $V$ and the initial function $S_0$ are allowed to grow quadratically while $\rho_0$ is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity--measure solution $(S,\rho)$.

Keywords:

Mathematics Subject Classification: 35F25, 35L67, 35R05.

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