Viscous conservation laws in 1D with measure initial data

Abstract

The one-dimensional viscous conservation law is considered on the whole line u t + f ( u ) x = ε u x x , ( x , t ) ∈ R × R + ¯ , ε > 0 , \begin{equation*} u_t + f(u)_x=\varepsilon u_{xx},\quad (x,t)\in \mathbb {R}\times \overline {\mathbb {R}_{+}},\quad \varepsilon >0, \end{equation*} subject to positive measure initial data. The flux f ∈ C 1 ( R ) f\in C^1(\mathbb {R}) is assumed to satisfy a p − p- condition, a weak form of convexity. In particular, any flux of the form f ( u ) = ∑ i = 1 J a i u m i f(u)=\sum _{i=1}^Ja_iu^{m_i} is admissible if a i > 0 , m i > 1 , i = 1 , 2 , … , J . a_i>0,\,m_i>1,\,\,i=1,2,\ldots ,J. The only case treated hitherto in the literature is f ( u ) = u m f(u)=u^m [Arch. Rat. Mech. Anal. 124 (1993), pp. 43–65] and the initial data is a “single source”, namely, a multiple of the delta function. The corresponding solutions have been labeled as “source-type” and the treatment made substantial use of the special form of both the flux and the initial data. In this paper existence and uniqueness of solutions is established. The method of proof relies on sharp decay estimates for the viscous Hamilton-Jacobi equation. Some estimates are independent of the viscosity coefficient, thus leading to new estimates for the (inviscid) hyperbolic conservation law.