Integrable viscous conservation laws

On integrable conservation laws.

We study normal forms of scalar integrable dispersive (not necessarily Hamiltonian) conservation laws, via the Dubrovin-Zhang perturbative scheme. Our computations support the conjecture that such normal forms are parametrized by infinitely many arbitrary functions that can be identified with the coefficients of the quasi-linear part of the equation. Moreover, in general, we conjecture that two...

Viscous Conservation Laws, Part I: Scalar Laws

Viscous conservation laws are the basic models for the dissipative phenomena. We aim at a systematic presentation of the basic ideas for the quantitative study of the nonlinear waves for viscous conservation laws. The present paper concentrates on the scalar laws; an upcoming Part II will deal with the systems. The basic ideas for scalar viscous conservation laws originated from two sources: th...

Stability of systems of viscous conservation laws

Viscous System of Conservation Laws: Singular Limits

Abstract. We continue our analysis of the Cauchy problem for viscous system of conservation, under natural assumptions. We examine in which way does the existence time depend upon the viscous tensor B(u). In particular, we consider singular limits, where the rank of the symbol B(u; ξ) drops at the limit. This covers a lot of situations, for instance that of the limit of the Navier-Stokes-Fourie...

Scalar Non-linear Conservation Laws with Integrable Boundary Data

We consider the initial-boundary value problem for a scalar non-linear conservation law u t + f(u)] x = 0; u(0; x) = u(x); u(; 0) = ~ u(t); () on the domain = f(t; x) 2 R 2 : t 0; x 0g. Here u = u(t; x) is the state variable, u; ~ u are integrable (possibly unbounded) initial and boundary data, and f is assumed to be strictly convex and superlinear. We rst derive an explicit formula for a solut...