The generalized BBM-Burger equations with non-linear dissipative term: existence and convergence results

Abstract

We study the global existence of solutions for certain equations of the form u _t + f(u) _x = γB(u _x ) _x + δu _xxt − αu _xxxx , as γ > 0, δ > 0 and α > 0 aproach zero, and f and B are sufficiently smooth functions satisfying certain appropriate assumptions. We consider solutions of hyperbolic conservation laws regularized of this equations. Following a pioneering work by Schonbek and a work by LeFloch and Natalini, we establish the convergence of the regularized solutions towards discontinuous solutions of the hyperbolic conservation law.

Keywords:

• existence and convergence of the smooth solutions
• partial differential equations
• dissipative term
• hyperbolic conservation law

AMS Subject Classifications:

• primary: 35L65
• secondary: 76N10

Acknowledgement

The second author was supported by Capes, Brazil.