Samokhin A. Gradient catastrophes for Burgers equation on a finite interval. Numerical and qualitative study, talk Workshop Geom. of PDEs and Integrability, 14-18 Oct 2013, Teplice nad Becvou, Czech Rep. (abstract)

Speaker: Alexey Samokhin

Title: Gradient catastrophes for Burgers equation on a finite interval. Numerical and qualitative study

Abstract:
We consider initial value-boundary problem (IVBP) for the Burgers equation

      $u_{t}(x,t)=u_{xx}(x,t)+2\eta u(x,t)u_{x}(x,t)$

on a finite interval:

      $u(x,0)=f(x),\quad u(\alpha ,t)=l(t),\quad u(\beta ,t)=r(t),\quad x\in [\alpha ,\beta ]$.

The case of constant boundary conditions $u(\alpha ,t)=A,\quad u(\beta ,t)=B$ and its asymptotics is of special interest here. For such a IVBP viscosity usually produces asymptotic stationary solution which is invariant for some subalgebra of the full symmetry algebra of the equation. But the evolution may also result in a stable gradient catastrophe.