Long-time behavior of scalar conservation laws with critical dissipation

  • Dallas Albritton

    New York University, USA
  • Rajendra Beekie

    New York University, USA
Long-time behavior of scalar conservation laws with critical dissipation cover
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Abstract

The critical Burgers equation is a toy model for the competition between transport and diffusion with regard to shock formation in fluids. It is well known that smooth initial data does not generate shocks in finite time. Less is known about the long-time behavior for ‘shock-like’ initial data: as . We describe this long-time behavior in the general setting of multidimensional critical scalar conservation laws when the initial data has limits at infinity. The asymptotics are given by certain self-similar solutions, whose stability we demonstrate with the optimal diffusive rates.

Cite this article

Dallas Albritton, Rajendra Beekie, Long-time behavior of scalar conservation laws with critical dissipation. Ann. Inst. H. Poincaré Anal. Non Linéaire 39 (2022), no. 1, pp. 225–243

DOI 10.4171/AIHPC/7