Finite time singularities and global well-posedness for fractal Burgers equations

Abstract

Burgers equations with fractional dissipation on ℝ x ℝ+ or on ℝ1 x ℝR+ are studied. In the supercritical dissipative case, we show that with very generic initial data, the equation is locally well-posed and its solution develops gradient blow-up in finite time. In the critical dissipative case, the equation is globally well-posed with arbitrary initial data in H1/2. Finally, in the subcritical dissipative case, we prove that with initial data in the scaling-invariant Lebesgue space, the equation is globally well-posed. Moreover, the solution is spatial analytic and has optimal Gevrey regularity in the time variable.