On convergence of solutions of fractal Burgers equation toward rarefaction waves

Abstract

In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation $u_t+(-{\partial }_x^2{)}^{\alpha /2}u+u{u}_x=0$ with $\alpha \in (1,2)$ is studied. It is shown that if the nondecreasing initial datum approaches the constant states $u_{\pm }$ ($u_{-}<u_{+}$) as $x\to \pm \infty$, respectively, then the corresponding solution converges toward the rarefaction wave, {\it i.e.} the unique entropy solution of the Riemann problem for the nonviscous Burgers equation.

15 pages

References in corpus (1)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations

Cited by in corpus (1)

• Global well-posedness of the critical Burgers equation in critical Besov spaces