On Convergence of Solutions of Fractal Burgers Equation toward Rarefaction Waves

Source abstract

In this paper, the large time behavior of solutions of the Cauchy problem for the one-dimensional fractal Burgers equation $u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0$ 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, i.e., the unique entropy solution of the Riemann problem for the nonviscous Burgers equation.