Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: , 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.
['Grzegorz Karch', 'Changxing Miao', 'Xiaojing Xu']
I am writing a research paper on a idea ON CONVERGENCE OF SOLUTIONS OF FRACTAL BURGERS EQUATION TOWARD RAREFACTION WAVES
[{"from": "human", "value": "I am writing a research paper on a idea ON CONVERGENCE OF SOLUTIONS OF FRACTAL BURGERS EQUATION TOWARD RAREFACTION WAVES"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: , 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."}]