主持人：叶东
报告摘要：
We investigate the large time behavior of solutions to the two-dimensional viscous Burgers equation $u_t+uu_x+uu_y=\Delta u$, toward a non-self-similar rarefaction wave of inviscid Burgers equation with two initial constant states, separated by a curve $y=\varphi(x)$, and prove that the above 2D non-self-similar rarefaction wave is time-asymptotically stable. This is the first result on the nonlinear time-asymptotic stability of non-self-similar rarefaction waves. Furthermore, we can get the decay rate. Both the rarefaction wave strength and the initial perturbation can be large.
The main difficulty comes from the fact that the initial discontinuity of 2D non-self-similar rarefaction wave is a curve $y=\varphi(x)$. Fortunately, we uncover a novel property that the non-self-similar inviscid rarefaction wave is also asymptotically stable with respect to the discontinuity curve $y=\varphi(x)$. More precisely, let $u_i^R(x,y,t), i=1,2$ be the corresponding non-self-similar rarefaction wave with the initial discontinuity curve $y=\varphi_i(x)$, then $\|u_1^R-u_2^R\|_{L^\infty}\le \frac{C}{t}$ if $\|\varphi_1(x)-\varphi_2(x)\|_{L^\infty}$ is bounded. Based on this property, we prove that the asymptotic stability of non-self-similar rarefaction wave corresponding to the general initial discontinuity $y=\varphi(x)$ is equivalent to that of the non-self-similar rarefaction wave with an initial discontinuity given by the modification curve of a polyline, and the slopes on the left and right of the polyline are $k_\pm=\lim_{x\to \pm\infty}\frac{\varphi(x)}{x}$.
Then we construct the approximate smooth rarefaction wave of the viscous Burgers equation based on the above modification curve, and convert this wave to the self-similar planar rarefaction wave through a new nonlinear coordinate transformation with the price that the 2D viscous Burgers equation is changed into a parabolic equation with variable and mixed derivative viscosities. Another advantage is that the main error terms generated by the new approximate smooth rarefaction in the new parabolic equation are integrable in $\mathbb R^2$. These new approaches enable us to overcome the main difficulty mentioned above. Finally, the time-asymptotic stability analysis is carried out to the viscous rarefaction wave and the transformed 2D viscous Burgers equation by using suitable time-weighted $L^p$-energy estimates.
报告人简介：
黄飞敏，华罗庚首席研究员，现任中国科学院数学与系统科学研究院副院长、中国数学会党委书记暨副理事长，中国工业与应用数学学会会士，国家杰出青年基金获得者。曾获2004年美国工业与应用数学学会杰出论文奖，2013年国家自然科学奖二等奖（第一完成人）等重要奖项，2013年入选国家杰出青年科学基金20周年巡礼。主要从事非线性偏微分方程的理论研究，在非线性双曲守恒律、可压缩Navier-Stokes方程、Boltzmann方程等重要领域取得一系列突破性成果，发表学术论文100多篇并被引用3000余次，现任《Communications in Mathematical Sciences》、《Nonlinear Analysis: Real World Applications》、《中国科学数学》、《应用数学学报》、《数学物理学报》等国内外学术杂志编委。
