论文标题
各向异性稳定非局部扩散问题的大型行为与对流
Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection
论文作者
论文摘要
我们研究了具有非线性对流术语的非局部热方程式库奇问题的非负溶液的大型行为。扩散算子是稳定的Lévy过程的无限发电机，可能是高度各向异性的。假定初始数据是有界和整合的。溶液的质量是沿进化的保守的，源行为是由源类型溶液具有该质量方程的质量方程式，取决于对流和扩散的相对强度。当扩散比对流更强时，原始方程将渐近地简化为纯粹的扩散非局部热方程。当对流统治时，仅在对流方向上才能进行，并且极限方程在与该方向的子空间正交中仍然是扩散的，而原始一个方向的扩散操作员是在子空间上的``投影''。该预测的确定是本文的主要问题之一。当对流和扩散的顺序相同时，极限方程与原始方程一致。 即使在各向同性的情况下，我们的大多数结果都是新的，其中扩散操作员是分数拉普拉斯。只要保留质量，我们就可以涵盖两个缓慢和快速对流的情况。快速对流对应于不是局部Lipschitz的对流非线性，而是仅在非局部扩散设置中尚未考虑过局部Hölder。
We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution with this mass of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a ``projection'' of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one. Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.
