对流扩散方程的解

关注如下的对流扩散方程 $$ u_{t}=\text{div}(|\nabla u^{m}|^{p-2}\nabla u^{m})+\sum_{i=1}^{N}\frac{\partial b_{i}(u^{m})}{\partial x_{i}} $$ 的初边值问题. 若 $p>1+\frac{1}{m}$, 通过考虑正则化问题的解 $u_{k}$, 利用 Moser 迭代技巧, 得到了$u_{k}$ 的 $L^{\infty}$ 模与 梯度 $\nabla u_{k}$ 的 $L^{p}$ 模的局部有界性. 利用紧致性定理, 得到了对流扩散方程本身解的存在性. 若 $p<1+\frac{1}{m},\ p>2$ 或者 $p=1+\frac{1}{m}$, 利用类似的方法可以得到解的存在性. 证明了解的唯一性, 同时讨论了正性和熄灭性等解的性质.

Solutions to a Convection Diffusion Equation

This paper concerns the initial-boundary value problem of the following convection diffusion equation: $$ u_{t}=\text{div}(|\nabla u^{m}|^{p-2}\nabla u^{m})+\sum_{i=1}^{N}\frac{\partial b_{i}(u^{m})}{\partial x_{i}}. $$ If $p>1+\frac{1}{m}$, by considering the solution $u_{k}$ to the regularized problem and using the Moser iteration technique, the local bounded properties of the $L^{\infty}$-norm of $u_{k}$ and those of the $L^{p}$-norm of the gradient $\nabla u_{k}$ are obtained. By the compactness theorem, the existence of solution to the convection diffusion equation itself is proved. If $p<1+\frac{1}{m},\ p>2$ or $p=1+\frac{1}{m}$, the existence of solutions is obtained in a similar way. The uniqueness of solutions is also known. At the same time, some properties of solutions, such as the positivity, the extinction, etc. are discussed.