退化抛物方程的可控性

In this research project, we will investigate the problem of the controllability for degenerate equations:(1) the null controllability of one dimensional degenerate parabolic equations with the gradient terms, (2)the approximate controllability of multi-dimensional nonlinear degenerate parabolic equations with the gradient terms. Comparing with the non-degenerate parabolic equations, some degenerate parabolic equations can exactly describe the diffusion phenomena that exist widely in the nature. Hence, it has important significance to investigate these problems. At present, in the study of null controllability of one dimensional degenerate parabolic equations with gradient terms, the gradient terms are dependent on the degree of degradation of the equation. And the gradient terms we want to study are independent of it, which will bring difficulties to establish the Carleman estimates. As for multi-dimensional nonlinear degenerate parabolic equations with the gradient terms, the weighted space is more complex than that for the nonlinear equations without the gradient terms by applying Kakutani fixed point theorem. Hence, we need more accurately compact estimates for the linearized problems. Our work can not only enrich the controllability theory of the degenerate parabolic equations, but also provide effective technical methods for the research of the controllability of the degenerate parabolic equations.

本项目拟研究退化抛物方程的可控性问题:(1)一维含梯度项的非线性退化抛物方程的零可控性，(2)高维含梯度项的非线性退化抛物方程的近似可控性。相对于一致抛物系统，一些具有退化性的抛物方程能够更加精确地描述自然界广泛存在的扩散现象。因此，对这类问题的研究具有重要的意义。目前，在一维含梯度项的退化抛物方程的零可控性问题的研究结果中，梯度项都是依赖于方程的退化度的，而我们要研究的梯度项不依赖于方程的退化度，这会给Carleman估计的建立带来一定的困难。而对于高维含梯度项的非线性退化抛物方程，在应用Kakutani不动点定理时，需选取比不含梯度项时更为复杂的加权解空间。因此，需要对线性化问题做更为精确的紧性估计。我们的工作不仅能够丰富退化抛物方程的控制理论，而且也为研究退化抛物方程的可控性提供有效的技术方法。

本项目主要研究了具有退化奇异性质的抛物方程，得到了以下五个结果。 第一，我们证明了一类高维的带梯度项的线性退化抛物方程的近似可控性。第二，我们研究了一类一维的含对流项的退化抛物方程的边界控制问题并得出其近似可控性。第三，我们证明了一类高维耦合退化抛物方程组的近似可控性。第四，我们考虑了一类一维带梯度项的退化抛物方程组的零可控性。第五，我们还研究了一类具有退化性质的耦合方程组的解的长时间渐近行为。我们的工作不仅能够丰富退化抛物方程的控制理论，而且也为研究退化抛物方程的可控性提供有效的技术方法。