几类无界域上耗散演化方程解的长时间行为研究
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摘要
现实世界中很多物理现象可以由具有能量耗散的非线性发展方程描述.对这些发展方程解的长时间行为研究有助于我们理解非线性科学中的复杂现象.全局吸引子是无穷维动力系统中的核心概念.我们主要关心全局吸引子的存在性,正则性,分形维数估计等等.对于吸引子的高正则性,弱耗散系统,非局部耗散等,这些问题将变得更加困难.本文主要致力于处理这几类问题,具体安排如下.
第一章,我们对无穷维动力系统的主要内容作简要介绍,包括吸引子存在性的充分必要刻画,紧性描述以及相关论题.
第二章,考虑具有任意增长阶非线性的反应扩散方程,通过一系列解的估计,得出单位算子在吸收集内是L2-H2∩L2p-2Holder连续的,结合解半群在L2中的渐近紧性,证明了L2-H2∩L2p-2全局吸引子的存在性.由于方程的平衡点至多属于上述空间,因此该结果已经达到最优.
第三章,研究了广义KdV方程解的渐近行为.通过构造三个守恒律能量方程,证明了H2中吸收集的存在性.利用Ball的能量方法,证明了解半群在H2中的渐近紧性,从而得到全局吸引子的存在性.我们还给出了一个平凡吸引子的充分条件.结果表明当外力项相对阻尼系数小时,吸引子是平凡的,即只包含一个平衡点.
第四章,分析了QG方程解的渐近行为.首先,利用分数拉普拉斯算子的正性不等式,交换子估计以及Besov空间技巧,证明了无界域上阻尼QG方程在LP中全局吸引子的存在性.
第五章,利用Chueshov.Lasiecka拟稳定估计方法,得出2维环域上全局吸引子的分形维数依赖耗散系数的精确估计.维数上界随着耗散系数增大而减小,这与物理事实是对应的.
第六章,考虑了具有白噪声扰动的分数拉普拉斯反应扩散方程.用∈正则性方法得到解的适定性.通过尾端估计方法,得出系统在L2中随机吸引子的存在性.
It's known to us that many phenomena in physics can be described by dissipative non-linear evolution equation. The study of long time behavior of solutions of these equations will enhance our understanding of the complexity in nonlinear science. The central topic of infinite dimensional dynamical system is the global attractor. Topics like existence of the global attractor, further regularity and fractal dimension are mainly concerned. Especially, if one works on global attractor in regular spaces, weak dissipative or nonlocal dissipative equations, new difficulties will arise. In the dissertation, we are devoted to deal with prob-lems of this kind. The outline is as follows.
In Chapter1, we introduce the main concepts and theorems in the field of infinite dimensional dynamical system briefly, including the sufficient and necessary conditions of the existence of global attractor, the description of compactness and related topics.
In Chapter2, we consider the long-time behavior of solution of reaction diffusion equa-tion with arbitrary growth nonlinear term. Based on several a priori estimates, it can be shown that the unitary operator is Holder continuous from L2to H2∩L2p-2. Since the so-lution semigroup is asymptotically compact in L2, we obtain the L2－H2∩L2p-2asymp-totical compactness of semigroup, and then the L2-H2∩L2p-2global attractor. This result is sharp in the sense that the stationary point is at most in H2∩L2p-2.
In Chapter3, we investigate the long-time behavior of solution of generalized KdV equation on the real line. We obtain the existence of an absorbing set in H2by establishing three energy equations. Using the Ball's idea, we prove the asymptotical compactness of solution semigroup in H2. Moreover, we analyze the structure of attractor in some special cases. Precisely, it can be shown that the global attractor will reduce to a single stationary point if the force is relatively small compared to the dissipative coefficient.
In Chapter4, we consider the surface Quasi-Geostrohic equation on the whole space. At first, applying positive lemma of fractional Laplacian, commutator estimates and Besov spaces technique, we prove the existence of global attractor in LP.
In Chapter5, we obtain the precise estimate of fractal dimension of global attractor on torus by quasi-stable method of Chueshov&Lasiecka. The upper bound is decreasing function of dissipative coefficient, which conforms to physical intuition.
In Chapter6, a fractional reaction diffusion equation with white noise is studied. The well-posedness of solution is proved by∈-regular approach. Applying the idea of tailed estimates, we show the existence of global random attractor in L2.
第一章,我们对无穷维动力系统的主要内容作简要介绍,包括吸引子存在性的充分必要刻画,紧性描述以及相关论题.
第二章,考虑具有任意增长阶非线性的反应扩散方程,通过一系列解的估计,得出单位算子在吸收集内是L2-H2∩L2p-2Holder连续的,结合解半群在L2中的渐近紧性,证明了L2-H2∩L2p-2全局吸引子的存在性.由于方程的平衡点至多属于上述空间,因此该结果已经达到最优.
第三章,研究了广义KdV方程解的渐近行为.通过构造三个守恒律能量方程,证明了H2中吸收集的存在性.利用Ball的能量方法,证明了解半群在H2中的渐近紧性,从而得到全局吸引子的存在性.我们还给出了一个平凡吸引子的充分条件.结果表明当外力项相对阻尼系数小时,吸引子是平凡的,即只包含一个平衡点.
第四章,分析了QG方程解的渐近行为.首先,利用分数拉普拉斯算子的正性不等式,交换子估计以及Besov空间技巧,证明了无界域上阻尼QG方程在LP中全局吸引子的存在性.
第五章,利用Chueshov.Lasiecka拟稳定估计方法,得出2维环域上全局吸引子的分形维数依赖耗散系数的精确估计.维数上界随着耗散系数增大而减小,这与物理事实是对应的.
第六章,考虑了具有白噪声扰动的分数拉普拉斯反应扩散方程.用∈正则性方法得到解的适定性.通过尾端估计方法,得出系统在L2中随机吸引子的存在性.
It's known to us that many phenomena in physics can be described by dissipative non-linear evolution equation. The study of long time behavior of solutions of these equations will enhance our understanding of the complexity in nonlinear science. The central topic of infinite dimensional dynamical system is the global attractor. Topics like existence of the global attractor, further regularity and fractal dimension are mainly concerned. Especially, if one works on global attractor in regular spaces, weak dissipative or nonlocal dissipative equations, new difficulties will arise. In the dissertation, we are devoted to deal with prob-lems of this kind. The outline is as follows.
In Chapter1, we introduce the main concepts and theorems in the field of infinite dimensional dynamical system briefly, including the sufficient and necessary conditions of the existence of global attractor, the description of compactness and related topics.
In Chapter2, we consider the long-time behavior of solution of reaction diffusion equa-tion with arbitrary growth nonlinear term. Based on several a priori estimates, it can be shown that the unitary operator is Holder continuous from L2to H2∩L2p-2. Since the so-lution semigroup is asymptotically compact in L2, we obtain the L2－H2∩L2p-2asymp-totical compactness of semigroup, and then the L2-H2∩L2p-2global attractor. This result is sharp in the sense that the stationary point is at most in H2∩L2p-2.
In Chapter3, we investigate the long-time behavior of solution of generalized KdV equation on the real line. We obtain the existence of an absorbing set in H2by establishing three energy equations. Using the Ball's idea, we prove the asymptotical compactness of solution semigroup in H2. Moreover, we analyze the structure of attractor in some special cases. Precisely, it can be shown that the global attractor will reduce to a single stationary point if the force is relatively small compared to the dissipative coefficient.
In Chapter4, we consider the surface Quasi-Geostrohic equation on the whole space. At first, applying positive lemma of fractional Laplacian, commutator estimates and Besov spaces technique, we prove the existence of global attractor in LP.
In Chapter5, we obtain the precise estimate of fractal dimension of global attractor on torus by quasi-stable method of Chueshov&Lasiecka. The upper bound is decreasing function of dissipative coefficient, which conforms to physical intuition.
In Chapter6, a fractional reaction diffusion equation with white noise is studied. The well-posedness of solution is proved by∈-regular approach. Applying the idea of tailed estimates, we show the existence of global random attractor in L2.
引文
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