Global time estimates for solutions to equations of dissipative type
Global time estimates for solutions to equations of dissipative type
复制标题
耗散型方程解的全局时间估计
DOI：
10.5802/jedp.23
复制
发表时间：
2006
期刊：
影响因子：
--
通讯作者：
James Smith
中科院分区：
文献类型：
--
作者：
Michael Ruzhansky;James Smith
考虑一般严格双曲偏微分方程解的 L p − L q 范数的全局时间估计。本文特别感兴趣的情况是表现出耗散行为的方程。结果用于讨论福克-普朗克方程和具有负质量的波型方程的时间衰减估计。本文致力于研究一般形式的常系数严格双曲方程解的 L p − L q 范数的时间衰减。众所周知，这种估计导致了 Strichartz 估计，这是处理非线性问题时的一种强大技术。我们假设方程的主要部分是严格双曲的。由于较低阶项，完整方程可能具有可变的重数。一个令人感兴趣的问题是确定此类方程的属性，这些属性决定了解的时间衰减率。另一个有趣的问题是当存在多个特征根时会发生什么。高阶方程出现在许多应用中。特别是，它们作为双曲系统的色散方程出现，例如在非平衡热力学中的 Fokker-Planck 方程和 Grad 系统的研究中。此外，在福克-普朗克方程解的近似中，相应系统的阶数趋于无穷大。然而，事实证明仍然可以确定其解的衰减率。这些例子所表现出的行为与耗散波动方程的行为类似，即特征根位于复数上半平面中，并以单根和孤立点的形式到达原点。这就是为什么在本文中我们将集中讨论定理 2.2 中的此类方程，尽管我们还将提出更一般的定理 2.1。此处描述的结果是针对标量方程制定的。然而，它们可以很容易地扩展到系统中。它们还产生半线性方程的适定性结果。此类分析的详细信息将在其他地方出现。
Global time estimates of L p − L q norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass. The paper is devoted to the time decay of L p − L q norms of solutions to constant coefficients strictly hyperbolic equations of general form.It is known that such esti- mates lead to Strichartz estimates which are a powerful technique when dealing with nonlinear problems. We will assume that the principal part of the equation is strictly hyperbolic. The full equation may have variable multiplicities because of the lower order terms. One question of interest is to identify properties of such equations which determine the time decay rate of solutions. Another question of interest is what happens when there are multiple characteristic roots. Equations of higher orders appear in many applications. In particular, they arise as dispersion equations for hyperbolic systems, for example in the study of the Fokker- Planck equation and Grad systems in nonequilibrium thermodynamics. Moreover, in approximations of solutions to the Fokker-Planck equation the order of the cor- responding system tends to infinity. However, it turns out to still be possible to determine the decay rate of its solutions. The behaviour exhibited by these exam- ples is similar to the behaviour of the dissipative wave equation in the sense that characteristic roots lie in the complex upper half plane and come to the origin as single roots and at isolated points. That is why in this paper we will concentrate on equations of such type in Theorem 2.2, although we will also present a more general Theorem 2.1. Results described here are formulated for scalar equations. However, they can be easily extended to systems. They also yield the well-posedness results for semilinear equations. Details of such analysis will appear elsewhere.