Nonlinear partial differential equations in heterogeneous frameworks

我建议的研究涉及某些偏微分方程（PDE）在异构框架的确定性分析。建议的第一部分是专门的反应对流扩散方程的系数取决于空间/时间变量的域穿孔。虽然抛物型偏微分方程理论中的一些数学里程碑可以追溯到20世纪30年代，但直到2000年，所考虑的设置都是相对均匀的：具有常数扩散和无漂移的半线性抛物方程。这就是这些偏微分方程表现出行波解的地方。 非均质域和系数出现在反应-平流-扩散方程中，用于模拟受扩散影响的化学品或种群的密度、反应过程以及底层流动的运输等量的演变。然而，行波解不再存在于这样的设置。在2000年初的一个显着的进步推广了行波的概念脉动行进的前沿。在2002年，Berestycki等人证明，在KPP非线性（以Kolmogorov，Petrovskii和Piscunov命名）的情况下，脉动行进锋存在于称为KPP最小速度的阈值之外。 拟议研究的第一部分有三个方面。我们首先关注湍流平流的作用：强流可能会提高反应速率，或者在某些情况下，阻止传播。我们的目标是找到尖锐的标准来表征加快传播的流量。现有的理论在这方面还远远不够完善，特别是在三维环境中。从PDE域继承的复杂性和3维流表现出的混沌行为带来了从功能分析，动力系统和测量理论的研究工具。第二组问题在第一部分的拟议研究涉及同质化。即，当区域的体积趋于零时，最小速度和解的渐近性。这部分的第三行集中在域的几何形状和PDE的系数之间的相互作用。KPP速度的变分公式显示其作为传播方向和扩散/反应系数的函数的连续性。这启动了对脉动前沿传播最快的最佳方向的搜索。 该建议的第二部分是致力于一类非线性椭圆型偏微分方程的扰动几何和/或系数。偏微分方程是非自伴的，没有变分结构。这些例子是“扰动莱恩-埃姆登”方程在无限圆柱。这类偏微分方程解的存在性和正则性本身就是一个重要的研究课题。此外，我们在这一部分开发的技术将是有用的研究反应扩散方程在随机区域。

My proposed research deals with the deterministic analysis of certain partial differential equations (PDEs) in heterogeneous frameworks. The first part of the proposal is devoted to reaction-advection-diffusion equations with coefficients that depend on space/time variables in domains with perforations. Although some mathematical milestones in the theory of parabolic PDEs date back to the 1930's, the setting considered until the year 2000 was relatively homogeneous: semi-linear parabolic equations with constant diffusion and no drift. That is where those PDEs exhibit traveling wave solutions. Heterogeneous domains and coefficients appear in reaction-advection-diffusion equations when intended to model the evolution of quantities such as densities of chemicals or populations subject to diffusion, a reactive process as well as transport by an underlying flow. However, traveling wave solutions no longer exist in such settings. A remarkable advancement in the early 2000's generalized the notion of traveling waves to pulsating traveling fronts. In 2002, Berestycki et al. proved that, in the case of KPP nonlinearity (named after Kolmogorov, Petrovskii and Piscunov), pulsating traveling fronts exist beyond a threshold known as the KPP minimal speed. The first part of the proposed research is three-fold. We first focus on the role of turbulent advection: a strong flow may enhance the rate of reaction or, in some situations, block the propagation. Our goal is to find sharp criteria to characterize the flows that speed-up the propagation. Existing theory is far from complete in this regard, especially in 3-dimensional settings. The complexity inherited from the domain of the PDE and the chaotic behaviours exhibited by 3-dimensional flows bring tools from functional analysis, dynamical systems, and measure theory to the study. The second set of questions in part one of the proposed research deals with homogenization. Namely, the asymptotics of the minimal speed and solutions as the volume of the domain's periodicity-cell goes to zero. The third line in this part focuses on the interaction between the geometry of the domain and the coefficients of the PDE. Variational formulations of the KPP speed show its continuity as a function of the direction of propagation and diffusion/reaction coefficients. This initiates a search for the optimal direction(s) in which pulsating fronts propagate fastest. The second part of the proposal is devoted to a class of nonlinear elliptic PDEs with perturbed geometries and/or coefficients. The PDEs are non-self-adjoint and have no variational structure. Examples of these are “perturbed Lane-Emden” equations in infinite cylinders. The existence and regularity of solutions for this class of PDEs is an important study on its own. Moreover, the techniques that we develop in this part will be useful for the study of reaction-diffusion equations in random domains.