Nonlinear partial differential equations in heterogeneous frameworks

我提出的研究是关于异质框架中某些偏微分方程的确定性分析。建议的第一部分致力于反应-平流-扩散方程，其系数依赖于穿孔区域中的空间/时间变量。抛物型偏微分方程组理论中的一些数学里程碑可以追溯到1930年的S，但直到2000年，研究的背景是相对均匀的：具有常数扩散和无漂移的半线性抛物型方程。这就是那些PDE展示行波解决方案的地方。 非均相域和系数出现在反应-平流-扩散方程中，用于模拟扩散的化学品或种群的密度、反应过程以及底层流动的输运等量的演变。然而，行波解决方案不再存在于这样的设置中。2000年初，S将行波的概念推广到脉动的行波前沿，这是一个显著的进步。2002年，Berestycki et al.证明了，在KPP非线性(以Kolmogorov、Petrovskii和Piscunov命名)的情况下，脉动行波的存在超过了被称为KPP最小速度的阈值。 拟议研究的第一部分包括三个方面。我们首先关注湍流平流的作用：强烈的气流可能会加快反应速度，或者在某些情况下，会阻碍传播。我们的目标是找到准确的标准来描述加速传播的流动。现有的理论在这方面还远远不完整，特别是在三维环境下。从偏微分方程域继承的复杂性和三维流动表现出的混沌行为为研究带来了泛函分析、动力系统和测度论的工具。拟议研究的第一部分中的第二组问题涉及同质化。也就是说，当区域周期单元的体积趋于零时，最小速度和解的渐近性。这一部分的第三行着重于域的几何形状和偏微分方程的系数之间的相互作用。KPP速度的变分公式显示了它作为传播方向和扩散/反应系数的函数的连续性。这引发了对脉动锋面传播速度最快的最佳方向(S)的探索。 第二部分研究了一类具有扰动几何和/或系数的非线性椭圆型偏微分方程解。偏微分方程组是非自伴的，没有变分结构。这些问题的例子是无限长圆柱体中的“摄动Lane-Emden”方程。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，我们在这一部分中发展的技术将有助于研究随机区域中的反应扩散方程。

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.