Pathwise mild solutions for quasilinear stochastic partial differential equations
Pathwise mild solutions for quasilinear stochastic partial differential equations
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DOI：
10.1016/j.jde.2020.01.032
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发表时间：
2018-02
影响因子：
2.4
通讯作者：
C. Kuehn;A. Neamţu
中科院分区：
文献类型：
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作者：
C. Kuehn;A. Neamţu
随机偏微分方程（SPDE）已成为一种重要的建模工具.然而，有许多类的SPDEs，其中的存在性和正则性理论还没有完全发展。在这里，我们有助于这方面，并证明了温和的解决方案的存在性广泛的一类拟线性柯西问题，包括-除其他外-交叉扩散系统作为一个关键的应用。我们的解决方案是本地的时间，并通过一个不动点参数在适当的函数空间。其核心思想是将确定性拟线性抛物型偏微分方程的经典理论与发展半群理论以适当的方式联合收割机。我们还展示了如何将我们的理论应用于Shigesada-Kawasaki-Teramoto（SKT）模型。此外，我们提供了爆破和不适定的运营商，这可能发生在有限时间后，显示解决方案只能是本地的一般准线性SPDE的时间，而他们可能是全球的特殊子类的问题的时间。
Stochastic partial differential equations (SPDEs) have become a key modeling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, including - among others - cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine in a suitable way the classical theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time showing that solutions can only be local-in-time for general quasilinear SPDEs, while they might be global-in-time for special subclasses of problems.