Regularity for solutions to quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations (SPDEs) driven by nonlinear multipli

本文旨在建立拟线性退化抛物-双曲型随机偏微分方程（SPDEs）解在时间和空间上的新的正则性估计。我们的研究将集中在具有一般乘法噪声和非线性扩散系数的方程的解上。这些方程的经典例子是随机标量守恒定律，它在广泛的应用中出现，包括描述理想流体在多孔介质中的对流扩散等现象。除了这些方程的确定性部分（即偏微分方程）之外，随机噪声的存在通常用于描述数值、经验或物理的不确定性。在文献中，涉及这类方程的初值问题的适定性通常是通过将原（非线性）方程转化为新的线性方程来证明的。后者被称为原始方程的动力学公式，它的优点是从数学的角度更容易处理。这些拟线性退化抛物双曲SPDEs解的正则性将利用上述动力学方法以及傅里叶分析技术和平均引理来研究。第一步将包括开发由非线性乘性时空白噪声驱动的多孔介质方程解的最优正则性估计。证明这些新结果的一种可能方法是推广由空间白噪声驱动的退化抛物型安德森模型的正则性估计。一旦完成了第一步，下一步将是推导一般拟线性退化抛物-双曲SPDEs的最优正则性估计。当时空白噪声被空间上的规则噪声所取代，并由时间上的粗糙路径驱动时，这类方程解的正则性如何变化，可能是进一步研究的方向。所考虑的所有方程都在其他研究领域的几个应用中出现。描述零范围过程中关于其流体动力极限的波动流体动力学方程或描述由不可压缩牛顿液体组成的薄膜在平面d维衬底上的演化方程都具有我们项目中研究的spde的相同形式。研究这些解的解析性质（如正则性估计）将有助于更好地理解这些现象。该项目由巴斯EPSRC统计应用数学CDT （SAMBa）资助。如上所述，这项研究有可能应用于不同的数学学科，这是SAMBa的目标之一。

We aim to establish new regularity estimates in time and space for solutions to quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations (SPDEs). Our study will be focused on the solutions of equations having a general multiplicative noise and a nonlinear diffusion coefficient. Classical examples of these equations are stochastic scalar conservation laws that arise in a wide range of applications including the description of phenomena as the convection-diffusion of an ideal fluid in porous media. The presence of a stochastic noise in addition to the deterministic part of these equations (namely to the PDEs) is often used to describe numerical, empirical or physical uncertainties. In literature, the well-posedness for initial value problems involving such type of equations is often proved by transforming the original (nonlinear) equation into a new linear equation. The latter is known as the kinetic formulation of the original equation and it has the advantage that it is easier to handle from a mathematical point of view.The regularity of solutions of these quasilinear degenerate parabolic-hyperbolic SPDEs will be studied by exploiting the kinetic approach described above along with Fourier analytic techniques and averaging Lemmata. A first step will consist in developing optimal regularity estimates for solutions of porous medium equations driven by a nonlinear multiplicative space-time white noise. A possible way of proving such new results could consist in generalising regularity estimates for a degenerate parabolic Anderson model driven by a spatial white noise.Once finished the first step, the next step would consist in deriving optimal regularity estimates for general quasilinear degenerate parabolic-hyperbolic SPDEs. A possible further direction of the research may be the study of how the regularity of solutions for these kind of equations changes when the space-time white noise is replaced by a noise regular in space and driven by a rough path in time.All equations considered arise in several applications across other research fields. The equation that describes the fluctuating hydrodynamics of the zero range process about its hydrodynamic limit or the equation describing the evolution of a thin film consisting of an incompressible Newtonian liquid on a flat d-dimensional substrate have all the same form of the SPDEs studied in our project. The study of the analytical properties for these solutions (like the regularity estimates) would be beneficial for a better understanding of these phenomena. The project is funded through the EPSRC CDT in Statistical Applied Mathematics at Bath (SAMBa). As mentioned above, this research has potential to be applied across different mathematical disciplines, which is one of the objectives of SAMBa.