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

我们的目标是建立新的正则性估计的时间和空间的解拟线性退化抛物-双曲随机偏微分方程（SPDE）。我们的研究将集中在具有一般乘性噪声和非线性扩散系数的方程的解。这些方程的经典例子是随机标量守恒律，其出现在广泛的应用中，包括描述多孔介质中理想流体的对流扩散现象。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通过将原（非线性）方程转化为新的线性方程来证明。后者被称为动力学公式的原始方程，它的优点是，它是更容易处理从数学的观点来看，这些准线性退化抛物双曲SPDE的解决方案的规律性将研究利用动力学的方法与傅立叶分析技术和平均Lemmata上面所描述的沿着。第一步将包括开发最佳的正则性估计的非线性乘性时空白色噪声驱动的多孔介质方程的解决方案。证明这些新结果的一个可能的方法是推广空间白色噪声驱动的退化抛物安德森模型的正则性估计，一旦完成了第一步，下一步将是导出一般拟线性退化抛物-双曲SPDE的最优正则性估计.进一步的研究方向可能是研究当时空白色噪声被空间上规则的噪声所取代时，这类方程解的正则性如何变化，这些方程都出现在其他研究领域的一些应用中。描述零程过程关于其流体动力学极限的波动流体动力学的方程或描述由不可压缩牛顿液体组成的薄膜在平坦d维基底上的演化的方程都具有与我们所研究的SPDE相同的形式。研究这些解的分析性质（如正则性估计）将有助于更好地理解这些现象。该项目是通过EPSRC CDT在巴斯统计应用数学（桑巴舞）资助。如上所述，这项研究有可能应用于不同的数学学科，这是桑巴舞的目标之一。

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.