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

由非线性乘法驱动的拟线性简并抛物双曲随机偏微分方程 (SPDE) 解的正则性

• 批准号: 1939627
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1939627
• 关键词: Regularity; solutions; quasilinear; degenerate; parabolic

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.

我们旨在在时间和空间上建立新的规律性估计，以解决准线性退化的抛物线抛物性氧化纤维随机偏微分方程（SPDE）。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解决方案上。这些方程式的经典示例是随机标量保护定律，在广泛的应用中出现，包括将现象描述为多孔介质中理想流体的对流扩散。除了这些方程的确定性部分（即PDES）外，还存在随机噪声的存在，通常用于描述数值，经验或物理不确定性。在文献中，通常通过将原始（非线性）方程转换为新的线性方程来证明涉及这种类型方程的初始值问题的良好性。后者被称为原始方程式的动力学表述，它的优点是从数学角度更容易处理。这些准线性变性抛物线抛物线抛物线 - 透明质 - 透明质SPDE的规律性将通过上面描述的动力学方法以及Fourier Analytic Techniques and Averaging Lemmata来研究。第一步将包括为由非线性乘法时空白噪声驱动的多孔介质方程解决方案的最佳规则估计。证明这种新结果的一种可能的方法可以包括对由空间白噪声驱动的退化抛物线安德森模型的普遍性估计。完成第一步，下一步将包括对一般的准线性变性抛物线抛物性抛物性抛物性透明质量spdes得出最佳的规律性估计。研究的进一步方向可能是研究当时空白噪声被空间中规则定期的噪声替换并在艰难时期驱动时，这些方程式的规律性如何变化。在其他研究领域的几种应用中都会出现所有方程式。描述了零范围过程中有关其流体动力极限或描述薄膜演化的方程式的流体动力学的方程式，该薄膜在我们项目中研究的平坦D维底物上不可压缩的牛顿液体在平坦的D维底物上的演变具有相同的SPDE形式。对这些解决方案的分析特性的研究（如规律性估计）将有助于更好地理解这些现象。该项目通过EPSRC CDT在BATH（SAMBA）的统计应用数学中资助。如上所述，这项研究有可能在不同的数学学科上应用，这是桑巴舞的目标之一。