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

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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.