Published · Journal of the European Mathematical Society, 23 (2021) 2, pp. 425-465. We prove optimal regularity for solutions to porous media equations in Sobolev spaces, based on velocity averaging techniques. In particular, the obtained regularity is consistent with the optimal regularity in the linear…
Published · Journal of functional analysis, 277 (2019) 5, pp. 1469-1498. We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results…
Published · Annales de l’Institut Henri Poincaré / C, 36 (2019) 2, pp. 505-521. We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced,…
Published · Communications in partial differential equations, 43 (2018) 12, pp. 1702-1736. We consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are…
Published · The annals of probability, 46 (2018) 5, pp. 2495-2544. We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging…
Published · Stochastic processes and their applications, 127 (2017) 9, pp. 2961-3004. We introduce the notion of pathwise entropy solutions for a class of degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity and fluxes with rough time dependence and prove their well-posedness. In the case of…
Published · Probability theory and related fields, 173 (2019) 3/4, pp. 1063-1098. We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path $L^{\infty}$ bounds for the second derivative of solutions to such PDE are shown. These bounds are expressed…
Published · SIAM journal on numerical analysis, 54 (2016) 4, pp. 2187-2209. We develop a semi-discretization approximation for scalar conservation laws with multiple rough time dependence in inhomogeneous fluxes. The method is based on Brenier’s transport-collapse algorithm and uses characteristics defined in the setting…
Published · Discrete and Continuous Dynamical Systems B 21(9), 3003–3014 (2016). We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of…
Published · SIAM journal on mathematical analysis, 48 (2016) 6, pp. 4094-4125. Ergodicity for local and nonlocal stochastic singular $p$-Laplace equations is proven, without restriction on the spatial dimension and for all $p\in[1,2)$. This generalizes previous results from [Gess, Tölle; J. Math. Pures Appl.,…
Published · Journal of differential equations, 260 (2016) 6, pp. 4973-5025. We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift…
Published · The annals of probability, 45 (2017) 2, pp. 1325-1350. We provide sufficient conditions for weak synchronization by noise for order-preserving random dynamical systems on Polish spaces. That is, under these conditions we prove the existence of a weak point attractor consisting…
