Phd in MATHEMATICS

Thesis title: Nonlinear parabolic stochastic evolution equations in critical spaces

In this thesis we develop a new approach to nonlinear stochastic partial differential equations (SPDEs) with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. One of the main contributions of this thesis is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. This leads to new results even in the classical $L^2$-settings, which we illustrate for a parabolic SPDE and for the stochastic Navier-Stokes equations in two dimensions. Our theory is formulated in an $L^p$-setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to local-well posedness to several concrete problems and their quasilinear variants are given. This includes Stochastic Navier-Stokes equations, Burger's equation, the Allen-Cahn equation, the Cahn-Hilliard equation, reaction-diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. Most of the previous equations will be considered with a gradient-noise term. The thesis is divided into three parts. The first one concerns local well-posedness for stochastic evolution equations. Here, we study stochastic maximal $L^p$-regularity for semigroup generators, and in particular, we prove a sharp time-space regularity result for stochastic convolutions which will play a basic role for the nonlinear theory. Next, we show local existence of solutions to stochastic evolution equations with rough initial data which allows us to define `critical spaces' in an abstract way. The proofs are based on weighted maximal regularity techniques for the linearized problem as well as on a combination of several sophisticated splitting and truncation arguments. The local-existence theory developed here can be seen as a stochastic version of the theory of critical spaces due to Prüss-Simonett-Wilke (2018). We conclude the first part by applying our main result to several SPDEs. In particular, we check that critical spaces defined abstractly coincide with the critical spaces from a PDEs perspective, i.e. spaces invariant under the natural scaling of the SPDE considered. The second part is devoted to the study of blow-up criteria and instantaneous regularization. Here we prove several blow-up criteria for stochastic evolution equations. Some of them were not known even in the determinstic setting. For semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss-Simonett-Wilke (2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in the first part, maximal regularity techniques and weights in time play a central role in the proofs. Next we present a new abstract bootstrapping method to show Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This fact will be illustrated for a concrete SPDE. In the third part, we apply the previous results to study quasilinear reaction-diffusion equations and stochastic Navier-Stokes equations with gradient noise. As regards the former, we show global well-posedness and instantaneous regularization of solutions employing suitable dissipative conditions. Here we also prove a suitable stochastic version of the parabolic DeGiorgi-Nash-Moser estimates by employing a standard reduction method. The last chapter concerns stochastic Navier-Stokes equations and in the three dimensional case we prove local existence with data in the critical spaces $L^3$ and $B^{\frac{3}{q}-1}_{q,p}$. In addition, we prove a blow-up criterium for solutions with paths in $L^p(L^q)$ where $\frac{2}{p}+\frac{3}{q}=1$ (resp. $C(L^3)$) which extends the usual Serrin blow-up criteria (resp. its end-point version) to the stochastic setting. Finally, we prove existence of global solutions in two dimensions under minimal assumptions on the noise term and on the initial data.