Pathwise mild solutions for quasilinear stochastic partial differential equations

Abstract

Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, includingamong others -cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time.Here we aim to develop a theory for quasilinear stochastic evolution equations (1.1) using a semigroup approach. The main theme is to extend the very general deterministic theory of quasilinear Cauchy problems [3,61,63]. The key idea is to employ a modified definition of mild solutions [57] for the quasilinear case in comparison to the more classical parabolic SPDE setting [54]. Before we describe our approach in more detail, we briefly review some other techniques and solution concepts used for (certain subclasses of) the SPDE (1.1). Instead of mild solutions, one may instead use weak, or martingale, solutions [13,20,24,35] of (1.1); here weak solution is interpreted in the classical PDE sense while these solutions are also sometimes referred to as strong solutions from a probabilistic perspective [55]. There are also several works exploiting the additional assumption of monotone coefficients [43] particularly in the case of the stochastic porous medium equation [9,10], where A(u) = ∆(a(u)) for a maximal monotone map a and F ≡ 0. Other approaches to quasilinear SPDEs are based upon a gradient structure [29], approximation methods [38,39], kinetic solutions [20,30], or directly looking at strong (in the PDE sense) solutions [36].One may ask, why one might want to prove the existence of pathwise mild solutions obtained by a suitable variations-of-constants/Duhamel formula [34] instead of working with weak solutions obtained in a formulation via test functions [26]? One reason is that a mild formulation is often more natural to work with in the context of (random) dynamical systems for SPDEs [18,34]. In fact, many classical results regarding dynamics and long-time behavior of semilinear SPDEs are often crucially based upon the mild formulation and semigroups [34]. We expect this theory to generalize a lot easier also in the quasilinear case if one does not have to work with weak(er) solutions. If we take the SKT system again as a motivation, then there are deterministic results regarding the existence of attractors using weak [53] as well as mild [62] solutions concepts. We intend to investigate the existence of random attractors for the stochastic SKT equation ...

Dynamics of Stochastic Reaction-Diffusion Equations

“…Assumption 3) represents a classical Lipschitz continuity, which can be weakened according to [123]. One can easily incorporate a locally Lipschitz drift term.…”

“…In the nonautonomous case, one can also use the forward integral of Russo-Vallois [153] to define the convolution (2.18) and to construct a solution to the corresponding SPDE which coincides with the pathwise mild solution as argued in [151,Section 4.5]. The statement can be proved using fixed-point arguments as in [123]. This can be achieved in two steps.…”

“…The second step is to prove that the mapping Φ(v) := u, for v ∈ P T maps P T into itself and is a contraction if the time-horizon T is chosen sufficiently small. As applications we mention the stochastic Shigesada-Kawasaki-Teramoto model from mathematical biology, which was introduced to analyze population segregation by induced cross-diffusion, see [123,Section 4]. Written in divergence form, this is given by…”

“…The first three subsections 2.1-2.3 provide a concise survey of several broadly used classical solution concepts for semi-linear SPDEs such as mild, weak, strong, martingale and variational solutions [51,150,118]. In Section 2.4 we investigate more complicated quasilinear SPDEs and their pathwise mild solutions [123]. Section 2.5 presents an alternative to the Itô calculus via the random field approach [52,53] based on Walsh integration theory [164].…”

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

“…There are other solution concepts such as strong, kinetic [42], martingale [31], pathwise mild [106], or renormalized [71] solutions. Strong solutions rarely exist [148] due to the roughness of the noise.…”