Grade-two fluids on non-smooth domain driven by multiplicative noise: Existence, uniqueness and regularity

Abstract

In this paper we present a systematic study of a stochastic PDE with multiplicative noise modeling the motion of viscous and inviscid grade-two fluids on a bounded domain O of R 2. We aim to identify the minimal conditions on the boundary smoothness of the domain for the well-posedness and time regularity of the solution. In particular, we found out that the existence of a H 1 pOq weak martingale solution holds for any bounded Lipschitz domain O. When O is a convex polygon the solution u lives in the Sobolev space W 2,r pOq for some r ą 2 and rotpu´α∆uq is continuous in L 2 pOq with respect to the time variable. Moreover, pathwise uniqueness of solution holds. The existence result is new for the stochastic inviscid model and improves previous results for the viscous one. The time continuity result is new, even for the deterministic case when the domain O is a convex polygon.

Viscosity limit and deviations principles for a grade-two fluid driven by multiplicative noise

“…We also recall the following properties of J k , see for instance [18] and also [52,Proposition 6.3 as k Ñ 8.…”

“…Now we can pass to the limit to complete the existence of a solution u 0 h P Cpr0, T s; DpAqq X L 8 p0, T ; Wq. The continuity of u 0 h : r0, T s Ñ W was established in [44] and in the recent paper [52]. The uniqueness of the solution can also be established as in [8,Section 4], see also [15,Theorem 3.6].…”

“…(d) Item (a) was very important in [52] for the proof of the existence of weak martingale solution to (1.2).…”

“…For δ " 0 the problem (1.6) reduces to the stochastic model for grade-two fluid. Under Assumption (G0) it is proved in [52], see also [49], that for δ " 0 problem (1.6) has a weak martingale solution satisfying (3.8) and which is pathwise unique. Thus, by the Watanabe-Yamada's theorem, see [48], it has a unique strong solution; see also [51] for a direct proof of the existence and uniqueness of a strong solution.…”

“…65, Lemma 5.2] for the stochastic case and Theorem B.1 for the deterministic case. Thanks to Assumption (G0) one can prove by arguing as in[52] that if it has a solution then it is pathwise unique. In fact, under the theorem assumption it is not difficult to check that u ε,1 :" ε´1 2 λ´1pεqru ε´u s satisfies (1.8) and hence the only solution.Remark 3.7.…”