Existence, uniqueness and regularity for the stochastic Ericksen-Leslie equation

Abstract

We investigate existence and uniqueness for the liquid crystal flow driven by colored noise on the two-dimensional torus. After giving a natural uniqueness criterion, we prove local solvability in L p -based spaces, for every p > 2. Thanks to a bootstrap principle together with a Gyöngy-Krylov-type compactness argument, this will ultimately lead us to prove the existence of a particular class of global solutions which are partially regular, strong in the probabilistic sense, and taking values in the "critical space" L 2 × H 1 .

Global weak solutions to the Stochastic Ericksen--Leslie equations in dimension two

“…For the mathematically modeling, taking the stochastic terms into account reflects the influence of environmental noises, measurement uncertainties or thermal fluctuations. Analogously, Bouard-Hocquet-Prohl obtained the Struwe-like global solution to (1.1) in [2] by a bootstrap argument together with Gyöngy-Krylov L p estimates [14]. Very recently, Brzeźniak, Deugoué, and Razafimandimby in [3] proved the existence of short time strong solutions to the simplified stochastic Ericksen-Leslie system.…”

Concentration-cancellation in the Ericksen-Leslie model

“…In [16, p. 1108] and [2, p. 290] this issue was highlighted as an open problem. This limit passage is also of interest for numerical approximations [34], in the stochastic Ericksen-Leslie system [8] or the flow of magnetoviscoelastic materials (see [29]). The singular limit problem ǫ → 0 + for the harmonic map heat flow into spheres and more general manifolds was first studied by Chen and Struwe in [5,6] (see also [1] for the related Landau-Lifshitz equation).…”