Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form

Brzeźniak, Zdzisław; Manna, Utpal; Panda, Akash Ashirbad

doi:10.1016/j.jde.2018.11.001

Cited by 28 publications

(32 citation statements)

References 35 publications

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“…Now, we follow closely the framework of Brzeźniak et al For any p ∈ [1, ∞ ) and

k \in double-struckN

, let

false(L^{p} false(double-struckO false), false| \cdot |_{L^{p}} false)

and

false(W^{k, p} false(double-struckO false), false‖ \cdot ‖_{W^{k, p}} false)

be Lebesgue and Sobolev space of

R^{2}

‐valued functions, respectively. For p = 2, set W k ,2 = H k .…”

Section: The Mathematical Frameworkmentioning

confidence: 99%

“…To obtain the global well posedness of , we introduce the following hypotheses stated in Brzeźniak et al…”

Section: The Mathematical Frameworkmentioning

confidence: 99%

“…Proof Combing Theorem in our paper and proposition 5.4 to 5.5 proved by Brzeźniak et al, we can obtain the estimates of and . Let

ϕ_{ϵ} \in U^{M}

, we have

\begin{matrix} ũ^{ϵ} false(t false) & = u_{0} - \int_{0}^{t} A_{1} ũ^{ϵ} false(s false) d s - \int_{0}^{t} B false(ũ^{ϵ} false(s false) false) d s \\ - \int_{0}^{t} M false({\overset{θ}{true}}^{ϵ} false(s false) false) d s + \int_{0}^{t} \int_{X} G false(s, ũ^{ϵ} false(s false), v false) false(ϕ_{ϵ} false(s, v false) - 1 false) ϑ false(d v false) d s false) \\ + \int_{0}^{t} \int_{X} ϵ G false(s, ũ^{ϵ} false(s false), v false) false(N^{ϵ^{- 1} ϕ_{ϵ}} false(d s d v false) - ϵ^{- 1} ϕ_{ϵ} false(s, v false) ϑ false(d v false) d s false) \\ : = J_{ϵ}^{1} + J_{ϵ}^{2} + J_{ϵ}^{3} + J_{ϵ}^{4} + J_{ϵ}^{5} + J_{ϵ}^{6}, \\ {\overset{θ}{true}}^{ϵ} false(t false) & = θ_{0} - \int_{0}^{t} A_{2} {\overset{θ}{true}}^{ϵ} false(s false) d s - \int_{0}^{t} \overset{B}{true} false(ũ^{ϵ} false(s false), {\overset{θ}{true}}^{ϵ} false(s false) false)<...> \end{matrix}

…”

Section: Large Deviationsmentioning

confidence: 99%

“…Also, they proved the existence of global strong solution in the 2D case. Moreover, Brzeźniak et al studied the nematic liquid crystals driven by pure jump noise in both 2D and 3D cases. They proved the global well posedness of strong solution in the 2D case and established the existence of weak martingale solution of this model in the 3D case, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of the large deviations will be based on the weak convergence approach introduced in Budhiraja et al and Budhiraja et al As an important part of the proof, we need to obtain global well posedness of the so‐called skeleton equation. For the uniqueness, we adopt the method introduced in Brzeźniak et al For the existence, we first apply the Faedo‐Galerkin approximation method to construct a sequence of approximating equations as in Brzeźniak et al We then show that the family of the solutions of the approximating equations is compact in an appropriate space, and that any limit of the approximating solutions gives rise to a solution of the skeleton equation. To complete the proof of the large deviation principle (LDP), we also need to study the weak convergence of the perturbations of the system in the random directions of the Cameron‐Martin space of the driving Brownian motions.…”

Section: Introductionmentioning

confidence: 99%

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