Abstract:In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic … Show more
“…Therefore, fixing > 0 and letting → ∞ and arguing similarly as in Brzeźniak et al, 55 we infer that for any Φ ∈ L 2 ( Ω, , P; L 2 (D) ) ,…”
Section: Existence Of Probabilistic Weak Solutionmentioning
confidence: 55%
“…The stochastic process W = {W(t) ∶ t ∈ [0, T]} is a K-cylindrical Wiener process evolving on L 2 (D), and for any , t ∈ [0, T], the increments W(t) − W( ) is independent of the -algebra generated by (u( ), W( )) for any∈ [0, ].Proof. The proof of this result is a verbatim repetition of the proof in Brzeźniak et al,8 , lemma 5.2, p25 the proofs of Proposition 4.11 in Breit and Hofmanova,54 , p. 1211 and Proposition 3.23 in Brzeźniak et al55 …”
mentioning
confidence: 51%
“…Indeed, this filtration satisfies the usual conditions. Following, for instance, previous studies, 8,54,55 we can easily check that for each , the sequences of stochastic processes (W m (t); t ∈ [0, T]) m∈N form K-valued -Wiener processes defined on the probability space (Ω, ; P) and that for each…”
In this paper, we study a certain class of stochastic quasilinear parabolic equations describing a generalized polytropic elastic filtration in the framework of variable exponents Lebesgue and Sobolev spaces. We establish an existence result in the infinite dimensional framework of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions, and the governing equations are subjected to cylindrical Wiener processes. We use a Galerkin method, derive crucial a priori estimates for the approximate solutions, and combine profound analytic and probabilistic compactness results in order to pass to the limit. Several difficulties arise in obtaining these uniform bounds and passing to the limit since the nonlinear elliptic part of the leading operator admits nonstandard growth. Apart from adapting the above essential tools, we extend classical methods of monotonicity to the present situation. KEYWORDS generalized weak solution, monotone method, non-Newtonian polytropic filtration, stochastic partial differential equations, stochastic systems MSC CLASSIFICATION 60H15; 35D30; 35K59; 76A05; 93E03where Q T = (0, T) × D, the function u = u(t, x) is an unknown defined in Q T , and f (t, u) and G(t, u) are random external forces. Moreover, W is a cylindrical Wiener process evolving on L 2 (D), which enters the equation as an unknown, the function u 0 ∈ L 2 (D), and A is a nonlinear operator in the divergence form
“…Therefore, fixing > 0 and letting → ∞ and arguing similarly as in Brzeźniak et al, 55 we infer that for any Φ ∈ L 2 ( Ω, , P; L 2 (D) ) ,…”
Section: Existence Of Probabilistic Weak Solutionmentioning
confidence: 55%
“…The stochastic process W = {W(t) ∶ t ∈ [0, T]} is a K-cylindrical Wiener process evolving on L 2 (D), and for any , t ∈ [0, T], the increments W(t) − W( ) is independent of the -algebra generated by (u( ), W( )) for any∈ [0, ].Proof. The proof of this result is a verbatim repetition of the proof in Brzeźniak et al,8 , lemma 5.2, p25 the proofs of Proposition 4.11 in Breit and Hofmanova,54 , p. 1211 and Proposition 3.23 in Brzeźniak et al55 …”
mentioning
confidence: 51%
“…Indeed, this filtration satisfies the usual conditions. Following, for instance, previous studies, 8,54,55 we can easily check that for each , the sequences of stochastic processes (W m (t); t ∈ [0, T]) m∈N form K-valued -Wiener processes defined on the probability space (Ω, ; P) and that for each…”
In this paper, we study a certain class of stochastic quasilinear parabolic equations describing a generalized polytropic elastic filtration in the framework of variable exponents Lebesgue and Sobolev spaces. We establish an existence result in the infinite dimensional framework of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions, and the governing equations are subjected to cylindrical Wiener processes. We use a Galerkin method, derive crucial a priori estimates for the approximate solutions, and combine profound analytic and probabilistic compactness results in order to pass to the limit. Several difficulties arise in obtaining these uniform bounds and passing to the limit since the nonlinear elliptic part of the leading operator admits nonstandard growth. Apart from adapting the above essential tools, we extend classical methods of monotonicity to the present situation. KEYWORDS generalized weak solution, monotone method, non-Newtonian polytropic filtration, stochastic partial differential equations, stochastic systems MSC CLASSIFICATION 60H15; 35D30; 35K59; 76A05; 93E03where Q T = (0, T) × D, the function u = u(t, x) is an unknown defined in Q T , and f (t, u) and G(t, u) are random external forces. Moreover, W is a cylindrical Wiener process evolving on L 2 (D), which enters the equation as an unknown, the function u 0 ∈ L 2 (D), and A is a nonlinear operator in the divergence form
“…He is very grateful for the financial support he received from the International Centre for Mathematical Sciences (ICMS) Edinburgh. Last, but not the least, the authors wish to thank Professor Guoli Zhou for pointing out some gaps in the previous version of this paper [8].…”
Section: Proof Of Part (Iii)mentioning
confidence: 99%
“…However, it is pointed out in [23,Chapter 5] that the fluid flow disturbs the alignment and conversely a change in the alignment will induce a flow in the nematic liquid crystal. It is this gap in knowledge that is the motivation for our mathematical study which was initiated in the old unpublished preprints [7] and [8], see also the recent papers [6] and [5].…”
In this paper, we prove the existence of a unique maximal local strong solutions to a stochastic system for both 2D and 3D penalised nematic liquid crystals driven by multiplicative Gaussian noise. In the 2D case, we show that this solution is global. As a by-product of our investigation, but of independent interest, we present a general method based on fixed point arguments to establish the existence and uniqueness of a maximal local solution of an abstract stochastic evolution equations with coefficients satisfying local Lipschitz condition involving the norms of two different Banach spaces.
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