Existence of a unique solution and invariant measures for the stochastic Landau–Lifshitz–Bloch equation

Brzeźniak, Zdzisław; Gołdys, Beniamin; Le, Kim Ngan

doi:10.1016/j.jde.2020.06.061

Cited by 20 publications

(23 citation statements)

References 15 publications

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“…All coefficients in (5.1) are locally Lipschitz (see, for example [9]). For the control term (F 5 n ), we use Assumption 3.1.…”

Section: Existence Of a Solution For The Relaxed Control Problemmentioning

confidence: 99%

“…We give a sketch of the proof. For detailed arguments for a similar proof, we refer the reader to [9]. Consider the equality (5.1).…”

Section: Uniform Energymentioning

confidence: 99%

“…Following the works [7], [9], we introduce the noise (and later, as an example, also control, see Remark 3.4) in the effective field. One of the aims of studying the said models is to understand the transition phenomenon due to thermal fluctuations within the material [4].…”

Section: Introductionmentioning

confidence: 99%

“…Brzeźniak and Le [9] showed the existence of a martingale solution to the stochastic LLB equation. For dimensions 1, 2, they showed that the solution is pathwise unique, and hence the problem admits a unique, strong solution.…”

Section: Introductionmentioning

confidence: 99%

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Wong–Zakai approximations for the stochastic Landau–Lifshitz–Bloch equations

Gokhale

Manna

2022

J. Math. Phys.

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The stochastic Landau–Lifshitz–Bloch equation describes the evolution of spins in ferromagnetic materials for temperatures both below and above the Curie temperature. We consider both exchange and anisotropy energy in dimensions 1 and 2. The equation is transformed into a non-linear partial differential equation with random coefficients, which has a pathwise unique solution. This solution depends continuously on the considered Wiener process. We transform back to the original equation to conclude that the solution to the original equation depends continuously on the Wiener process considered.

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“…All coefficients in (5.1) are locally Lipschitz (see, for example [9]). For the control term (F 5 n ), we use Assumption 3.1.…”

Section: Existence Of a Solution For The Relaxed Control Problemmentioning

confidence: 99%

“…We give a sketch of the proof. For detailed arguments for a similar proof, we refer the reader to [9]. Consider the equality (5.1).…”

Section: Uniform Energymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Wong–Zakai approximations for the stochastic Landau–Lifshitz–Bloch equations

Gokhale

Manna

2022

J. Math. Phys.

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“…Recently, Ayouch-Benmouane-Essouf [1] established the uniqueness and local existence of the LLB equation in a bounded domain of R 3 . For the results of the stochastic LLB equation, see [5,20,26,34] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Strong solutions of the Landau-Lifshitz-Bloch equation in Besov space

Peng¹,

Wang²

2022

Preprint

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We focus on the existence and uniqueness of the three-dimensional Landau-Lifshitz-Bloch equation supplemented with the initial data in Besov space Ḃ 3 2 2,1 . Utilizing a new commutator estimate, we establish the local existence and uniqueness of strong solutions for any initial data in Ḃ 3 2 2,1 . When the initial data is small enough in Ḃ 3 22,1 , we obtain the global existence and uniqueness. Furthermore, we also establish a blow-up criterion of the solution to the Landau-Lifshitz-Bloch equation and then we prove the global existence of strong solutions in Sobolev space under a new condition based on the blow-up criterion.

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Stochastic Landau–Lifshitz–Bloch Equation with Transport Noise: Well-Posedness, Dissipation Enhancement

Qiu,

Sun

2024

J Stat Phys

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Existence of a unique solution and invariant measures for the stochastic Landau–Lifshitz–Bloch equation

Cited by 20 publications

References 15 publications

Wong–Zakai approximations for the stochastic Landau–Lifshitz–Bloch equations

Wong–Zakai approximations for the stochastic Landau–Lifshitz–Bloch equations

Strong solutions of the Landau-Lifshitz-Bloch equation in Besov space

Stochastic Landau–Lifshitz–Bloch Equation with Transport Noise: Well-Posedness, Dissipation Enhancement

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