Smooth Solutions of the Landau-Lifshitz-Bloch Equation

Li, Qiaoxin; Guo, Boling; Zeng, M.

doi:10.11948/20200376

Cited by 3 publications

(3 citation statements)

References 17 publications

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“…Using the preceeding discussion, along with (1.2) and (1.3), we can write equation (1.1) as Le [33] proved the existence of a weak solution in a bounded domain for d = 1, 2, 3. The reader can refer to [3,29,34,42,47] and references within for some recent developments. The (deterministic) LLB equation turns out to be insufficient, for example, to capture the dispersion of individual trajectories at high temperatures.…”

Section: Introductionmentioning

confidence: 99%

Relaxed optimal control for the stochastic Landau-Lifshitz-Bloch equation with jump noise

Gokhale

2023

Preprint

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We study the stochastic Landau-Lifshitz-Bloch equation perturbed by pure jump noise. In order to understand and also control the fluctuations and jumps observed in the hysteresis loop, we add noise and an external control to the effective field. We consider the control operator as one that can depend, even nonlinearly on both the control and the associated solution process. We reformulate the problem as a relaxed control problem by using random Young measures and show that the relaxed problem admits an optimal control. The proof employs the Aldous condition for establishing tightness of laws and a modified version of the Skorohod Theorem for obtaining subconvergence of laws.

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Section: Introductionmentioning

confidence: 99%

Relaxed optimal control for the stochastic Landau-Lifshitz-Bloch equation with jump noise

Gokhale

2023

Preprint

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“…Later, Guo-Li [22] established the global existence of smooth solution in one-dimensional space. After that, Li-Guo-Zeng [29] obtained the global existence of smooth solution in R 2 for any initial data, and in R 3 for small initial data. Recently, Ayouch-Benmouane-Essouf [1] established the uniqueness and local existence of the LLB equation in a bounded domain of R 3 .…”

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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“…The LLB equation has a wide range of applications such as laser, magnetic recording, magnetization switching, control domain walls, etc [2]. The existence of global weak solutions was studied by Le in [8] recently and the existence of a smooth solution to the LLB equation in R 2 when θ = 1 was studied by Li et al in [9]. In a special case, the second author studied the global well-posedness of the Landau-Lifshitz-Bloch equation with a helicity term [10].…”

Section: Introductionmentioning

confidence: 99%