Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation

Jiu, Quansen; Suo, Xiaoxiao; Wu, Jiahong; Yu, Huan

doi:10.4310/cms.2020.v18.n4.a5

Cited by 14 publications

(10 citation statements)

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“…Materials presented here will be used in the proofs of Theorems 1.1 and 1.2. The definition of the Besov space and related simple facts can be found in Bahouri et al 22 Lemma 2.6 is taken from Jiu et al 23 , Lemma A.5 In what follows,

scriptS false(ℝ^{d} false)

denotes the Schwartz class and

{scriptS}^{'} false(ℝ^{d} false)

the tempered distribution.…”

Section: Preparations: Besov Spacesmentioning

confidence: 99%

“…The definition of the Besov space and related simple facts can be found in [1]. Lemma 2.6 is taken from [15,Lemma A.5]. In what follows, S(R d ) denotes the Schwartz class and S ′ (R d ) the tempered distribution.…”

Section: Preparations: Besov Spacesmentioning

confidence: 99%

“…These bounds will be used in the proofs of Theorems 1.1 and 1.2. We refer the reader to Lemma A.5 in [15] for a detailed proof of the following lemma. Lemma 2.6.…”

Section: Preparations: Besov Spacesmentioning

confidence: 99%

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Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

Said

2020

Math Methods in App Sciences

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This article examines the existence and uniqueness of weak solutions to the d-dimensional micropolar equations (d = 2 or d = 3) with general fractional dissipation (−Δ) u and (−Δ) w. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when ≥ 1 2 and ≥ 1 2 , any initial data (u 0 , w 0) in the critical Besov space u 0 ∈ B 1+ d 2 −2 2,1 (R d) and w 0 ∈ B 1+ d 2 −2 2,1 (R d) yields a unique weak solution. For ≥ 1 and = 0, any initial data u 0 ∈ B 1+ d 2 −2 2,1 (R d) and w 0 ∈ B d 2 2,1 (R d) also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, = = 1, have a unique weak solution for (u 0 , w 0) ∈ B 0 2,1. The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.

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scriptS false(ℝ^{d} false)

denotes the Schwartz class and

{scriptS}^{'} false(ℝ^{d} false)

the tempered distribution.…”

Section: Preparations: Besov Spacesmentioning

confidence: 99%

Section: Preparations: Besov Spacesmentioning

confidence: 99%

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Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

Said

2020

Math Methods in App Sciences

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“…When θ is a constant, the system (1.1) reduces to the MHD-type equations. For MHD equations, Jiu et al [21] established the local existence and uniqueness of weak solutions with the minimal initial regularity assumption and for the largest possible range of α's. Naturally, we wonder that whether the tropical climate model can use the minimum initial regularity hypothesis to obtain the existence and uniqueness of the weak solution in the maximum possible range of α, β.…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by [21,22], the main goal of this paper is to establish the unique weak solutions to (1.1) in a weakest possible functional setting for the largest possible ranges of α and β. The difficulty of this paper lies in the treatment of nonlinear terms, which is due to the lack of free divergence conditions of v and the absence of thermal diffusion in the equation of θ.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of local weak solution of d-dimensional tropical climate model without thermal diffusion in inhomogeneous Besov space

Yuan,

Zhang

2020

Preprint

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Local existence for the d‐dimensional magneto‐micropolar equations with fractional dissipation in Besov spaces

Qiu

Xiao

Yao

2023

Math Methods in App Sciences

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In this paper, we consider the Cauchy problem of the d-dimensional magnetomicropolar equations (d = 2 or d = 3) with general fractional dissipation. The aim of this paper is to obtain the existence and uniqueness of solutions in the weakest possible inhomogeneous Besov spaces. Using the technical tools of Litttlewood-Paley decomposition and Besov spaces theory, we obtain the local existence in the functional setting of inhomogeneous Besov spaces. Furthermore, such solutions are unique only in 2D case.

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Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation

Cited by 14 publications

References 0 publications

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

Existence and uniqueness of local weak solution of d-dimensional tropical climate model without thermal diffusion in inhomogeneous Besov space

Local existence for the d‐dimensional magneto‐micropolar equations with fractional dissipation in Besov spaces

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