A uniqueness result for the d-dimensional magnetohydrodynamics equations with fractional dissipation in Besov spaces

Hua Qiu (), Xia Wang () and Zheng-an Yao ()
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Hua Qiu: South China Agricultural University
Xia Wang: South China Agricultural University
Zheng-an Yao: Sun Yat-sen University

Partial Differential Equations and Applications, 2025, vol. 6, issue 1, 1-27

Abstract: Abstract In this paper, we consider the Cauchy problem of d-dimensional magnetohydrodynamics equations $$(d\ge 2)$$ ( d ≥ 2 ) with fractional dissipation $$(-\Delta )^{\alpha }u$$ ( - Δ ) α u and fractional magnetic diffusion $$(-\Delta )^{\beta }b.$$ ( - Δ ) β b . The aim of this paper is to establish the uniqueness of weak solutions under the $$L^p$$ L p framework in sense of the weakest possible inhomogeneous Besov spaces. We obtain the local existence and uniqueness in the functional setting $$u\in L_T^{\infty }(B_{p,1}^{\frac{d}{p}+1-2\alpha }({\mathbb {R}}^d))$$ u ∈ L T ∞ ( B p , 1 d p + 1 - 2 α ( R d ) ) and $$b\in L_T^{\infty }(B_{p,1}^{\frac{d}{p}}({\mathbb {R}}^d))$$ b ∈ L T ∞ ( B p , 1 d p ( R d ) ) when $$\alpha $$ α and $$\beta $$ β satisfy certain conditions by using the iterative scheme and compactness arguments.

Keywords: Magnetohydrodynamics equations; Local solution; Besov space; Uniqueness; 35A02; 35Q35; 76D03 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00311-8

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