On Current Developments in Partial Differential Equations

Lin, Fanghua

doi:10.4208/cmr.2020-0004

Cited by 8 publications

(2 citation statements)

References 89 publications

(104 reference statements)

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“…where Re and Rm stand for the hydrodynamic and magnetic Reynolds numbers, respectively. The MHD system is well-explored when the fluid region is the whole space, seeing for instance the survey paper [25] and the references therein. Here we assume that the fluid is in the half-space…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption

Li¹,

Yang²

2020

Preprint

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We establish the well-posedness of the MHD boundary layer system in Gevrey function space without any structural assumption. Compared to the classical Prandtl equation, the loss of tangential derivative comes from both the velocity and magnetic fields that are coupled with each other. By observing a new type of cancellation mechanism in the system for overcoming the loss derivative degeneracy, we show that the MHD boundary layer system is well-posed with Gevrey index up to 3/2 in both two and three dimensional spaces.

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Section: Introductionmentioning

confidence: 99%

Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption

Li¹,

Yang²

2020

Preprint

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“…The regularity estimates for second order elliptic equations and systems in Lipschitz domains have been studied extensively. We refer the reader to [11,4,12,1,18,7,16,8,17,15,6,5] for references. For operators not involving drift terms, W 1,p estimate for Laplace equation with Dirichlet boundary condition was established by D. Jerison and C. Kenig in [11] for 3 2 − ε < p < 3 + ε if d ≥ 3 and 4 3 − ε < p < 4 + ε when d = 2 on Lipschitz domains, and the ranges of p are sharp.…”

mentioning

confidence: 99%

$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

Shi¹

2022

DCDS

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<p style='text-indent:20px;'>In this paper, we establish the <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M3">\begin{document}$ \frac{3}{2}-\varepsilon<p<3+\varepsilon $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}$ d\geq3 $\end{document}</tex-math></inline-formula>, and the range for <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is sharp. For elliptic systems, we prove that the <inline-formula><tex-math id="M6">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M7">\begin{document}$ \frac{2d}{d+1}-\varepsilon<p<\frac{2d}{d-1}+\varepsilon $\end{document}</tex-math></inline-formula> under the assumption that the Lipschitz constant of the domain is small.</p>

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Global Existence and the Decay of Solutions to the Prandtl System with Small Analytic Data

Paicu

Zhang

2021

Arch Rational Mech Anal

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In this paper, we prove the global existence and the large time decay estimate of solutions to Prandtl system with small initial data, which is analytical in the tangential variable. The key ingredient used in the proof is to derive sufficiently fast decay-in-time estimate of some weighted analytic energy estimate to a quantity, which consists of a linear combination of the tangential velocity with its primitive one, and which basically controls the evolution of the analytical radius to the solutions. Our result can be viewed as a globalin-time Cauchy-Kowalevsakya result for Prandtl system with small analytical data.

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On Current Developments in Partial Differential Equations

Cited by 8 publications

References 89 publications

Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption

Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption

$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

Global Existence and the Decay of Solutions to the Prandtl System with Small Analytic Data

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