On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics

Jiang, Song; Zhang, Jianwen

doi:10.1088/1361-6544/aa82f2

Cited by 44 publications

(15 citation statements)

References 29 publications

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“…Motivated by the studies for the Navier-Stokes equations (cf. [17,19,36]), we here introduce the following so-called "effective viscous flux G(x, t)":…”

Section: Remarkmentioning

confidence: 99%

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Boundary layers and stabilization of the singular Keller-Segel system

Peng

Wang

Zhao

et al. 2018

Kinetic &Amp; Related Models

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The original Keller-Segel system proposed in [23] remains poorly understood in many aspects due to the logarithmic singularity. As the chemical consumption rate is linear, the singular Keller-Segel model can be converted, via the Cole-Hopf transformation, into a system of viscous conservation laws without singularity. However the chemical diffusion rate parameter ε now plays a dual role in the transformed system by acting as the coefficients of both diffusion and nonlinear convection. In this paper, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing appropriate conditions on the boundary data, we show that boundary layer profiles are present as ε → 0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the "effective viscous flux" technique to establish the uniform-in-ε estimates to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic L 1estimate. We the obtain the corresponding results for the original Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications.

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“…Motivated by the studies for the Navier-Stokes equations (cf. [17,19,36]), we here introduce the following so-called "effective viscous flux G(x, t)":…”

Section: Remarkmentioning

confidence: 99%

“…Proof of Theorem 2.3 (ii). Inspired by a recent work[19], we first establish the following lemma by the weighted L 2 -method dedicating to the boundary layer solutions.…”

mentioning

confidence: 99%

Boundary layers and stabilization of the singular Keller-Segel system

Peng

Wang

Zhao

et al. 2018

Kinetic &Amp; Related Models

View full text Add to dashboard Cite

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“…It should be addressed that recently Zhang and Zhao [23] proved the global solvability of strong solutions to the initial-boundary value problem for (1.5)-(1.8) with general heat conductivity coefficient and arbitrarily large data, by making a full use of the effective viscous flux, the material derivative and the structure of the equations; while the global well-posedness of strong solutions for isentropic fluids was verified by Jiang and Zhang [11]. When initial vacuum is allowed, Fan and Hu [8] proved the global existence of strong solutions to the problem (1.5)-(1.8) under certain technical assumptions concerning the growth condition of the heat conductivity coefficient and the ratio between the initial magnetic field and the initial density; see also [21] for a similar result dealing with the isentropic regime, where some technical hypotheses are imposed on the initial magnetic field and the initial density.…”

Section: Introductionmentioning

confidence: 99%

Global weak solutions and long time behavior for 1D compressible MHD equations without resistivity

Sun

2019

Journal of Mathematical Physics

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“…In fact, the lack of diffusion mechanism for the magnetic field brings extra obstacle in constructing global solutions. Even in the one-dimensional case, global existence and uniqueness of smooth solutions with large data has recently been obtained by Jiang and Zhang [21]; see also [23] for the treatment of planar non-resistive MHD equations. Wu and Wu [36] showed the global solvability of small smooth solutions to the two-dimensional compressible non-resistive MHD equations; the verification in three space dimensions can be found in Jiang and Jiang [19,20].…”

Section: Introductionmentioning

confidence: 99%

Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids

Sun

2019

Journal of Differential Equations

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We consider a two-dimensional MHD model describing the evolution of viscous, compressible and electrically conducting fluids under the action of vertical magnetic field without resistivity. Existence of global weak solutions is established for any adiabatic exponent γ ≥ 1. Inspired by the approximate scheme proposed in [15], we consider a two-level approximate system with artificial diffusion and pressure term. At the first level, we prove global well-posedness of the regularized system and establish uniform-in-ε estimates to the regular solutions. At the second level, we show global existence of weak solutions to the system with artificial pressure by sending ε to 0 and deriving uniform-in-δ estimates. Then global weak solution to the original system is constructed by vanishing δ. The key issue in the limit passage is the strong convergence of approximate sequence of the density and magnetic field. This is accomplished by following the technique developed in [15,26] and using the new technique of variable reduction developed by Vasseur et al. [33] in the context of compressible two-fluid model so as to handle the cross terms.

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On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics

Cited by 44 publications

References 29 publications

Boundary layers and stabilization of the singular Keller-Segel system

Boundary layers and stabilization of the singular Keller-Segel system

Global weak solutions and long time behavior for 1D compressible MHD equations without resistivity

Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids

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