Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

Charles L. Fefferman; David S. McCormick; James C. Robinson; Jose L. Rodrigo

doi:10.1007/s00205-016-1042-7

Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

Charles L. Fefferman, David S. McCormick, James C. Robinson, Jose L. Rodrigo

Mathematics

Research output: Contribution to journal › Article › peer-review

99 Scopus citations

Abstract

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in R^d, where d = 2, 3, with initial data B₀∈ H^s(R^d) and u₀∈ H^s ^- ¹ ⁺ ^ϵ(R^d) for s> d/ 2 and any 0 < ϵ< 1. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ϵ= 0 is explained by the failure of solutions of the heat equation with initial data u₀∈ H^s ^- ¹ to satisfy u∈ L¹(0 , T; H^s ⁺ ¹) ; we provide an explicit example of this phenomenon.

Original language	English (US)
Pages (from-to)	677-691
Number of pages	15
Journal	Archive for Rational Mechanics and Analysis
Volume	223
Issue number	2
DOIs	https://doi.org/10.1007/s00205-016-1042-7
State	Published - Feb 1 2017

All Science Journal Classification (ASJC) codes

Analysis
Mathematics (miscellaneous)
Mechanical Engineering

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title = "Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces",

abstract = "This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rd, where d = 2, 3, with initial data B0∈ Hs(Rd) and u0∈ Hs - 1 + ϵ(Rd) for s> d/ 2 and any 0 < ϵ< 1. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ϵ= 0 is explained by the failure of solutions of the heat equation with initial data u0∈ Hs - 1 to satisfy u∈ L1(0 , T; Hs + 1) ; we provide an explicit example of this phenomenon.",

author = "Fefferman, \{Charles L.\} and McCormick, \{David S.\} and Robinson, \{James C.\} and Rodrigo, \{Jose L.\}",

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doi = "10.1007/s00205-016-1042-7",

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T1 - Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

AU - Fefferman, Charles L.

AU - McCormick, David S.

AU - Robinson, James C.

AU - Rodrigo, Jose L.

PY - 2017/2/1

Y1 - 2017/2/1

N2 - This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rd, where d = 2, 3, with initial data B0∈ Hs(Rd) and u0∈ Hs - 1 + ϵ(Rd) for s> d/ 2 and any 0 < ϵ< 1. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ϵ= 0 is explained by the failure of solutions of the heat equation with initial data u0∈ Hs - 1 to satisfy u∈ L1(0 , T; Hs + 1) ; we provide an explicit example of this phenomenon.

AB - This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rd, where d = 2, 3, with initial data B0∈ Hs(Rd) and u0∈ Hs - 1 + ϵ(Rd) for s> d/ 2 and any 0 < ϵ< 1. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ϵ= 0 is explained by the failure of solutions of the heat equation with initial data u0∈ Hs - 1 to satisfy u∈ L1(0 , T; Hs + 1) ; we provide an explicit example of this phenomenon.

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Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

Abstract

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