Ill-posedness of the Navier-Stokes and magneto-hydrodynamics systems

2015/10/20 by Alexey Cheskidov, Cheskidov, Alexey, Mimi Dai +1
Mathematics · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1510.05733

Abstract

We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we construct initial data which has finite energy and is small in certain spaces, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time t=0 in such spaces. An analogous result is also obtained for the Navier-Stokes equation which extends the previous result of ill-posedness in B^-1_∞,∞ by Cheskidov and Shvydkoy to spaces that are not necessarily critical. The region of the spaces where the norm inflation occurs almost touches L².

Citations

• Inéquations en thermoélasticité et magnétohydrodynamique

Related

• Ill-posedness of the incompressible magneto-hydrodynamics system in B_∞,∞^-1
• Ill-posedness of the Navier-Stokes equations in a critical space in 3D
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• Sharp and strong non-uniqueness for the magneto-hydrodynamic equations
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