scholarly journals Mass concentration phenomenon to the two-dimensional Cauchy problem of the compressible Magnetohydrodynamic equations

2020 ◽  
Vol 19 (10) ◽  
pp. 4973-4994
Author(s):  
Yongfu Wang ◽  
Keyword(s):  
Cauchy Problem ◽  
Mass Concentration ◽  
Two Dimensional ◽  
Magnetohydrodynamic Equations ◽  
Concentration Phenomenon

scholarly journals MASS CONCENTRATION WITH MIXED NORM FOR A NONELLIPTIC SCHRÖDINGER EQUATION

2012 ◽  
Vol 93 (3) ◽  
pp. 238-258
Author(s):  
SEHEON HAM
Keyword(s):  
Schrödinger Equation ◽  
Finite Time ◽  
Schrodinger Equation ◽  
Mass Concentration ◽  
Mixed Norm ◽  
Two Dimensional ◽  
Concentration Phenomenon

AbstractThis paper is concerned with a mass concentration phenomenon for a two-dimensional nonelliptic Schrödinger equation. It is well known that this phenomenon occurs when the ${L}^{4} $-norm of the solution blows up in finite time. We extend this result to the case where a mixed norm of the solution blows up in finite time.

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

scholarly journals The influence of resistivity gradients on shock conditions for a Petschek reconnection geometry

2016 ◽  
Vol 34 (4) ◽  
pp. 421-425
Author(s):  
Christian Nabert ◽  
Karl-Heinz Glassmeier
Keyword(s):  
Magnetic Field ◽  
Necessary Conditions ◽  
The Other ◽  
Diffusion Region ◽  
Two Dimensional ◽  
Characteristic Velocity ◽  
Magnetohydrodynamic Equations ◽  
One Dimensional ◽  
The Magnetic Field ◽  
Alfven Velocity

Abstract. Shock waves can strongly influence magnetic reconnection as seen by the slow shocks attached to the diffusion region in Petschek reconnection. We derive necessary conditions for such shocks in a nonuniform resistive magnetohydrodynamic plasma and discuss them with respect to the slow shocks in Petschek reconnection. Expressions for the spatial variation of the velocity and the magnetic field are derived by rearranging terms of the resistive magnetohydrodynamic equations without solving them. These expressions contain removable singularities if the flow velocity of the plasma equals a certain characteristic velocity depending on the other flow quantities. Such a singularity can be related to the strong spatial variations across a shock. In contrast to the analysis of Rankine–Hugoniot relations, the investigation of these singularities allows us to take the finite resistivity into account. Starting from considering perpendicular shocks in a simplified one-dimensional geometry to introduce the approach, shock conditions for a more general two-dimensional situation are derived. Then the latter relations are limited to an incompressible plasma to consider the subcritical slow shocks of Petschek reconnection. A gradient of the resistivity significantly modifies the characteristic velocity of wave propagation. The corresponding relations show that a gradient of the resistivity can lower the characteristic Alfvén velocity to an effective Alfvén velocity. This can strongly impact the conditions for shocks in a Petschek reconnection geometry.

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