Existence of Vortex Sheets with Reflection Symmetry in Two Space Dimensions

2001 ◽  
Vol 158 (3) ◽  
pp. 235-257 ◽  
Author(s):  
M. C. Lopes Filho ◽  
H. J. Nussenzveig Lopes ◽  
Zhouping Xin
Keyword(s):  
Reflection Symmetry ◽  
Vortex Sheets ◽  
Two Space Dimensions

scholarly journals Nonlinear compressible vortex sheets in two space dimensions

2008 ◽  
Vol 41 (1) ◽  
pp. 85-139 ◽  
Author(s):  
Jean-François Coulombel ◽  
Paolo Secchi
Keyword(s):  
Vortex Sheets ◽  
Two Space Dimensions

scholarly journals Vortex sheets with reflection symmetry in exterior domains

2006 ◽  
Vol 229 (1) ◽  
pp. 154-171 ◽  
Author(s):  
M.C. Lopes Filho ◽  
H.J. Nussenzveig Lopes ◽  
Zhouping Xin
Keyword(s):  
Reflection Symmetry ◽  
Exterior Domains ◽  
Vortex Sheets

scholarly journals The stability of compressible vortex sheets in two space dimensions

2004 ◽  
Vol 53 (4) ◽  
pp. 941-1012 ◽  
Author(s):  
Jean-Francois Coulombel ◽  
Paolo Secchi
Keyword(s):  
Vortex Sheets ◽  
Two Space Dimensions ◽  
The Stability

scholarly journals Anisotropic regularity of linearized compressible vortex sheets

2020 ◽  
Vol 17 (03) ◽  
pp. 443-458
Author(s):  
Paolo Secchi
Keyword(s):  
Weighted Sobolev Spaces ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Frequency Space ◽  
Inviscid Fluids ◽  
Classical Stability ◽  
Pseudo Differential Equation ◽  
Constant Coefficients ◽  
Two Space Dimensions ◽  
Weakly Stable

We consider supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, Morando et al. recently derived a pseudo-differential equation that describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if [Formula: see text], and the well-posedness holds in standard weighted Sobolev spaces. Our aim in this paper is to improve this result, by showing the existence in functional spaces with additional weighted anisotropic regularity in the frequency space.

TWO-DIMENSIONAL VORTEX SHEETS FOR THE NONISENTROPIC EULER EQUATIONS: LINEAR STABILITY

2008 ◽  
Vol 05 (03) ◽  
pp. 487-518 ◽  
Author(s):  
ALESSANDRO MORANDO ◽  
PAOLA TREBESCHI
Keyword(s):  
Linear Stability ◽  
Euler Equations ◽  
Compressible Flows ◽  
Tangential Velocity ◽  
Vortex Sheets ◽  
Isentropic Euler Equations ◽  
Contact Discontinuities ◽  
Two Space Dimensions ◽  
Loss Of Regularity ◽  
Additional Loss

We study the linear stability of contact discontinuities for the nonisentropic compressible Euler equations in two space dimensions. Assuming the jump of the tangential velocity across the discontinuity surface is sufficiently large, we derive a suitable energy estimate for the linearized boundary value problem. The found estimate extends to nonisentropic compressible flows the main result of Coulombel–Secchi for the isentropic Euler equations. Compared with this latter case, when the jump of the tangential velocity of the unperturbed flow attains a certain critical value in the region of weak stability, here an additional loss of regularity appears; this is related to the presence of a double root of the Lopatinskii determinant associated to the problem.

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