Vortex sheets and diffeomorphism groupoids

Existence and Stability for the 3D Linearized Constant-Coefficient Incompressible Current-Vortex Sheets

We consider the free boundary problem for current-vortex sheets in ideal incompressible magnetohydrodynamics. The problem of current-vortex sheets arises naturally, for instance, in geophysics and astrophysics. We prove the existence of a unique solution to the constant-coefficient linearized problem and an a priori estimate with no loss of derivatives. This is a preliminary result to the study...

Stability of Compressible Vortex Sheets in Steady Supersonic Euler Flows over Lipschitz Walls

We are concerned with the stability of compressible vortex sheets in two-dimensional steady supersonic Euler flows over Lipschitz walls under a BV boundary perturbation, since steady supersonic Euler flows are important in many physical situations. It is proved that steady compressible vortex sheets in supersonic flow are stable in structure globally, even under the BV perturbation of the Lipsc...

Symplectic structure on vortex sheets

We present a Lie algebraic framework for vortex sheets as singular 2-forms with support of codimension 1, i.e. singular elements of a completion of the dual to the Lie algebra of divergence-free vector fields. This framework allows one to define the Poisson and symplectic structures on the space of vortex sheets, which interpolate between the corresponding structures on filaments and smooth vor...

Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics

Compressible vortex sheets are fundamental waves, along with shocks and rarefaction waves, in entropy solutions to multidimensional hyperbolic systems of conservation laws. Understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, th...

Singularity formation in vortex sheets and interfaces