Large time behavior for vortex evolution in the half-plane

In this article we study the long-time behavior of incompressible ideal flow in a half plane from the point of view of vortex scattering. Our main result is that certain asymptotic states for half-plane vortex dynamics decompose naturally into a nonlinear superposition of soliton-like states. Our approach is to combine techniques developed in the study of vortex confinement with weak convergence tools in order to study the asymptotic behavior of a self-similar rescaling of a solution of the incompressible 2D Euler equations on a half plane with compactly supported, nonnegative initial vorticity.Comment: 30 pages, no figure

Global existence of a weak solution of the incompressible Euler equations with helical symmetry and Lp vorticity

CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPERJ - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DO RIO DE JANEIROFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOWe prove the global existence of a helical weak solution of the 3D Euler equations, in full space, for an initial velocity with helical symmetry, without swirl, and whose initial vorticity is compactly supported in the axial plane and belongs to L-p, for some p > 4/3. This result is an extension of the existence part of the work of B. Ettinger and E. Titi [9], who studied well-posedness of the Euler equations with helical symmetry without swirl, with bounded initial vorticity, in a helical pipe.We prove the global existence of a helical weak solution of the 3D Euler equations, in full space, for an initial velocity with helical symmetry, without swirl, and whose initial vorticity is compactly supported in the axial plane and belongs to L-p, for some p > 4/3. This result is an extension of the existence part of the work of B. Ettinger and E. Titi [9], who studied well-posedness of the Euler equations with helical symmetry without swirl, with bounded initial vorticity, in a helical pipe.641309341CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPERJ - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DO RIO DE JANEIROFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPERJ - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DO RIO DE JANEIROFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPq [306331/2010-1, 303089/2010-5, 236994/2012-3]FAPESP [2007/51490-7, 05/58136-9]306331/2010-1; 303089/2010-5; 236994/2012-3E-26/103.197/20122007/51490-7; 05/58136-

Vanishing Viscosity Limits and Boundary Layers for Circularly Symmetric 2D Flows

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [LMN], on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in $L^2$-norm as long as the prescribed angular velocity $\alpha(t)$ of the boundary has bounded total variation. Here we establish convergence in stronger $L^2$ and $L^p$-Sobolev spaces, allow for more singular angular velocities $\alpha$, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently. [LMN] Lopes Filho, M. C., Mazzucato, A. L. and Nussenzveig Lopes, H. J., Vanishing viscosity limit for incompressible flow inside a rotating circle, preprint 2006

Propagation of support and singularity formation for a class of 2D quasilinear hyperbolic systems

In this paper we consider a class of quasilinear, non-strictly hyperbolic 2 \Theta 2 systems in two space dimensions. Our main result is finite speed of propagation of the support of smooth solutions for these systems. As a consequence, we establish non-existence of global smooth solutions for a class of sufficiently large, smooth initial data. The nonexistence result applies to systems in conservation form, which satisfy a convexity condition on the fluxes. We apply the non-existence result to a prototype example, obtaining an upper bound on the lifespan of smooth solutions with small amplitude initial data. We exhibit explicit smooth solutions for this example, obtaining the same upper bound on the lifespan and illustrating loss of smoothness through blow-up and through shock formation. AMS Subject Classification: 35L40, 35L67, 35L80 35L65. Keywords and phrases: propagation of support, lifespan of classical solutions, partially aligned systems. Consider a quasilinear, hyperbolic sy..