Approximate solutions of the incompressible Euler equations with no concentrations

We present a sharp local condition for the lack of concentrations in (and hence the L-2 convergence of) sequences of approximate solutions to the incompressible Euler equations. We apply this characterization to greatly simplify known existence results for 2D flows in the full plane (with special emphasis on rearrangement invariant regularity spaces), and obtain new existence results of solutions without energy concentrations in any number of spatial dimensions. Our results identify the 'critical' regularity which prevents concentrations, regularity which is quantified in terms of Lebesgue, Lorentz, Orlicz and Morrey spaces. Thus, for example, the strong convergence criterion cast in terms of circulation logarithmic decay rates due to DiPerna and Majda is simplified (removing the weak control of the vorticity at infinity) and extended (to any number of space dimensions). Our approach relies on using a generalized div-curl lemma (interesting for its own sake) to replace the role that elliptic regularity theory has played previously in this problem. (C) 2000 Editions scientifiques et medicales Elsevier SAS.17337141

Transport of interfaces with surface tension by 2D viscous flows

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)We consider the problem of finding a global weak solution for two-dimensional, incompressible viscous flow on a torus, containing a surface-tension bearing curve transported by the flow. This is the simplest case of a class of two-phase flows considered by Plotnikov in [16] and Abels in [1]. Our work complements Abels' analysis by examining this special case in detail. We construct a family of approximations and show that the limit of these approximations satisfies, globally in time, an incomplete set of equations in the weak sense. In addition, we examine criteria for closure of the limit system, we find conditions which imply nontrivial dependence of the limiting solution on the surface tension parameter, and we obtain a new system of evolution equations which models our flow-interface problem, in a form that may be useful for further analysis and for numerical simulations.1212344NSF [DMS-0926378, DMS-0405066]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)NSF [DMS-0926378, DMS-0405066]CNPq [303.301/2007-4, 302.214/2004-6]FAPESP [2007/51490-7

A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution

In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system, and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-Rott descriptions of vortex sheet dynamics.35994125414

AN ESTIMATE ON THE HAUSDORFF DIMENSION OF A CONCENTRATION SET FOR THE INCOMPRESSIBLE 2-D EULER EQUATIONS

A concentration set is the locus of loss of kinetic energy in a weakly convergent sequence of L(2) functions. In this paper we construct a concentration set for an approximate solution sequence of the incompressible 2-D Euler equations. S;Ve estimate its classical Hausdorff dimension and show that it is less than or equal to D, for some D < 3. This is related to the problem of existence of a solution to these equations with very singular initial data through the work of R. DiPerna and A. Majda [4], [5], [6].43252153

A refined estimate of the size of concentration sets for 2D incompressible inviscid flow

In this paper we consider approximate solution sequences of the incompressible 2D Euler equations with vortex sheet initial data, that is, initial Rows with locally bounded kinetic energy such that the initial vorticity has bounded total mass. We show that, for every gamma > 0, there exists a concentration set (in the sense of DiPerna and Majda) for these approximate solution sequences with Hausdorff dimension at most 2+gamma, in the following fluid dynamics situations: Row in a bounded domain, periodic flow, and flow in the full plane but with non-negative vorticity.46116518

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 x 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 nonexistence 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 nonexistence 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. Consider a quasilinear, hyperbolic system of differential equations of the form: [GRAPHICS] where U(x, y, t) = (u(x, y, t), v(x, y, t)) is the state vector and A(.), B(.) are smooth, 2 x 2 matrix-valued functions of the state vector. Hyperbolicity means that the matrix C(xi, U) = xi(1)A(U) + xi(2)B(U) has real eigenvalues for any U in state space and for any xi = (xi(1), xi(2)) is an element of S-1. A partially aligned state U-0 for the system above is a state such that A(U-0) and B(U-0) have a common eigenvector. A partially aligned system is one for which all states are partially aligned. This class of systems was introduced by the authors in [8], along with some basic properties and a classification of different kinds of partial alignment. In this article we will use the results and terminology from [8]. An earlier version of the results contained here was announced in [10].57222924

Pointwise blow-up of sequences bounded in L-1

Given a sequence of functions bounded in L-1([0, 1]), is it possible to extract a subsequence that is pointwise bounded almost everywhere? The main objective of this note is to present an example showing that this is not possible in general. We will also prove a pair of positive results. We show that if the sequence of functions consists of multiples of characteristic functions of measurable sets, the answer is yes. We also show that it is always possible to extract a subsequence that is pointwise bounded on a countable, dense set of points. (C) 2001 Academic Press.263244745

An extension of Marchioro's bound on the growth of a vortex patch to flows with L-p vorticity

We observe that C. Marchioro's cubic-root bound in time on the growth of the diameter of a patch of vorticity [Comm. Math. Phys, 164 (1994), pp. 507-524] can be extended to incompressible two-dimensional Euler flows with compactly supported initial vorticity in L-p, p > 2, and with a distinguished sign.29359659

Existence of a weak solution for the semigeostrophic equation with integrable initial data

In this article we present a proof of existence of a weak solution for the semigeostrophic system of equations, formulated as an active scalar transport equation with Monge-Ampere coupling, with initial data in L-c(1) (R-3). This is an extension of a 1998 result due to Benamou and Brenier, who proved existence with initial data in L-c(p)(R-3), p > 3.o TEXTO COMPLETO DESTE ARTIGO, ESTARÁ DISPONÍVEL À PARTIR DE AGOSTO DE 2015.132232933