50 research outputs found
Weak vorticity formulation for the incompressible Euler equations in domains with boundary
In this article we examine the interaction of incompressible 2D flows with
compact material boundaries. Our focus is the dynamic behavior of the
circulation of velocity around boundary components and the possible exchange
between flow vorticity and boundary circulation in flows with vortex sheet
initial data We begin by showing that the velocity can be uniquely
reconstructed from the vorticity and boundary component circulations, which
allows to recast 2D Euler evolution using vorticity and the circulations as
dynamic variables. The weak form of this vortex dynamics formulation of the
equations is called the weak vorticity formulation. The main result in this
article is the equivalence between the weak velocity and weak vorticity
formulations, without sign assumptions. Next, we focus on weak solutions
obtained by mollifying initial data and passing to the limit, with the portion
of vorticity singular with respect to the Lebesgue measure assumed to be
nonnegative. For these solutions we prove that the circulations around each
boundary component cannot be smaller than the initial data circulation, so that
nonnegative vorticity may be absorbed by the boundary, but not produced by the
boundary. In addition, we prove that if the weak solution conserves circulation
at the boundary components it is a boundary coupled weak solution, a stronger
version of the weak vorticity formulation. We prove existence of a weak
solution which conserves circulation at the boundary components if the initial
vorticity is integrable. In addition, we discuss the definition of the
mechanical force which the flow exerts on material boundary components and its
relation with conservation of circulation. Finally, we describe the
corresponding results for a bounded domain with holes, and the adaptations
required in the proofs.Comment: 37 page
Pointwise Blow-up of Sequences Bounded in L1
AbstractGiven a sequence of functions bounded in L1([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
Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions
The second-grade fluid equations are a model for viscoelastic fluids, with
two parameters: , corresponding to the elastic response, and , corresponding to viscosity. Formally setting these parameters to
reduces the equations to the incompressible Euler equations of ideal fluid
flow. In this article we study the limits of solutions of
the second-grade fluid system, in a smooth, bounded, two-dimensional domain
with no-slip boundary conditions. This class of problems interpolates between
the Euler- model (), for which the authors recently proved
convergence to the solution of the incompressible Euler equations, and the
Navier-Stokes case (), for which the vanishing viscosity limit is
an important open problem. We prove three results. First, we establish
convergence of the solutions of the second-grade model to those of the Euler
equations provided , as , extending
the main result in [19]. Second, we prove equivalence between convergence (of
the second-grade fluid equations to the Euler equations) and vanishing of the
energy dissipation in a suitably thin region near the boundary, in the
asymptotic regime ,
as . This amounts to a convergence criterion similar to the
well-known Kato criterion for the vanishing viscosity limit of the
Navier-Stokes equations to the Euler equations. Finally, we obtain an extension
of Kato's classical criterion to the second-grade fluid model, valid if , as . The proof of all these results
relies on energy estimates and boundary correctors, following the original idea
by Kato.Comment: 20pages,1figur