
Milton Lopes FilhoFederal University of Rio de Janeiro | UFRJ · Instituto de Matemática (IM)
Milton Lopes Filho
Berkeley, 1990
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88
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April 2012 - present
May 1992 - April 2012
Publications
Publications (88)
We prove the existence of solutions to the Kuramoto–Sivashinsky equation with low regularity data in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data that are in a pseudomeasure space of...
We consider the three-dimensional Navier–Stokes equations, with initial data having second derivatives in the space of pseudomeasures. Solutions of this system with such data have been shown to exist previously by Cannone and Karch. As the Navier–Stokes equations are a parabolic system, the solutions gain regularity at positive times. We demonstrat...
We prove existence of solutions to the Kuramoto-Sivashinsky equation with low-regularity data, in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data which is in a pseudomeasure space of neg...
Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae [Proc. Amer. Math. Soc. 143 (2015), pp. 2887–2892], and th...
We study the three-dimensional incompressible Navier-Stokes equations in a smooth bounded domain $\Omega$ with initial velocity $u_0$ square-integrable, divergence-free and tangent to $\partial \Omega$. We supplement the equations with the Navier friction boundary conditions $u \cdot n = 0$ and $[(2Su)n + \alpha u]_{tang} = 0$, where $n$ is the uni...
Lei and Lin have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae, and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times....
In this article we consider the $\alpha$--Euler equations in the exterior of a small fixed disk of radius $\epsilon$. We assume that the initial potential vorticity is compactly supported and independent of $\epsilon$, and that the circulation of the unfiltered velocity on the boundary of the disk does not depend on $\epsilon$. We prove that the so...
Cheskidov et al. (2016 Commun. Math. Phys. 348 , 129–143. ( doi:10.1007/s00220-016-2730-8 )) proved that physically realizable weak solutions of the incompressible two-dimensional Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that can be obtained as limits of vanishing viscosity. The key hypothes...
In [Commun Math Phys 348(1), 129-143, 2016], Cheskidov et al. proved that physically realizable weak solutions of the incompressible 2D Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that can be obtained as limits of vanishing viscosity. The key hypothesis was boundedness of the initial vorticity...
In this article we study the limit when $\alpha \to 0$ of solutions to the $\alpha$-Euler system in the half-plane, with no-slip boundary conditions, to weak solutions of the 2D incompressible Euler equations with non-negative initial vorticity in the space of bounded Radon measures in $H^{-1}$. This result extends the analysis done in arXiv:1611.0...
We consider a sequence of Leray-Hopf weak solutions of the 2D Navier-Stokes equations on a bounded domain, in the vanishing viscosity limit. We provide sufficient conditions on the associated vorticity measures, away from the boundary, which ensure that as the viscosity vanishes the sequence converges to a weak solution of the Euler equations. The...
We consider a sequence of Leray-Hopf weak solutions of the 2D Navier-Stokes equations on a bounded domain, in the vanishing viscosity limit. We provide sufficient conditions on the associated vorticity measures, away from the boundary, which ensure that as the viscosity vanishes the sequence converges to a weak solution of the Euler equations. Thes...
In this article we consider weak solutions of the Euler-α equations in the full plane. We take, as initial unfiltered vorticity, an arbitrary nonnegative, compactly supported, bounded Radon measure. Global well-posedness for the corresponding initial value problem is due M. Oliver and S. Shkoller. We show that, for all time, the support of the unfi...
In this article we consider weak solutions of the Euler-$\alpha$ equations in the full plane. We take, as initial unfiltered vorticity, an arbitrary nonnegative, compactly supported, bounded Radon measure. Global well-posedness for the corresponding initial value problem is due M. Oliver and S. Shkoller. We show that, for all time, the support of t...
In this paper, we are concerned with the vanishing viscosity problem for the three-dimensional Navier–Stokes equations with helical symmetry, in the whole space. We choose viscosity-dependent initial u0ν with helical swirl, an analogue of the swirl component of axisymmetric flow, of magnitude O(ν) in the L2 norm; we assume u0ν→u0 in H1. The new ing...
In this paper, we are concerned with the vanishing viscosity problem for the three-dimensional Navier-Stokes equations with helical symmetry, in the whole space. We choose viscosity-dependent initial $\bu_0^\nu$ with helical swirl, an analogue of the swirl component of axisymmetric flow, of magnitude $\mathcal{O}(\nu)$ in the $L^2$ norm; we assume...
The focus of this paper is on the analysis of the boundary layer and the associated vanishing viscosity limit for two classes of flows with symmetry, namely, Plane-Parallel Channel Flows and Parallel Pipe Flows. We construct explicit boundary layer correctors, which approximate the difference between the Navier-Stokes and the Euler solutions. Using...
The focus of this paper is on the analysis of the boundary layer and the associated vanishing viscosity limit for two classes of flows with symmetry, namely, Plane-Parallel Channel Flows and Parallel Pipe Flows. We construct explicit boundary layer correctors, which approximate the difference between the Navier-Stokes and the Euler solutions. Using...
This note addresses the issue of energy conservation for the 2D Euler system with an Lp-control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if \({\omega = \nabla \times u \in L^{\frac{3}{2}}}\). An example of a 2D field in the class \({\omega \in L^{...
In this article, we study the homogenization limit of a family of solutions
to the incompressible 2D Euler equations in the exterior of a family of $n_k$
disjoint disks with centers $\{z^k_i\}$ and radii $\varepsilon_k$. We assume
that the initial velocities $u_0^k$ are smooth, divergence-free, tangent to the
boundary and that they vanish at infini...
In this article, we study the homogenization limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of $n_k$ disjoint disks with centers $\{z^k_i\}$ and radii $\varepsilon_k$. We assume that the initial velocities $u_0^k$ are smooth, divergence-free, tangent to the boundary and that they vanish at infini...
This note addresses the question of energy conservation for the 2D Euler
system with an $L^p$-control on vorticity. We provide a direct argument, based
on a mollification in physical space, to show that the energy of a weak
solution is conserved if $\omega = \nabla \times u \in L^{\frac32}$. An example
of a 2D field in the class $\omega \in L^{\fra...
We consider the $\alpha$-Euler equations on a bounded three-dimensional
domain with frictionless Navier boundary conditions. Our main result is the
existence of a strong solution on a positive time interval, uniform in
$\alpha$, for $\alpha$ sufficiently small. Combined with the convergence result
in a previous article by the same authors, this imp...
We consider the $\alpha$-Euler equations on a bounded three-dimensional domain with frictionless Navier boundary conditions. Our main result is the existence of a strong solution on a positive time interval, uniform in $\alpha$, for $\alpha$ sufficiently small. Combined with the convergence result in a previous article by the same authors, this imp...
In Ann. Math., 170 (2009), 1417-1436, C. De Lellis and L. Sz\'ekelyhidi Jr.
constructed wild solutions of the incompressible Euler equations using a
reformulation of the Euler equations as a differential inclusion together with
convex integration. In this article we adapt their construction to the system
consisting of adding the transport of a pass...
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 $p>\frac{4}{3}$. This result is an
extension of the existence part of the work of B. Etti...
The second-grade fluid equations are a model for viscoelastic fluids, with
two parameters: $\alpha > 0$, corresponding to the elastic response, and $\nu >
0$, corresponding to viscosity. Formally setting these parameters to $0$
reduces the equations to the incompressible Euler equations of ideal fluid
flow. In this article we study the limits $\alp...
In this article we consider the Euler-$\alpha$ system as a regularization of
the incompressible Euler equations in a smooth, two-dimensional, bounded
domain. For the limiting Euler system we consider the usual non-penetration
boundary condition, while, for the Euler-$\alpha$ regularization, we use
velocity vanishing at the boundary. We also assume...
We prove existence and uniqueness of a weak solution to the incompressible 2D
Euler equations in the exterior of a bounded smooth obstacle when the initial
data is a bounded divergence-free velocity field having bounded scalar curl.
This work completes and extends the ideas outlined by P. Serfati for the same
problem in the whole-plane case. With n...
This article is concerned with the limiting behavior of incompressible flow past a small obstacle. Previous work on this problem has dealt with flows with vanishing velocity at infinity. We examine this limit for flows that are constant at infinity in the simplest case, that of two-dimensional, ideal flow past an obstacle. This extends the work in...
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 ca...
Helical symmetry is invariance under a one-dimensional group of rigid motions
generated by a simultaneous rotation around a fixed axis and translation along
the same axis. The key parameter in helical symmetry is the step or pitch, the
magnitude of the translation after rotating one full turn around the symmetry
axis. In this article we study the l...
The vortex-wave system is a model for the evolution of 2D incompressible
fluids in which the vorticity is split into a finite sum of Dirac masses plus
an Lp part. Existence of a weak solution for this system was recently proved by
Lopes Filho, Miot and Nussenzveig Lopes, for p > 2, but their result left open
the existence and basic properties of th...
In this article, we prove nonlinear orbital stability for steadily
translating vortex pairs, a family of nonlinear waves that are exact solutions
of the incompressible, two-dimensional Euler equations. We use an adaptation of
Kelvin's variational principle, maximizing kinetic energy penalised by a
multiple of momentum among mirror-symmetric isovort...
In this article we consider weak solutions of the three-dimensional
incompressible fluid flow equations with initial data admitting a
one-dimensional symmetry group. We examine both the viscous and inviscid cases.
For the case of viscous flows, we prove that Leray-Hopf weak solutions of the
three-dimensional Navier-Stokes equations preserve initial...
In this article we study the limit α→0α→0 of solutions of the α-Euler equations and the limit α,ν→0α,ν→0 of solutions of the second grade fluid equations in a bounded domain, both in two and in three space dimensions. We prove that solutions of the complex fluid models converge to solutions of the incompressible Euler equations in a bounded domain...
The vortex-wave system is a coupling of the two-dimensional vorticity equation with the point-vortex system. It is a model
for the motion of a finite number of concentrated vortices moving in a distributed vorticity background. In this article,
we prove existence of a weak solution to this system with an initial background vorticity in L
p
, p>2,...
In this article we study a variational formulation, proposed by V. I. Arnold and by Y. Brenier, for the incompressible Euler equations with variable density. We consider the problem of minimizing an action functional, the integral in time of the kinetic energy, among fluid motions considered as trajectories in the group of volume-preserving diffeom...
In recent years, the author and his research team have obtained several results concerning the limiting behavior of incompressible flow around small obstacles, both in the inviscid and viscous cases. These results showcase the difficulties and mathematical issues surrounding the description of fluid-solid interaction at large Reynolds number. In th...
We explore the relationship between the hydrodynamic Green’s function and the Dirichlet Green’s function in a bounded domain
with holes. This relationship is expressed using the harmonic measures on the domain, following the work of Flucher and Gustafsson
(Vortex motion in two-dimensional hydrodynamics, energy renormalization and stability of vorte...
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 P. I. Plotnikov [Sib. Math. J. 34, No. 4, 704–716 (1993); translation from Sib. Mat. Zh. 34,...
We study the limiting behavior of viscous incompressible flows when the fluid domain is allowed to expand as the viscosity vanishes. We describe precise conditions under which the limiting flow satisfies the full space Euler equations. The argument is based on truncation and on energy estimates, following the structure of the proof of Kato's criter...
In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition
at the surface of the obstacle. We seek conditions under which solutions of the Navier-Stokes system in the exterior domain
converge to solutions of the Euler system in the full space when both viscosity and the size of the obs...
In [1], Cullen and Feldman proved existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in $L^p$, $p>1$. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial potential vorticities in $L^1$ converges strongly in $L^q$,...
In [A. Shnirelman, On the non-uniqueness of weak solutions of Euler equations, Comm. Pure Appl. Math. L (1997) 1261–1286], Shnirelman described the construction of a weak solution of the 2D incompressible Euler equations on a torus, with compact support in time. In this article, we use computational tools to obtain an explicit approximation of Shni...
In this article we consider circularly symmetric incompressible viscous flow in a disk. The boundary condition is no-slip with respect to a prescribed time-dependent rotation of the boundary about the center of the disk. We prove that, if the prescribed angular velocity of the boundary has finite total variation, then the Navier–Stokes solutions co...
In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the Navier-Stokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obs...
This manuscript is a survey on results related to boundary layers and the vanishing viscosity limit for incompressible flow. It is the lecture notes for a 10 hour minicourse given at the Morningside Center, Academia Sinica, Beijing, PRC from 11/28 to 12/07, 2007. The main topics covered are: a derivation of Prandtl's boundary layer equation; an out...
We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [10] 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 α(t) of the boundary has bounded total variation. Here we establish con...
The purpose of this work is to prove existence of a weak solution of the two dimensional incompressible Euler equations on a noncylindrical domain consisting of a smooth, bounded, connected and simply connected domain undergoing a prescribed motion. We prove existence of a weak solution for initial vorticity in $L^p$, for $p>1$. This work complemen...
The purpose of this work is to prove existence of a weak solution of the two dimensional incompressible Euler equations on a noncylindrical domain consisting of a smooth, bounded, connected and simply connected domain undergoing a prescribed motion. We prove existence of a weak solution for initial vorticity in $L^p$, for $p>1$. This work complemen...
We examine the asymptotic behavior of solutions of the incompressible two-dimensional Euler equations on a domain with several holes, when one of the holes becomes small. We show that the limit flow satisfies a modified Euler system in the domain with the small hole removed. In vorticity form, the limit system is the usual equation for transport of...
In this paper we prove the existence of a weak solution of the incompressible 2D Euler equations in the exterior of a reflection symmetric smooth bluff body with symmetric initial flow corresponding to vortex sheet type data whose vorticity is of distinguished sign on each side of the symmetry axis. This work extends the results proved for full pla...
In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the
problem of vorticity confinement in the exterior of a smooth bounded domain.
The main result in Marchioro's paper is that solutions of the incompressible 2D
Euler equations with compactly supported nonnegative initial vorticity in the
exterior of a connected bounded region have...
In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro's paper is that solutions of the incompressible 2D Euler equations with compactly supported nonnegative initial vorticity in the exterior of a connected bounded region have...
We consider a special configuration of vorticity that consists of a pair of externally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Eul...
In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω0 and the circulation γ of the initial flow around the obstacle. We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier–Stokes system, with initial vorticity ω...
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...
In [Nonlinearity 11, 1625–1636 (1998; Zbl 0911.76014)], A. Clopeau, T. Mikelić, and R. Robert studied the inviscid limit of the two-dimensional incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations, and their re...
Enstrophy, half the integral of the square of vorticity, plays a role in 2D turbulence theory analogous to that played by kinetic energy in the Kolmogorov theory of 3D turbulence. It is therefore interesting to obtain a description of the way enstrophy is dissipated at high Reynolds number. In this article we explore the notions of viscous and tran...
In [1], T. Clopeau, A. Mikeli\'c, and R. Robert studied the inviscid limit of the 2D incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations and their result ultimately includes flows generated by bounded initial...
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...
In this paper we prove two results regarding the large-time behavior of vortex dynamics in the full plane. In the first result we show that the total integral of vorticity is confined in a region of diameter growing at most like the square root of time. In the second result we show that if a dynamic rescaling of the absolute value of vorticity with...
We show the the voricity distribution obtained by minimizing the induced drag on a wing, the so called Prandtl-Munk vortex sheet, is not a travelling-wave weak solution of the Euler equations, contrary to what has been claimed by a number of authors. Instead, it is a weak solution of a non-homogeneous Euler equation, where the forcing term represen...
In this article we study the asymptotic behavior of incompressible, ideal, time-dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the equation satisfied by the limit flow. We will see that the asymptotic behavior depends on $\gamma$, the circu...
The main purpose of this work is to establish the existence of a weak solution to the incompressible 2D Euler equations with initial vorticity consisting of a Radon measure with distinguished sign in H
− 1, compactly supported in the closed right half-plane, superimposed on its odd reflection in the left half-plane. We make use of a new a priori es...
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 1 c (R 3 ). This is an extension of a 1998 result due to J.-D. Benamou and Y. Brenier, who proved existence with initial data in L p c (R...
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 space...
We consider approximate solution sequences of the 2D incompressible Euler equations obtained by mollifying compactly supported initial vorticities in L p , 1≤p≤2, or bounded measures in H loc -1 and exactly solving the equations. For these solution sequences we obtain uniform estimates on the evolution of the mass of vorticity and on the measure of...
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 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 o...
A confined eddy is a circularly symmetric flow with vorticity of compact support and zero net circulation. Confined eddies with disjoint supports can be superimposed to generate stationary weak solutions of the two-dimensional incompressible inviscid Euler equations. In this work, we consider the unique weak solution of the twodimensional incompres...
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...
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 Lp, p > 2, and with a distinguished sign.
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 Lp, p > 2, and with a distinguished sign.
We study singularity formation for the 2×2 systemut+(u2)x+(uv)y=0 andvt+(uv)x+(v2)y=0. Our analysis is based on the argument, due to J. Keller and L. Ting (1966,Comm. Pure. Appl. Math.19, 371–420), about the evolution along a characteristic of the compression rate of nearby characteristics. This system is one of a class of systems, called partially...
We recall the definition of DiPerna-Majda concentration sets and their role in the study of existence for the incompressible 2D Euler equations with singular initial data. We review examples of concentration-cancellation, focusing on the issue of dynamic kinetic energy defects for the limit flows. We also describe a recent refined estimate of the d...
We discuss smooth solutions for a class of quasilinear non-strictly hyperbolic 2×2 systems in two space dimensions that have a distinct characteristic structure consisting of pairs of curves through each point in physical space. We survey the elementary properties of these systems, present an example of interest for certain applications and describ...
In this paper we validate the generalized geometric entropy criterion for admissibility of shocks in systems which change type. This condition states that a shock between a state in a hyperbolic region and one in a nonhyperbolic region is admissible if the Lax geometric entropy criterion, based on the number of characteristics entering the shock, h...
Thesis (Ph. D. in Mathematics)--University of California, Berkeley, May 1990. Includes bibliographical references (leaf 44).