arXiv subjects

D. Iftimie

Publications and source records attributed to D. Iftimie.

7 recordsLinked to original sources

arXiv2020

The limit $α\to 0$ of the $α$-Euler equations in the half plane with no-slip boundary conditions and vortex sheet initial data

In this article we study the limit when $α\to 0$ of solutions to the $α$-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:1611.05300 and arXiv:1403.5682. It requires a substantially distinct approach, analogous to that used for Delort's Theorem, and a new detailed investigation of the relation between (no-slip) filtered velocity and potential vorticity in the half-plane.

math.AP↗

arXiv2015

Uniform time of existence for the alpha Euler equations

We consider the $α$-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 $α$, for $α$ sufficiently small. Combined with the convergence result in a previous article by the same authors, this implies convergence of solutions of the $α$-Euler equations to solutions of the incompressible Euler equations when $α\to 0$. In addition, we obtain a new result on local existence of strong solutions for the incompressible Euler equations on bounded three-dimensional domains. The proofs are based on new {\it a priori} estimates in conormal spaces.

math.AP↗

arXiv2008

Incompressible flow around a small obstacle and the vanishing viscosity limit

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 obstacle vanish. We prove that this convergence is true assuming two hypothesis: first, that the initial exterior domain velocity converges strongly in $L^2$ to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This work complements the study of incompressible flow around small obstacles, which has been carried out in [1,2,3] [1] D. Iftimie and J. Kelliher, {\it Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid.} Preprint available at http://math.univ-lyon1.fr/~iftimie/ARTICLES/viscoushrink3d.pdf . [2] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. {\it Two dimensional incompressible ideal flow around a small obstacle.} Comm. Partial Differential Equations {\bf 28} (2003), no. 1-2, 349--379. [3] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. {\it Two dimensional incompressible viscous flow around a small obstacle.} Math. Ann. {\bf 336} (2006), no. 2, 449--489.

math.AP↗

arXiv2006

Confinement of vorticity in two dimensional ideal incompressible exterior flow

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 vorticity support with diameter growing at most like $\mathcal{O}(t^{(1/2)+\vare})$, for any $\vare>0$. In addition, if the domain is the exterior of a disk, then the vorticity support is contained in a disk of radius $\mathcal{O}(t^{1/3})$. The purpose of the present article is to refine Marchioro's results. We will prove that, if the initial vorticity is even with respect to the origin, then the exponent for the exterior of the disk may be improved to 1/4. For flows in the exterior of a smooth, connected, bounded domain we prove a confinement estimate with exponent 1/2 (i.e. we remove the $\vare$) and in certain cases, depending on the harmonic part of the flow, we establish a logarithmic improvement over the exponent 1/2. The main new ingredients in our approach are: (1) a detailed asymptotic description of solutions to the exterior Poisson problem near infinity, obtained by the use of Riemann mappings; (2) renormalized energy estimates and bounds on logarithmic moments of vorticity and (3) a new {\it a priori} estimate on time derivatives of logarithmic perturbations of the moment of inertia.

math.AP↗

arXiv2005

Two-dimensional incompressible viscous flow around a small obstacle

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 $ω_0$ 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 $ω_0 + γδ$, where $δ$ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in [Comm P.D.E. 28 (2003) 349-379], where the effect of the small obstacle appears in the coefficients of the PDE and not only on the initial data. The main ingredients of the proof are $L^p-L^q$ estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato's fixed point method, energy estimates, renormalization and interpolation.

math.AP↗

arXiv2002

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.

physics.flu-dyn↗

arXiv2002

Two dimensional incompressible ideal flow around a small obstacle

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 $γ$, the circulation around the obstacle. For smooth flow around a single obstacle, $γ$ is a conserved quantity which is determined by the initial data. We will show that if $γ= 0$, the limit flow satisfies the standard incompressible Euler equations in the full plane but, if $γ\neq 0$, the limit equation acquires an additional forcing term. We treat this problem by first constructing a sequence of approximate solutions to the incompressible 2D Euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation. We use an explicit treatment of the Green's function of the exterior domain based on conformal maps, {\it a priori} estimates obtained by carefully examining the limiting process and the Div-Curl Lemma, together with a standard weak convergence treatment of the nonlinearity for the passage to the limit.

physics.flu-dyn↗