Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIncompressible flow around a small obstacle and the vanishing viscosity limit
420
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We show strong convergence of the vorticities in the vanishing viscosity limit for the incompressible Navier-Stokes equations on the two-dimensional torus, assuming only that the initial vorticity of the limiting Euler equations is in $L^p$ for some $p>1$. This substantially extends a recent result of Constantin, Drivas and Elgindi, who proved strong convergence in the case $p=infty$. Our proof, which relies on the classical renormalization theory of DiPerna-Lions, is surprisingly simple.
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 infinity. We allow, but we do not require, $n_k to infty$, and we assume $varepsilon_k to 0$ as $kto infty$. Let $gamma^k_i$ be the circulation of $u_0^k$ around the circle ${|x-z^k_i|=varepsilon_k}$. We prove that the homogenization limit retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) $omega_0^k = mbox{ curl }u_0^k$ has a uniform compact support and converges weakly in $L^{p_0}$, for some $p_0>2$, to $omega_0 in L^{p_0}_{c}(mathbb{R}^2)$, (2) $sum_{i=1}^{n_k} gamma^k_i delta_{z^k_i} rightharpoonup mu$ weak-$ast$ in $mathcal{BM}(mathbb{R}^2)$ for some bounded Radon measure $mu$, and (3) the radii $varepsilon_k$ are sufficiently small. Then the corresponding solutions $u^k$ converge strongly to a weak solution $u$ of a modified Euler system in the full plane. This modified Euler system is given, in vorticity formulation, by an active scalar transport equation for the quantity $omega=mbox{ curl } u$, with initial data $omega_0$, where the transporting velocity field is generated from $omega$ so that its curl is $omega + mu$. As a byproduct, we obtain a new existence result for this modified Euler system.
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 Katos criterion for the vanishing viscosity limit. This work complements previous work by the authors, see [Kelliher, Comm. Math. Phys. 278 (2008), 753-773] and [arXiv:0801.4935v1].
Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.
Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $big|u(t,cdot)-u^ve(t,cdot)big|_{L^1}= O(1)(1+t)cdot sqrtve|lnve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^ve$, letting the viscosity coefficient $veto 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^ve$ by taking a mollification $u*phi_{strut sqrtve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
