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Smooth solutions of the forced incompressible Euler equations satisfy an energy balance, where the rate-of-change in time of the kinetic energy equals the work done by the force per unit time. Interesting phenomena such as turbulence are closely linked with rough solutions which may exhibit {\it inviscid dissipation}, or, in other words, for which energy balance does not hold. This article provides a characterization of energy balance for physically realizable weak solutions of the forced incompressible Euler equations, i.e. solutions which are obtained in the limit of vanishing viscosity. More precisely, we show that, in the two-dimensional periodic setting, strong convergence of the zero-viscosity limit is both necessary and sufficient for energy balance of the limiting solution, under suitable conditions on the external force. As a consequence, we prove energy balance for a general class of solutions with initial vorticity belonging to rearrangement-invariant spaces, and going beyond Onsager's critical regularity.

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 demonstrate an improved gain of regularity at positive times as compared to that demonstrated by Cannone and Karch. We further demonstrate that the solutions are analytic at all positive times, with lower bounds given for the radius of analyticity., Comment: 14 pages

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 negative order. In two spatial dimensions, we also allow data which is in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence., Comment: 26 pages

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 $[(2Su)n + \alpha u]_{\rm tang} = 0$, where $n$ is the unit exterior normal to $\partial \Omega$, $Su = (Du + (Du)^t)/2$, $\alpha \in C^0(\partial\Omega)$ is the boundary friction coefficient and $[\cdot]_{\rm tang}$ is the projection of its argument onto the tangent space of $\partial \Omega$. We prove global existence of a weak Leray-type solution to the resulting initial-boundary value problem and exponential decay in energy norm of these solutions when friction is positive. We also prove exponential decay if friction is non-negative and the domain is not a solid of revolution. In addition, in the frictionless case $\alpha = 0$, we prove convergence of the solution to a steady rigid rotation, if the domain is a solid of revolution. We use the Galerkin method for existence, Poincar\'{e}-type inequalities, with suitable adaptations to account for the differential geometry of the boundary, and a global-in-time, integral Gronwall-type inequality.

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 solution of this problem converges, as $\epsilon \to 0$, to the solution of a modified $\alpha$--Euler equation in the full plane where an additional Dirac located at the center of the disk is imposed in the potential vorticity., Comment: 23 pages

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 $L^p$, $p>1$. In this work we extend their result, by adding forcing to the flow.

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., Comment: The statement of the main theorem has been improved

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. These assumptions are consistent with vortex sheet solutions of the Euler equations., Comment: 14 pages

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 the unfiltered vorticity is contained in a disk whose radius grows no faster than $\mathcal{O}((t\log t)^{1/4})$. This result is an adaptation of the corresponding result for the incompressible 2D Euler equations with initial vorticity compactly supported, nonnegative, and $p$-th power integrable, $p>2$, due to D. Iftimie, T. Sideris and P. Gamblin and, independently, to Ph. Serfati.

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 $\bu_0^\nu \to \bu_0$ in $H^1$. The new ingredient in our analysis is a decomposition of helical vector fields, through which we obtain the required estimates., Comment: 22pages

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 properties of these correctors, we establish convergence of the Navier-Stokes solution to the Euler solution as viscosity vanishes with optimal rates of convergence. In addition, we investigate vorticity production on the boundary in the limit of vanishing viscosity. Our work significantly extends prior work in the literature., Comment: 52 pages

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 $\alpha, \nu \to 0$ 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-$\alpha$ model ($\nu = 0$), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case ($\alpha = 0$), 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 $\nu = \mathcal{O}(\alpha^2)$, as $\alpha \to 0$, 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 $\nu = \mathcal{O}(\alpha^{6/5})$, $\nu/\alpha^2 \to \infty$ as $\alpha \to 0$. 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 $\alpha = \mathcal{O}(\nu^{3/2})$, as $\nu \to 0$. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato., Comment: 20pages,1figure

Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present a continuous data assimilation algorithm for the three-dimensional Navier-Stokes-$\alpha$ model. This algorithm consists of introducing a nudging process through general type of approximation interpolation operator (that is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of the unknown solutions the Navier-Stokes-$\alpha$ equations that corresponds to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, that is obtained from these collected data, converges to the unkown reference solution (physical reality) over time. These conditions are given in terms of some physical parameters, such as kinematic viscosity, the size of the domain and the forcing term.

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 that the initial velocities for the Euler-$\alpha$ system approximate, in a suitable sense, as the regularization parameter $\alpha \to 0$, the initial velocity for the limiting Euler system. For small values of $\alpha$, this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler-$\alpha$ system converge, as $\alpha \to 0$, to the corresponding solution of the Euler equations, in $L^2$ in space, uniformly in time. We also present an example involving parallel flows, in order to illustrate the indifference to the boundary layer of the $\alpha \to 0$ limit, which underlies our work., Comment: 22pages

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 non-decaying vorticity, the Biot-Savart integral does not converge, and thus velocity cannot be reconstructed from vorticity in a straightforward way. The key to circumventing this difficulty is the use of the Serfati identity, which is based on the Biot-Savart integral, but holds in more general settings., Comment: 50 pages

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 passive scalar to the two-dimensional incompressible Euler equations., Comment: 10 pages