Vanishing Viscosity Limit Research Articles

The vanishing viscosity limit for a 2D Cahn–Hilliard–Navier–Stokes system with a slip boundary condition

This paper studies the vanishing viscosity limit for the 2D Cahn–Hilliard–Navier–Stokes system in a bounded domain with a slip boundary condition. The result is proved globally in time.

The vanishing viscosity limit for a 3D model of electro-kinetic fluid in a bounded domain

This paper studies the vanishing viscosity limit for a 3D model of electrohydrodynamics in a bounded domain. The result has been proved locally in time.

The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow

We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray–Hopf weak solutions for the Navier–Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely determined and equals the shear flow solution of the Euler equations corresponding to this initial data. This simple example suggests that, also in more general situations, the vanishing viscosity limit of the Navier–Stokes equations could serve as a uniqueness criterion for weak solutions of the Euler equations.

A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation

In this Note, we explain how results on the cost of the null-controllability of the one-dimensional heat equation in small time can be used to bound from above the cost of the null-controllability of a one-dimensional transport-diffusion equation in the vanishing viscosity limit. We improve previous results about the minimal time needed to obtain the exponential decrease of the cost of the control and explain what would provide the usual conjecture concerning the cost of fast controls for the heat equation.

Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition

In this paper, we investigate the vanishing viscosity limit for solutions to the Navier-Stokes equations with a Navier slip boundary condition on general compact and smooth domains in $\mathbf{R}^3$. We first obtain the higher order regularity estimates for the solutions to Prandtl's equation boundary layers. Furthermore, we prove that the strong solution to Navier-Stokes equations converges to the Eulerian one in $C([0,T];H^1(\Omega))$ and $L^\infty((0,T)\times\o)$, where $T$ is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.

Degenerate regularization of forward-backward parabolic equations: the vanishing viscosity limit

We address the vanishing viscosity limit of the regularized problem studied in Smarrazzo and Tesei [Arch Rat Mech Anal 2012 (in press)]. We show that the limiting points in a suitable topology of the family of solutions of the regularized problem can be regarded as suitably defined weak measure-valued solutions of the original problem. In general, these solutions are the sum of a regular term, which is absolutely continuous with respect to the Lebesgue measure, and a singular term, which is a Radon measure singular with respect to the other. By using a family of entropy inequalities, we prove that the singular term is nondecreasing in time. We also characterize the disintegration of the Young measure associated with the regular term, proving that it is a superposition of two Dirac masses with support on the branches of the graph of the nonlinearity $${\varphi}$$ .

Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition

Abstract This article studies the partial vanishing viscosity limit of the 2D Boussinesq system in a bounded domain with a slip boundary condition. The result is proved globally in time by a logarithmic Sobolev inequality. 2010 MSC: 35Q30; 76D03; 76D05; 76D07.

Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions

This paper concerns the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space Lq with 2 < q ⩽ ∞, we show that the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In a special case when the vorticity is in L∞, an explicit convergence rate is obtained.

Uniform Controllability of Scalar Conservation Laws in the Vanishing Viscosity Limit

We deal with viscous perturbations of scalar conservation laws on a bounded interval with a general flux function $f$ and a small dissipation coefficient $\varepsilon$. Acting on this system on both endpoints of the interval, we prove global exact controllability to constant states with nonzero speed. More precisely, we construct boundary controls so that the solution is driven to the targeted constant state, and we moreover require these controls to be uniformly bounded as $\varepsilon \rightarrow 0^+$ in an appropriate space. For general (nonconvex) flux functions this can be done for sufficiently large time, and for convex fluxes $f$, we have a precise estimate on the minimal time needed to control.

Stability and interaction of vortices in two-dimensional viscous flows

The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the whole plane $\mathbb{R}$2, and assume that the initial vorticity is a finite measure. This general setting includes vortex patches, vortex sheets, and point vortices. We first prove the existence of a unique global solution, for any value of the viscosity parameter, and we investigate its long-time behavior. We next consider the particular case where the initial flow is a finite collection of point vortices. In that situation, we show that the solution behaves, in the vanishing viscosity limit, as a superposition of Oseen vortices whose centers evolve according to the Helmholtz-Kirchhoff point vortex system. The proof requires a careful stability analysis of the Oseen vortices in the large Reynolds number regime, as well as a precise computation of the deformations of the vortex cores due to mutual interactions.

On the Convergence Rate of Vanishing Viscosity Approximations for Nonlinear Hyperbolic Systems

In this paper we study the vanishing viscosity limit of strictly hyperbolic systems, extending the earlier result in [A. Bressan and T. Yang, Comm. Pure Appl. Math., 57 (2004), pp. 1075--1109] to systems where each characteristic field can be either genuinely nonlinear or linearly degenerate. For a given initial data with small total variation, our main estimate shows that the ${\bf L}^1$ distance between the exact solution $u$ and a viscous approximation $u^\varepsilon $ is bounded by $\|u(\tau,\cdot)-u^\varepsilon (\tau,\cdot)\|_{{\bf L}^1} = O(1)\cdot(1+\tau)\varepsilon^{1/4}.$ Under the additional assumptions that the integral curves of all linearly degenerate fields are straight lines, we obtain the sharper estimate $\|u(\tau,\cdot)-u^\varepsilon (\tau,\cdot)\|_{{\bf L}^1} = O(1)\cdot(1+\tau)\sqrt\varepsilon |\ln\varepsilon |.$

From Discrete Visco-Elasticity to Continuum Rate-Independent Plasticity: Rigorous Results

We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete-to-continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic lattice transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shape-memory alloys. The approach, however, is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models.

Uniform Regularity for the Navier–Stokes Equation with Navier Boundary Condition

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier-Stokes equation with Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in $L^\infty$. This allows to get the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.

Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions to a Riemann Problem

We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.

Some recent results concerning a class of forward-backward parabolic equations

We study a quasilinear parabolic equation of forward-backward type in one space dimension, under assumptions on the nonlinearity satisfied by important mathematical models (e.g., the one-dimensional Perona–Malik equation). We first address a degenerate pseudoparabolic regularization of the equation, which takes time delay e¤ects into account, proving existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures, as well as qualitative properties of such solutions. Then we address the vanishing viscosity limit of the regularized problem, proving that the limiting points (in a suitable topology) of the family of solutions of the regularized problem can be regarded as solutions of the original problem. In particular, we characterize the disintegration of the narrow limit of the family of Young measures associated with the absolutely continuous part of the solutions of the regularized problem, proving that it is a superposition of two Dirac masses with support on the branches of the graph of the nonlinearity φ . In the above study, the existence of a family of infinitely many entropy inequalities plays an important role.

Scale-wise coherent vorticity extraction for conditional statistical modeling of homogeneous isotropic two-dimensional turbulence

Classical statistical theories of turbulence have shown their limitations, in that they cannot predict much more than the energy spectrum in an idealized setting of statistical homogeneity and stationarity. We explore the applicability of a conditional statistical modeling approach: can we sort out what part of the information should be kept, and what part should be modeled statistically, or, in other words, “dissipated”? Our mathematical framework is the initial value problem for the two-dimensional (2D) Euler equations, which we approximate numerically by solving the 2D Navier–Stokes equations in the vanishing viscosity limit. In order to obtain a good approximation of the inviscid dynamics, we use a spectral method and a resolution going up to 8192 2. We introduce a macroscopic concept of dissipation, relying on a split of the flow between coherent and incoherent contributions: the coherent flow is constructed from the large wavelet coefficients of the vorticity field, and the incoherent flow from the small ones. In previous work, a unique threshold was applied to all wavelet coefficients, while here we also consider the effect of a scale by scale thresholding algorithm, called scale-wise coherent vorticity extraction. We study the statistical properties of the coherent and incoherent vorticity fields, and the transfers of enstrophy between them, and then use these results to propose, within a maximum entropy framework, a simple model for the incoherent vorticity. In the framework of this model, we show that the flow velocity can be predicted accurately in the L 2 norm for about 10 eddy turnover times.

Vanishing viscosity limit for a coupled Navier–Stokes/Allen–Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier–Stokes/Allen–Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen–Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier–Stokes and Euler equations, and assuming that the energy dissipation for Navier–Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier–Stokes/Allen–Cahn system converge to that of the Euler/Allen–Cahn system in a proper small time interval. In addition, for strong solutions of the Navier–Stokes/Allen–Cahn system in 2D, the convergence rate is c ν 1 / 2 .

Energy Dissipating Structures Produced by Walls in Two-Dimensional Flows at Vanishing Viscosity

We perform numerical experiments of a dipole crashing into a wall, a generic event in two-dimensional incompressible flows with solid boundaries. The Reynolds number (Re) is varied from 985 to 7880, and no-slip boundary conditions are approximated by Navier boundary conditions with a slip length proportional to Re(-1). Energy dissipation is shown to first set up within a vorticity sheet of thickness proportional to Re(-1) in the neighborhood of the wall, and to continue as this sheet rolls up into a spiral and detaches from the wall. The energy dissipation rate integrated over these regions appears to converge towards Re-independent values, indicating the existence of energy dissipating structures that persist in the vanishing viscosity limit.

Remarks on vanishing viscosity limits for the 3D Navier-Stokes equations with a slip boundary condition

The authors study vanishing viscosity limits of solutions to the 3-dimensional incompressible Navier-Stokes system in general smooth domains with curved boundaries for a class of slip boundary conditions. In contrast to the case of flat boundaries, where the uniform convergence in super-norm can be obtained, the asymptotic behavior of viscous solutions for small viscosity depends on the curvature of the boundary in general. It is shown, in particular, that the viscous solution converges to that of the ideal Euler equations in C([0, T];H1(Ω)) provided that the initial vorticity vanishes on the boundary of the domain.

BV solutions and viscosity approximations of rate-independent systems

In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of 'BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.