Suitable weak solutions: from compressible viscous to incompressible inviscid fluid flows

We establish the asymptotic limit of the compressible Navier–Stokes system in the regime of low Mach and high Reynolds number on unbounded spatial domains with slip boundary condition. The result holds in the class of suitable weak solutions satisfying a relative entropy inequality.     

Local times for solutions of the complex Ginzburg-Landau equation and the inviscid limit

We consider the behaviour of the distribution for stationary solutions of the complex Ginzburg-Landau equation perturbed by a random force. It was proved in S. Kuksin and A. Shirikyan (2004) [4] that if the random force is proportional to the square root of the viscosity ν>0, then the family of stationary measures possesses an accumulation point as ν→⁰⁺. We show that if μ is such a point, then the distributions of the L²-norm and of the energy possess a density with respect to the Lebesgue measure. The proofs are based on Itô?s formula and some properties of local time for semimartingales.     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

On the inviscid limit of the three dimensional incompressible chemotaxis-Navier–Stokes equations

In this paper, we study the inviscid limit of the 3D chemotaxis-Navier–Stokes equations and establish the convergence rate of the inviscid limit for vanishing diffusion.     

Viscous and inviscid magneto-hydrodynamics equations

In this paper we study the classical problem in turbulence for the magneto-hydrodynamics (MHD) equations: whether the solutions (u ^(v),B ^(v)) of the viscous MHD equations tend to the solutions (u ⁽⁰⁾,B ^(v)) of the inviscid MHD equations as the Reynolds numbersRe, Rm → ∞. As a preparation we first derive bounds for ||(u ⁽⁰⁾,B ⁽⁰⁾(t)||H ^m) (m ≥3) in terms of deformation tensor related quantities (0.1) {ie251-1} We then show that asRe → ∞ andRm → ∞, the difference {ie-251-2} {ie-251-3} converges to zero uniformly int as long as the quantities in (0.1) remain finite. The convergence rates are explicit. Supported by the NSF grant DMS 9304580 at IAS.     

Transonic viscous inviscid interactions in narrow channels

Unsteady as well as steady transonic flows through channels which are so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows is formulated for both perfect and dense gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes one-dimensional in the leading order approximation. The latter allows an interpretation of the flow in the core region by means of the theory of one-dimensional transonic inviscid flow through a Laval nozzle while preserving the essential features of the interaction problem associated with the internal structure of pseudoshocks. The sensitivity of a separation bubble caused by a pseudoshock of sufficient strength to perturbations under the condition of choked flow will be demonstrated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global weak solutions to one-dimensional compressible viscous hydrodynamic equations

In this article, we are concerned with the global weak solutions to the 1D compressible viscous hydrodynamic equations with dispersion correction δ~2ρ((φ(ρ))xxφ′(ρ))x withφ(ρ) = ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial conditions. The diffusion term εu xx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in[1](α =1/2) to 0 α≤1. In addition, we perform the limit ε→ 0 with respect to 0 α≤1/2.     

Nonrelativistic limit of solutions of radial quasipotential equations

Theoretical and Mathematical Physics -     

The combined inviscid and non-resistive limit for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions

In this paper, we establish the existence of the global weak solutions for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions for the velocity field and the magnetic field in a bounded domain ? ? R3. Furthermore,we prove that as the viscosity and resistivity coefficients go to zero simultaneously, these weak solutions converge to the strong one of the ideal nonhomogeneous incompressible magnetohydrodynamic equations in energy space.     

Cauchy problem for viscous rotating shallow water equations

We consider the Cauchy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modeling of motions for shallow water with free surface in a rotating sub-domain Marche (2007) [19]. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without Coriolis forces, see for instance Danchin (2000) [10], Haspot (2009) [16], the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, and additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.     

The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms

We study the behavior of solutions to the inviscid (A=0) and the viscous (A>0) hyperbolic conservation laws with stiff source terms

     

Shallow water equations for power law and Bingham fluids

In this note,we provide a consistant thin layer theory for power law and Bingham incompressible fluids flowing down an inclined plane under the effect of gravity.The derivation of such equations is based on formal asymptotic expansions of solutions of Cauchy momentum equations in the shallow water scaling and in the neighbourhood of steady solutions so that we can close the average equations on the fluid height h and the total discharge rate q.     

Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations

The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker-Planck interaction terms in the Wigner equation. The model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. This term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. The existence proof relies on the Faedo-Galerkin method together with a priori estimates from the energy functional.     

Local well-posedness of the incompressible Euler equations in B∞,11 and the inviscid limit of the Navier–Stokes equations

We prove the inviscid limit of the incompressible Navier–Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier–Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona–Smith type argument in the $L^p$ setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space $B_{\infty ,1}^1({\mathbb{R}}^d)$, $d\ge 2$, in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in [3], [4] and by Misio?ek and Yoneda in [12], [13], [14].     

Some solutions of continuum equations for an incompressible viscous medium

A new criterion is obtained for a weighted inequality to hold in the Lebesgue spaces for a Hardy-type integral operator with Oinarov’s kernel.     

On the classical solutions of two dimensional inviscid rotating shallow water system

We prove global existence and asymptotic behavior of classical solutions for two dimensional inviscid rotating shallow water system with small initial data subject to the zero relative vorticity condition. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system with quadratic nonlinearity, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal form method. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity.     

A two-phase flow with a viscous and an inviscid fluid

We study the free boundary between a viscous and an inviscid fluid satisfying the Navier-Stokes and Euler equations respectively. Surface tension is incorporated. We read the equations as a free boundary problem for one viscous fluid with a nonlocal boundary force. We decompose the pressure distribution in the inviscid fluid into two contributions. A positivity result for the first, and a compactness property for the second contribution are dervied. We prove a short time existence theorem for the two-phase problem     

Transonic viscous inviscid interactions in a slender Laval nozzle

Transonic, high Reynolds number flows through a Laval nozzle, which is so narrow that the classical boundary layer correction can no longer be considered to be an effect of higher order, are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows in a small nozzle is presented for perfect gases. Representative solutions including the internal structure of pseudo-shocks forming in the diffuser part of the nozzle and being strongly associated with the chocking phenomenon will be presented. The linear stability of the various flow regimes observed in the nozzle will be discussed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)