Ill-prepared Initial Data Research Articles

Fast rotation and inviscid limits for the SQG equation with general ill-prepared initial data

Fast rotation and inviscid limits for the SQG equation with general ill-prepared initial data

On the incompressible and non-resistive limit of 3D compressible magnetohydrodynamic equations in bounded domains

In this paper, we investigate the incompressible and non-resistive limit for the initial boundary value problem of isentropic compressible resistive magnetohydrodynamic equations with ill-prepared initial data in three-dimensional bounded domains. We establish the higher-order uniform estimates with respect to both the Mach number and the resistivity coefficient in the framework of new type of weighted Sobolev spaces. Then we obtain the strong convergence of the magnetic field and the divergence-free component of the velocity field, as both the Mach number and the resistivity coefficient tend to zero.

Small Alfvén number limit for shallow water magnetohydrodynamics

Small Alfvén number limit of the shallow water magnetohydrodynamic equations with ill-prepared initial data is proved by applying fast averaging method. It is rigorously justified that the solutions of the original system tend to the solution of a one-dimensional system coupled with a linear system as the Alfvén number goes to zero. The approach is also used to study the small Alfvén number limit of the three-dimensional ideal incompressible magnetohydrodynamic equations.

Asymptotics for the rotating fluids and primitive systems with large ill-prepared initial data in critical spaces

Asymptotics for the rotating fluids and primitive systems with large ill-prepared initial data in critical spaces

Incompressible limit of isentropic magnetohydrodynamic equations with ill-prepared data in bounded domains

This paper rigorously justifies the incompressible limit of strong solutions to isentropic compressible magnetohydrodynamic equations with ill-prepared initial data in a three-dimensional bounded domain as the Mach number goes to zero. In both cases of viscous and inviscid magnetic fields, we establish a new energy functional with weight to obtain uniform estimates for strong solutions with respect to the Mach number. Then, we prove the weak convergence of a velocity and the strong convergence of a magnetic field and the divergence-free component of a velocity field, which yields the corresponding incompressible limit.

Convergence of the relaxed compressible Navier–Stokes equations to the incompressible Navier–Stokes equations

In this paper, we study the combined low Mach number and the relaxation limits of the isentropic compressible Navier–Stokes equations with revised Maxwell’s law satisfying Galilean invariance. It is shown that, for the ill-prepared initial data, the solutions of the relaxed compressible Navier–Stokes equations converge to that of the incompressible Navier–Stokes equations as the Mach number and relaxation parameters tend to zero.

Rotation-dominant three-scale limit of the Cauchy problem to the inviscid rotating stratified Boussinesq equations

In this paper, we investigate the rotation-dominant three-scale singular limit of the initial-value problem to the inviscid rotating stratified Boussinesq equations. We first prove a dispersive decay estimate for the linear propagator of that limit regime, which requires novel techniques involving an elaborate frequency cut-off since the phase function is degenerate on both horizontal and vertical directions in some areas. Using the dispersive decay estimate, we then investigate rigorously the rotation-dominant limit of local strong solutions to Boussinesq equations with ill-prepared initial data. The limiting system is shown to be the two-dimensional incompressible Euler equations coupled with a two-dimensional transport equation but in three spatial dimensions.

Zero Mach number limit of the compressible primitive equations: Ill-prepared initial data

In the work, we consider the zero Mach number limit of compressible primitive equations in the domain R2×2T or T2×2T. We identify the limit equations to be the primitive equations with the incompressible condition. The convergence behaviors are studied in both R2×2T and T2×2T, respectively. This paper takes into account the high oscillating acoustic waves and is an extension of our previous work in [29].

Sharper dispersive estimates and asymptotics for a Boussinesq-type system with larger ill-prepared initial data

The aim of this article is to extend previous works about the asymptotics of an ill-prepared fast rotating, highly stratified incompressible Navier–Stokes system. Thanks to improved Strichartz and a priori estimates, we are able not only to cover a case which was unanswered in our previous work (allowing bigger ill-prepared initial data) but also to improve the convergence rates and reduce some assumptions on the initial data. In passing we also widen the range of some parameters and compare two methods to obtain dispersive estimates.

Low Mach Number Limit for the Degenerate Navier–Stokes Equations in Presence of Strong Stratification

In this paper, we investigate the low Mach and low Froude numbers limit for the compressible Navier–Stokes equations with degenerate, density-dependent, viscosity coefficient, in the strong stratification regime. We consider the case of a general pressure law with singular component close to vacuum, and general ill-prepared initial data. We perform our study in the three-dimensional periodic domain. We rigorously justify the convergence to the generalised anelastic approximation, which is used extensively to model atmospheric flows.

Zero-electron-mass limit of the compressible Navier–Stokes–Poisson equations with well/ill-prepared initial data

In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns.21 (1996), 363–394); Grenier (Commun. Pure Appl. Math.50 (1997), 821–865); Ukai (J. Math. Kyoto Univ.26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.

The combined quasineutral and incompressible limit for the Navier–Stokes–Poisson system in critical spaces

In this paper, we study the quasineutral and incompressible limit for the Navier–Stokes–Poisson system with ill-prepared initial data in whole space. Based on Strichartz estimates for the Klein–Gordon equation and the pseudo-scaling property of modified Besov norms, we prove that the density tends to a constant, the compressible part of velocity vanishes, and the incompressible part of velocity converges to the solution of the incompressible Navier–Stokes equation. We get a relation of the lifespan of the solution between the Navier–Stokes–Poisson and the incompressible Navier–Stokes equations for small Debye length and low Mach number.

Uniform regularity for the compressible Navier-Stokes system with low Mach number in domains with boundaries

We establish uniform with respect to the Mach number regularity estimates for the isentropic compressible Navier-Stokes system in smooth domains with Navier-slip condition on the boundary in the general case of ill-prepared initial data. To match the boundary layer effects due to the fast oscillations and the ill-prepared initial data assumption, we prove uniform estimates in an anisotropic functional framework with only one normal derivative close to the boundary. This allows to prove the local existence of a strong solution on a time interval independent of the Mach number and to justify the incompressible limit through a simple compactness argument.

Fast rotation limit for the 2-D non-homogeneous incompressible Euler equations

In the present paper, we study the fast rotation limit for the density-dependent incompressible Euler equations in two space dimensions with the presence of the Coriolis force. In the case when the initial densities are small perturbation of a constant profile, we show the convergence of solutions towards the solutions of a quasi-homogeneous incompressible Euler system. The proof relies on a combination of uniform estimates in high regularity norms with a compensated compactness argument for passing to the limit. This technique allows us to treat the case of ill-prepared initial data.

On the low Mach number limit of a two-fluid model with magnetic field and ill-prepared initial data in bounded domain

On the low Mach number limit of a two-fluid model with magnetic field and ill-prepared initial data in bounded domain

On the fast rotation asymptotics of a non-homogeneous incompressible MHD system

This paper is devoted to the analysis of a singular perturbation problem for a 2D incompressible MHD system with density variations and Coriolis force, in the limit of small Rossby numbers. Two regimes are considered. The first one is the quasi-homogeneous regime, where the densities are small perturbations around a constant state. The limit dynamics is identified as an incompressible homogeneous MHD system, coupled with an additional transport equation for the limit of the density variations. The second case is the fully non-homogeneous regime, where the densities vary around a general non-constant profile. In this case, in the limit, the equation for the magnetic field combines with an underdetermined linear equation, which links the limit density variation function with the limit velocity field. The proof is based on a compensated compactness argument, which enables us to consider general ill-prepared initial data. An application of Di Perna–Lions theory for transport equations allows to treat the case of density-dependent viscosity and resistivity coefficients.

A Multiscale Problem for Viscous Heat-Conducting Fluids in Fast Rotation

In the present paper, we study the combined incompressible and fast rotation limits for the full Navier-Stokes-Fourier system with Coriolis, centrifugal and gravitational forces, in the regime of small Mach, Froude and Rossby numbers and for general ill-prepared initial data. We consider both the isotropic scaling (where all the numbers have the same order of magnitude) and the multi-scale case (where some effect is predominant with respect to the others). In the case when the Mach number is of higher order than the Rossby number, we prove that the limit dynamics is described by an incompressible Oberbeck-Boussinesq system, where the velocity field is horizontal (according to the Taylor-Proudman theorem), but vertical effects on the temperature equation are not negligible. Instead, when the Mach and Rossby numbers have the same order of magnitude, and in absence of the centrifugal force, we show convergence to a quasi-geostrophic equation for a stream function of the limit velocity field, coupled with a transport-diffusion equation for a new unknown, which links the target density and temperature profiles. The proof of the convergence is based on a compensated compactness argument. The key point is to identify some compactness properties hidden in the system of acoustic-Poincar\'e waves. Compared to previous results, our method enables first of all to treat the whole range of parameters in the multi-scale problem, and also to consider a low Froude number regime with the somehow critical choice $Fr=\sqrt{Ma}$, where $Ma$ is the Mach number. This allows us to capture some (low) stratification effects in the limit.

Low stratification of a heat-conducting fluid in a time-dependent domain

We study the low Mach number limit of the full Navier–Stokes–Fourier system in the case of low stratification with ill-prepared initial data for the problem stated on a moving domain with a prescribed motion of the boundary. Similarly as in the case of a fixed domain, we recover as a limit the Oberback–Boussinesq system; however, we identify one additional term in the temperature equation of the limit system which is related to the motion of the domain and which is not present in the case of a fixed domain. One of the main ingredients in the proof is the properties of the Helmholtz decomposition on moving domains and the dependence of eigenvalues and eigenspaces of the Neumann Laplace operator on time.

Quasi-neutral limit and the initial layer problem of the electro-diffusion model arising in electro-hydrodynamics

In this paper we study quasi-neutral limit and the initial layer problem of the electro-diffusion model arising in electro-hydrodynamics which is the coupled Planck–Nernst–Poisson and Navier–Stokes equations. Different from other studies, we consider the physical case that the mobilities of the charges are different. For the generally smooth doping profile and for the ill-prepared initial data, under the assumption that the difference between the mobilities of two kinds of charges is very small, the quasi-neutral limit with an initial layer structure is rigorously proved by using the weighted energy method coupled with multi-scaling asymptotic expansions.

On a Singular Limit for the Compressible Rotating Euler System

The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in the case of ill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acoustic waves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in the asymptotic limit is eliminated.