In this article, we focus on a MHD model that takes into account the Coriolis force. In this case, the strength of the rotation is measured by the Rossby number ε (positive). Under the assumption that the rotation is strong (that is, when ε goes to zero) and the kinematic viscosity goes to zero as ε^α (α positive), we first establish the global convergence of weak solutions, and then prove the global existence and convergence of strong solutions. In particular, the Strichartz estimates used to obtain explicit convergence rates depend on the above parameter α and lead us to identify an admissible interval for α.

We recently proved that if their initial velocity and magnetic field both are the sum of a classical 3D-part and some 2D-part (i.-e. depending only on the horizontal space variables), the solutions of the 3D-rotating magnetohydrodynamic (MHD) system converge (when the Rossby number goes to zero) towards those of a 2D-MHD system with six components and an additional 3D magnetic field transported by the 2D limit velocity. We also provided explicit convergence rates for large ill-prepared initial data. In this article, thanks to new estimates, we improve the convergence rates and allow larger initial data. Finally these improvements are adapted to the rotating fluids system.

In this article, we consider the 3D-rotating magnetohydrodynamic (MHD) system when the initial velocity and magnetic field both feature some 2D-part (i.-e. depending only on the horizontal space variables). We prove for weak and strong solutions, that the limit system, when the Rossby number goes to zero (i.-e. for strong rotation), is a 2D-MHD system with three components. Moreover we are able to provide explicit global-in-time convergence rates thanks to adapted Strichartz estimates and with the help of an additional 3D magnetic field transported by the 2D limit velocity.

The asymptotics of the strongly stratified Boussinesq system when the Froude number goes to zero have been previously investigated, but the resulting limit system surprisingly did not depend on the thermal diffusivity v'. In this article we obtain richer asymptotics (depending on v') for more general ill-prepared initial data. As for the rotating fluids system, the only way to reach this limit consists in finding suitable non-conventional initial data: here, to a function classically depending on the full space variable, we add a second one only depending on the vertical coordinate. Thanks to a refined study of the structure of the limit system and to new adapted Strichartz estimates, we obtain convergence in the context of weak Leray-type solutions providing explicit convergence rates when possible. In the usually simpler case v = v' we are able to improve the Strichartz estimates and the convergence rates. The last part of the appendix is devoted to the proof of a new and crucial dispersion estimate, as classical methods fail. Finally, our theorems can also be rewritten as a global existence result and asymptotic expansion for the classical Boussinesq system near an explicit stationary solution and for large non-conventional vertically stratified initial data. (c) 2025 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In our previous work dedicated to the strongly stratified Boussinesq system, we obtained for the first time a limit system (when the froude number epsilon goes to zero) that depends on the thermal diffusivity nu ' (other works obtained a limit system only depending on the visosity nu). To reach those richer asymptotics we had to consider an unusual initial data which is the sum of a function depending on the full space variable and a function only depending on the vertical coordinate, and we studied the convergence of the weak Leray-type solutions. In the present article we extend these results to the strong Fujita-Kato-type solutions. In this setting, and compared to the case of weak solutions, we obtain far better convergence rates (in epsilon) for ill-prepared initial data with very large oscillating part of size some negative power of the small parameter epsilon. The main difficulties come from the anisotropy induced by the presence of x3-depending functions.

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.

In this article we study the lifespan and asymptotics (in the large rotation and stratification regime) for the Primitive system for highly ill-prepared initial data in critical spaces. Compared to our previous works, we simplified the proof and made it adaptable to the Rotating fluids system with highly ill-prepared initial data decomposed as a sum of 2D horizontal part and a very large 3D part. We also provide explicit convergence rates.

We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebraic time-decay estimates in critical Besov spaces (and even exponential decay for the high frequencies) for any derivatives of the solution. Our approach is partly inspired by the work of Bae, Biswas \& Tadmor [2] dedicated to the classical incompressible Navier-Stokes equations, and requires our establishing new bilinear estimates (of independent interest) involving the Gevrey regularity for the product or composition of functions. To the best of our knowledge, this is the first work pointing out Gevrey analyticity for a model of compressible fluids.

In this article we prove highly improved and flexible Strichartz-type estimates allowing us to generalize the asymptotics we obtained for a stratified and rotating incompressible Navier-Stokes system: for large (and less regular) initial data, we obtain global well-posedness, asymptotics (as the Rossby number $\epsilon$ goes to zero) and convergence rates as a power of the small parameter $\epsilon$. Our approach is lead by the special structure of the limit system: the 3D quasi-geostrophic system.

The present article is devoted to the 3D dissipative quasi-geostrophic system (QG). This system can be obtained as limit model of the Primitive Equations in the asymptotics of strong rotation and stratification, and involves a non-radial, nonlocal, homogeneous pseudo-differential operator of order 2 denoted by Γ (and whose semigroup kernel reaches negative values). After a refined study of the non-local part of Γ, we prove apriori estimates (in the general L setting) for the 3D QG-model. The main difficulty of this article is to study the commutator of Γ with a Lagrangian change of variable. An important application of these a priori estimates, providing bound from below to the lifespan of the solutions of the Primitive Equations for ill-prepared blowing-up initial data, can be found in a companion paper.

This article generalizes a previous work in which the author obtained a large lower bound for the lifespan of the solutions to the Primitive Equations, and proved convergence to the 3D quasi-geostrophic system for general and ill-prepared (possibly blowing-up) initial data that are regularization of vortex patches related to the potential velocity. These results were obtained for a very particular case when the kinematic viscosity $\nu$ is equal to the heat diffusivity $\nu '$, turning the diffusion operator into the classical Laplacian. Obtaining the same results without this assumption is much more difficult as it involves a non-local diffusion operator. The key to the main result is a family of a priori estimates for the 3D-QG system that we obtained in a companion paper.

J.-Y. Chemin proved the convergence (as the Rossby number $\epsilon$ goes to zero) of the solutions of the Primitive Equations to the solution of the 3D quasi-geostrophic system when the Froude number F = 1 that is when no dispersive property is available. The result was proved in the particular case where the kinematic viscosity $\nu$ and the thermal diffusivity $\nu$ ' are close. In this article we generalize this result for any choice of the viscosities, the key idea is to rely on a special feature of the quasi-geostrophic structure.

We study the inhomogeneous incompressible Navier--Stokes system endowed with a general capillary term. Thanks to recent methods based on Lagrangian change of variables, we obtain local well-posedness in critical Besov spaces (even if the integration index $p\neq 2$) and for variable viscosity and capillary terms. In the case of constant coefficients and for initial data that are perturbations of a constant state, we are able to prove that the lifespan goes to infinity as the capillary coefficient goes to zero, connecting our result to the global existence result obtained by Danchin and Mucha for the incompressible Navier--Stokes system with constant coefficients.

In the present article we consider a capillary compressible system introduced by C. Rohde after works of Bandon, Lin and Rogers, called the order-parameter model, and whose aim is to reduce the numerical difficulties that one encounters in the case of the classical local Korteweg system (involving derivatives of order three) or the non-local system (also introduced by Rohde after works of Van der Waals, and which involves a convolution operator). We prove that this system has a unique global solution for initial data close to an equilibrium and we precisely study the convergence of this solution towards the local Korteweg model.

The present article is devoted to the 3D dissipative quasi-geostrophic system (QG). This system can be obtained as limit model of the Primitive Equations in the asymptotics of strong rotation and stratification, and involves a non-radial, non-local, homogeneous pseudo-differential operator of order 2 denoted by $\Gamma$ (and whose semigroup kernel reaches negative values). After a refined study of the non-local part of $\Gamma$, we prove apriori estimates (in the general L^p setting) for the 3D QG-model. The main difficulty of this article is to study the commutator of $\Gamma$ with a Lagrangian change of variable. An important application of these a priori estimates, providing bound from below to the lifespan of the solutions of the Primitive Equations for ill-prepared blowing-up initial data, can be found in a companion paper.

In this article we study three capillary compressible models (the classical local Navier–Stokes–Korteweg system and two non-local models) for large initial data, bounded away from zero, and with a reference pressure state ρ¯ which is not necessarily stable (P′(ρ¯) can be non-positive). We prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non-local models towards the local Korteweg model. The results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients.

In the present article we are interested in further investigations for the barotropic compressible Navier-Stokes system endowed with a non-local capillarity we studied in [7]. Thanks to an accurate study of the associated linear system using a Lagrangian change of coordinates, we provide more precise energy estimates in terms of hybrid Besov spaces naturally depending on a threshold frequency (which is determined in function of the physical parameter) distinguishing the low and the high regimes. It allows us in particular to prove the convergence of the solutions from the non-local to the local Korteweg system. Another mathematical interest of this article is the study of the effect of the Lagrangian change on the non-local capillary term.

In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a $\gamma$ law for the pressure ($P(\rho)=a\rho^{\gamma}$, $\gamma>1$) to entropic solution of the compressible Euler equations. In particular it justifies that the Korteweg system is suitable for selecting the physical solutions in the case where the Euler system is strictly hyperbolic. The problem remains open for a Van der Waals pressure because in this case the system is not strictly hyperbolic and in particular the classical theory of Lax and Glimm (see \cite{Lax,G}) can not be used.

This paper is dedicated to the study of both viscous compressible barotropic fluids and Navier-Stokes equation with dependent density, when the viscosity coefficients are variable, in dimension $d\geq2$. We aim at proving the local and global well-posedness for respectively {\it large} and \textit{small} initial data having critical Besov regularity and more precisely we are interested in extending the class of initial data velocity when we consider the shallow water system, improving the results in \cite{CMZ1,H2} and \cite{arma}. Our result relies on the fact that the velocity $u$ can be written as the sum of the solution $u_{L}$ of the associated linear system and a remainder velocity term $\bar{u}$; then in the specific case of the shallow-water system the remainder term $\bar{u}$ is more regular than $u_{L}$ by taking into account the regularizing effects induced on the bilinear convection term. In particular we are able to deal with initial velocity in $\dot{H}^{\N-1}$ as Fujita and Kato for the incompressible Navier-Stokes equations (see \cite{FK}) with an additional condition of type $u_{0}\in B^{-1}_{\infty,1}$. We would like to point out that this type of result is of particular interest when we want to deal with the problem of the convergence of the solution of compressible system to the incompressible system when the Mach number goes to 0.