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Nader Masmoudi

Publications and source records attributed to Nader Masmoudi.

At least 55 records · Page 3Linked to original sources
arXiv2021

Incompressible limit for the free surface Navier-Stokes system

We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size $\sqrt{\varepsilon}$, $\varepsilon$ being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations.

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arXiv2021

On the diffusive limits of radiative heat transfer system I: well prepared initial and boundary conditions

We study the diffusive limit approximation for a nonlinear radiative heat transfer system that arises in the modeling of glass cooling, greenhouse effects and in astrophysics. The model is considered with the reflective radiative boundary conditions for the radiative intensity, and periodic, Dirichlet and Robin boundary conditions for the temperature. The global existence of weak solutions for this system is given by using a Galerkin method with a careful treatment of the boundary conditions. Using the compactness method, averaging lemma and Young measure theory, we prove our main result that the weak solution converges to a nonlinear diffusion model in the diffusive limit. Moreover, under more regularity conditions on the limit system, the diffusive limit is also analyzed by using a relative entropy method. In particular, we get a rate of convergence. The initial and boundary conditions are assumed to be well-prepared in the sense that no initial and boundary layer exist.

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arXiv2021

Boundary Layer Expansions for the Stationary Navier-Stokes Equations

This is the first part of a two paper sequence in which we prove the global-in-x stability of the classical Prandtl boundary layer for the 2D, stationary Navier-Stokes equations. In this part, we provide a construction of an approximate Navier-Stokes solution, obtained by a classical Euler- Prandtl asymptotic expansion. We develop here sharp decay estimates on these quantities. Of independent interest, we establish \textit{without} using the classical von-Mise change of coordinates, proofs of global in x regularity of the Prandtl system. The results of this paper are used in the second part of this sequence, [IM20] (arXiv:2008.12347), to prove the asymptotic stability of the boundary layer as $\eps \rightarrow 0$ and $x \rightarrow \infty$.

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arXiv2021

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

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.

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arXiv2021

Linear damping and depletion in flowing plasma with strong sheared magnetic fields

In this paper, we study the long-time behavior of the solution for the linearized ideal MHD around sheared velocity and magnetic field under Stern stability condition. We prove that the velocity and magnetic field will converge to sheared velocity and magnetic field as time approaches infinity. Moreover a new depletion phenomenon is proved: the horizontal velocity and magnetic field at the critical points will decay to 0 as time approaches infinity.

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arXiv2021

Global-in-$x$ Stability of Steady Prandtl Expansions for 2D Navier-Stokes Flows

In this work, we establish the convergence of 2D, stationary Navier-Stokes flows, $(u^ε, v^ε)$ to the classical Prandtl boundary layer, $(\bar{u}_p, \bar{v}_p)$, posed on the domain $(0, \infty) \times (0, \infty)$: \begin{equation*} \| u^ε - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrtε \langle x \rangle^{- \frac 1 4 + δ}, \qquad \| v^ε - \sqrtε \bar{v}_p \|_{L^\infty_y} \lesssim \sqrtε \langle x \rangle^{- \frac 1 2}. \end{equation*} This validates Prandtl's boundary layer theory \textit{globally} in the $x$-variable for a large class of boundary layers, including the entire one parameter family of the classical Blasius profiles, with sharp decay rates. The result demonstrates asymptotic stability in two senses simultaneously: (1) asymptotic as $ε\rightarrow 0$ and (2) asymptotic as $x \rightarrow \infty$. In particular, our result provides the first rigorous confirmation for the Navier-Stokes equations that the boundary layer cannot "separate" in these stable regimes, which is very important for physical and engineering applications.

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arXiv2021

Singularities and unsteady separation for the inviscid two-dimensional Prandtl's system

We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary layer separation. We then exhibit new Lagrangian formulae for backward self-similar profiles, and study them also with a different approach that was initiated by Elliott-Smith-Cowley and Cassel-Smith-Walker. One particular profile is at the heart of the so-called Van-Dommelen and Shen singularity, and we prove its generic appearance (that is, for an open and dense set of blow-up solutions) for any prescribed Eulerian outer flow. We comment on the connexion between these results and the full viscous Prandtl system. This paper combines ideas for transport equations, such as Lagrangian coordinates and incompressibility, and for singularity formation, such as self-similarity and renormalisation, in a novel manner, and designs a new way to study singularities for quasilinear transport equations.

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arXiv2021

Refined description and stability for singular solutions of the 2D Keller-Segel system

We construct solutions to the two dimensional parabolic-elliptic Keller-Segel model for chemotaxis that blow up in finite time $T$. The solution is decomposed as the sum of a stationary state concentrated at scale $λ$ and of a perturbation. We rely on a detailed spectral analysis for the linearized dynamics in the parabolic neighbourhood of the singularity performed by the authors, providing a refined expansion of the perturbation. Our main result is the construction of a stable dynamics in the full nonradial setting for which the stationary state collapses with the universal law $λ\sim 2e^{-\frac{2+γ}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln (T-t)|}{2}}}$ where $γ$ is the Euler constant. This improves on the earlier result by Raphael and Schweyer 2014 and gives a new robust approach to so-called type II singularities for critical parabolic problems. A by-product of the spectral analysis we developed is the existence of unstable blowup dynamics with speed $λ_\ell \sim C_0(T-t)^{\frac{\ell}{2}} |\ln(T-t)|^{-\frac{\ell}{2(\ell - 1)}}$ for $\ell \geq 2$ integer.

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arXiv2020

Singularity formation for Burgers equation with transverse viscosity

We consider Burgers equation with transverse viscosity $$\partial_tu+u\partial_xu-\partial_{yy}u=0, \ \ (x,y)\in \mathbb R^2, \ \ u:[0,T)\times \mathbb R^2\rightarrow \mathbb R.$$ We construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the $x$ variable, whose scaling parameters evolve according to parabolic equations along the $y$ variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.

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arXiv2020

Spectral analysis for singularity formation of the two dimensional Keller-Segel system

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as well as a coercivity estimate for the non-radial part. These results are used as key arguments in a new rigorous proof of the existence and refined description of singular solutions for the Keller-Segel problem by the authors. The present paper extends the result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [Physica D, 2012]. Two major difficulties arise in the analysis: this is a singular limit problem, and a degeneracy causes corrections not being polynomial but logarithmic with respect to the main parameter.

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arXiv2020

Well-posedness in Gevrey function space for 3D Prandtl equations without Structural Assumption

We establish the well-posedness in Gevrey function space with optimal class of regularity 2 for the three dimensional Prandtl system without any structural assumption. The proof combines in a novel way a new cancellation in the system with some of the old ideas to overcome the difficulty of the loss of derivatives in the system.This shows that the three dimensional instabilities in the system leading to ill-posedness are not worse than the two dimensional ones.

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arXiv2020

Relaxation to equilibrium in the one-dimensional thin-film equation with partial wetting

We investigate the large time behavior of compactly supported solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.

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arXiv2020

Linearized wave-damping structure of Vlasov-Poisson in $\mathbb R^3$

In this paper we study the linearized Vlasov-Poisson equation for localized disturbances of an infinite, homogeneous Maxwellian background distribution in $\mathbb R^3_x \times \mathbb R^3_v$. In contrast with the confined case $\mathbb T^d _x \times \mathbb R_v ^d$, or the unconfined case $\mathbb R^d_x \times \mathbb R^d_v$ with screening, the dynamics of the disturbance are not scattering towards free transport as $t \to \pm \infty$: we show that the electric field decomposes into a very weakly-damped Klein-Gordon-type evolution for long waves and a Landau-damped evolution. The Klein-Gordon-type waves solve, to leading order, the compressible Euler-Poisson equations linearized about a constant density state, despite the fact that our model is collisionless, i.e. there is no trend to local or global thermalization of the distribution function in strong topologies. We prove dispersive estimates on the Klein-Gordon part of the dynamics. The Landau damping part of the electric field decays faster than free transport at low frequencies and damps as in the confined case at high frequencies; in fact, it decays at the same rate as in the screened case. As such, neither contribution to the electric field behaves as in the vacuum case.

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arXiv2020

Optimal Prandtl expansion around concave boundary layer

We provide an optimal Gevrey stability result for general boundary layer expansions, under a mild concavity condition on the boundary layer profile. Our result generalizes (and even improves in the non strictly concave case) the one obtained in [Gerard-Varet et al, Duke Math. J. 167 (2018)], restricted to expansions of shear flow type.

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arXiv2019

On the Stability of Self-similar Blow-up for $C^{1,α}$ Solutions to the Incompressible Euler Equations on $\mathbb{R}^3$

We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,α}$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale-Kato-Majda criterion cannot be improved in the class of $C^{1,α}$ solutions.

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arXiv2019

On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov flows

We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier-Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient $ν$ is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale $O(ν^{-\frac12})$ for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale $O(ν^{-1})$. Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows.

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