We investigate the behavior of rotating incompressible flows near a non-flat horizontal bottom. In the flat case, the velocity profile is given explicitly by a simple linear ODE. When bottom variations are taken into account, it is governed by a nonlinear PDE system, with far less obvious mathematical properties. We establish the well-posedness of this system and the asymptotic behavior of the solution away from the boundary. In the course of the proof, we investigate in particular the action of pseudo-differential operators in non-localized Sobolev spaces. Our results extend the older paper [18], restricted to periodic variations of the bottom. It ponders on the recent linear analysis carried in [14].

Authors: A.-L. Dalibard
(LJLL), D. Gérard-Varet (IMJ-PRG)
Category: Research article
Date: 5 novembre 2015

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher m−fold symmetry with m≥3 the bifurcation does not occur. This answers to a problem left open in \cite{H-F-M-V}. Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in \cite{H-F-M-V}. The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation.

In this paper, we prove the existence of m-fold rotating patches for the Euler equations in the disc, for both simply-connected and doubly-connected cases. Compared to the planar case, the rigid boundary introduces rich dynamics for the lowest symmetries m=1 and m=2. We also discuss some numerical experiments highlighting the interaction between the boundary of the patch and the rigid one.

Authors: Francisco De
La Hoz, Zineb Hassainia, Taoufik Hmidi, Joan Mateu
Category: Research article
Date: 21 octobre 2015

In this article, we study the homogenization limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of nk disjoint disks with centers {zki} and radii εk. We assume that the initial velocities uk0 are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, nk→∞, and we assume εk→0 as k→∞. Let γki be the circulation of uk0 around the circle {|x−zki|=εk}. We prove that the homogenization limit retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) ωk0= curl uk0 has a uniform compact support and converges weakly in Lp0, for some p0>2, to ω0∈Lp0c(ℝ2), (2) ∑nki=1γkiδzki⇀μ weak-∗ in (ℝ2) for some bounded Radon measure μ, and (3) the radii εk are sufficiently small. Then the corresponding solutions uk converge strongly to a weak solution u of a modified Euler system in the full plane. This modified Euler system is given, in vorticity formulation, by an active scalar transport equation for the quantity ω= curl u, with initial data ω0, where the transporting velocity field is generated from ω so that its curl is ω+μ. As a byproduct, we obtain a new existence result for this modified Euler system.

Authors: C. Lacave, M. C. Lopes Filho, H. J. Nussenzveig Lopes
Category: Research article
Date: 20 octobre 2015

We study the inviscid multilayer Saint-Venant (or shallow-water) system in the limit of small density contrast. We show that, under reasonable hyperbolicity conditions on the flow, the system is well-posed on a large time interval, despite the singular limit. By studying the asymptotic limit, we provide a rigorous justification of the widely used rigid-lid and Boussinesq approximations for multilayered shallow water flows. The asymptotic behaviour is similar to that of the incompressible limit for Euler equations, in the sense that there exists a small initial layer in time for ill-prepared initial data, accounting for rapidly propagating " acoustic " waves (here, the so-called barotropic mode) which interact only weakly with the " incompressible " component (here, baroclinic).

In this paper we consider rotating doubly connected vortex patches for the Euler equations in the plane. When the inner interface is an ellipse we show that the exterior interface must be a confocal ellipse. We then discuss some relations, first found by Flierl and Polvani, between the parameters of the ellipses, the velocity of rotation and the magnitude of the vorticity in the domain enclosed by the inner ellipse.

In this paper, we are interested in the global persistence regularity for the 2D incompressible Euler equations in some function spaces allowing unbounded vorticities. More precisely, we prove the global propagation of the vorticity in some weighted Morrey-Campanato spaces and in this framework the velocity field is not necessarily Lipschitz but belongs to the log-Lipschitz class LαL, for some α∈(0,1).

We prove the existence of the V-states for the generalized inviscid SQG equations with α∈]0,1[. These structures are special rotating simply connected patches with m− fold symmetry bifurcating from the trivial solution at some explicit values of the angular velocity. This produces, inter alia, an infinite family of non stationary global solutions with uniqueness.

We introduce a new class of Green-Naghdi models for the propagation of internal waves between two (1+1)-dimensional layers of homogeneous, immiscible, ideal, incompressible, irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original Green-Naghdi model, and in particular to manage high-frequency Kelvin-Helmholtz instabilities. Our models preserve the Hamiltonian structure, symmetry groups and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well-posedness and stability results. These results apply in particular to the original Green-Naghdi model as well as to the Saint-Venant (hydrostatic shallow water) system with surface tension.

Authors: V. Duchene
(IRMAR), S. Israwi (U. Libanaise), R. Talhouk (U. Libanaise)
Category: Research article
Date: 13 octobre 2015

This study deals with asymptotic models for the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. We present a new Green-Naghdi type model in the Camassa-Holm (or medium amplitude) regime. This model is fully justified, in the sense that it is consistent, well-posed, and that its solutions remain close to exact solutions of the full Euler system with corresponding initial data. Moreover, our system allows to fully justify any well-posed and consistent lower order model; and in particular the so-called Constantin-Lannes approximation, which extends the classical Korteweg-de Vries equation in the Camassa-Holm regime.

Authors: V. Duchene
(IRMAR), S. Israwi (U. Libanaise), R. Talhouk (U. Libanaise)
Category: Research article
Date: 13 octobre 2015

In this paper, we prove the existence of doubly connected V-states for the generalized SQG equations with α∈]0,1[. They can be described by countable branches bifurcating from the annulus at some explicit "eigenvalues" related to Bessel functions of the first kind. Contrary to Euler equations \cite{H-F-M-V}, we find V-states rotating with positive and negative angular velocities. At the end of the paper we discuss some numerical experiments concerning the limiting V-states.

Authors: Francisco De
La Hoz, Zineb Hassainia, Taoufik Hmidi
Category: Research article
Date: 12 octobre 2015

Motivated by applications to vortex rings, we study the Cauchy problem for the three-dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the axisymmetric vorticity is integrable with respect to the two-dimensional measure dr dz, where (r,\theta,z) denote the cylindrical coordinates in R^3, we show the existence of a unique global solution, which converges to zero in L^1 norm as time goes to infinity. The proof of local well-posedness follows exactly the same lines as in the two-dimensional case, and our approach emphasizes the similarity between both situations. The solutions we construct have infinite energy in general, so that energy dissipation cannot be invoked to control the long-time behavior. We also treat the more general case where the initial vorticity is a finite measure whose atomic part is small enough compared to viscosity. Such data include point masses, which correspond to vortex filaments in the three-dimensional picture.

Authors: T. Gallay
(Institut Fourier), V. Sverak (U.Minnesota)
Category: Research article
Date: 5 octobre 2015

We consider rigid bodies moving under the influence of a viscous fluid and we study the asymptotic as the size of the solids tends to zero. In a bounded domain, if the solids shrink to " massive " pointwise particles, we obtain a convergence to the solution of the Navier-Stokes equations independently to any possible collision of the bodies with the exterior boundary. In the case of " massless " pointwise particles, we obtain a result for a single disk moving in the full plane. In this situation, the energy equality is not sufficient, and we obtain a uniform estimate for the solid velocity thanks to the optimal L p − L q decay estimates of the Stokes semigroup.

Authors: C. Lacave
(IMJ), T. Takahashi (IECL, INRIA SPHINX)
Category: Research article
Date: 29 septembre 2015

We characterize the set of functions u0∈L2(Rn) such that the solution of the problem ut=u in Rn×(0,∞) starting from u0 satisfy upper and lower bounds of the form c(1+t)−γ≤∥u(t)∥2≤c′(1+t)−γ.Here is in a large class of linear pseudo-differential operator with homogeneous symbol (including the Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will be discussed: in particular our characterization provides necessary and sufficient conditions on u0 for a solution of the Navier--Stokes system to satisfy sharp upper-lower decay estimates as above. In doing so, we will revisit and improve the theory of \emph{decay characters} by C.~Bjorland, C.~Niche, and M.E.~Schonbek, by getting advantage of the insight provided by the Littlewood--Paley analysis and the use of Besov spaces.

We study an unsteady non linear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoleastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure.

Author: C. Grandmont
(REO-LJLL), M. Hillairet (UM2)
Category: Research article
Date: 3septembre 2015

We consider the stationary Stokes problem in a three-dimensional fluid domain with non-homogeneous Dirichlet boundary conditions. We assume that this fluid domain is the complement of a bounded obstacle in a bounded or an exterior smooth container Ω. We compute sharp asymptotics of the solution to the Stokes problem when the distance between the obstacle and the container boundary is small.

Authors: M. Hillairet
(Montpellier), T. Kelai (IMJ-PRG)
Category: Research article
Date: 2 septembre 2015

The focusing cubic NLS is a canonical model for the propagation of laser beams. In dimensions 2 and 3, it is known that a large class of initial data leads to finite time blow-up. Now, physical experiments suggest that this blow-up does not always occur. This might be explained by the fact that some physical phenomena neglected by the standard NLS model become relevant at large intensities of the beam. Many ad hoc variants of the focusing NLS equation have been proposed to capture such effects. In this paper, we derive some of these variants from Maxwell's equations and propose some new ones. We also provide rigorous error estimates for all the models considered. Finally, we discuss some open problems related to these modified NLS equations.

Authors: Eric Dumas, David Lannes, Jeremie Szeftel
Category: Research article
Date: 2 septembre 2015

This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).

Authors: V. Bonnaillie-Noel
(DMA), M. Dambrine (LMAP), C. Lacave (IMJ)
Category: Research article
Date: 2 septembre 2015

We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in [9] that the motion of the waves could be described using an extended Green-Naghdi system. In this paper we propose an analysis of these equations, and show that they can be used to get some new insight into wave-current interactions. We show in particular that solitary waves may have a drastically different behavior in the presence of vorticity and show the existence of solitary waves of maximal amplitude with a peak at their crest, whose angle depends on the vorticity. We also propose a robust and simple numerical scheme validated on several examples. Finally, we give some examples of wave-current interactions with a non trivial vorticity field and topography effects.

Authors: D. Lannes
(IMB), F. Marche (I3M et INRIA LEMON)
Category: Research article
Date: 13 aout 2015

In this paper we consider the b-family equations on the torus ut−utxx+(b+1)uux=buxuxx+uuxxx=0, which for appropriate values of b reduces to well-known models, such as the Camassa-Holm equation or the Degasperis-Procesi equation. We establish a local-in-space blow-up criterion.

We prove the local existence for the Water Waves equations with large bathymetric variations on a time interval of size 1/\epsilon, where ϵ measures the amplitude of the wave. We just need the presence of surface tension.

We prove that the only global, strong, spatially periodic solution to the Degasperis-Procesi equation, vanishing at some point (t0, x0), is the identically zero solution. We also establish the analogue of such Liouville-type theorem for the Degasperis-Procesi equation with an additional dispersive term.

This paper is concerned with a complete asymptotic analysis as ν→0 of the Munk equation $\d\_x\psi-\nu \Delta^2 \psi= \tau$ in a domain Ω⊂R2, supplemented with homogeneous boundary conditions for ψ and ∂_nψ. This equation is a simple model for the circulation of currents in closed basins. A crude analysis shows that as ν→0, the weak limit of ψ satisfies the so-called Sverdrup transport equation inside the domain, namely $\d\_x \psi^0=\tau$, while boundary layers appear in the vicinity of the boundary. These boundary layers, which are the main center of interest of the present paper, exhibit several types of peculiar behaviour. First, the size of the boundary layer on the western and eastern boundary, which had already been computed by several authors, becomes formally very large as one approaches northern and southern portions of the boundary, i.e. pieces of the boundary on which the normal is vertical. This phenomenon is known as geostrophic degeneracy. In order to avoid such singular behaviour, previous studies imposed restrictive assumptions on the domain Ω and on the forcing term τ. Here, we prove that a superposition of two boundary layers occurs in the vicinity of such points: the classical western or eastern boundary layers, and some northern or southern boundary layers, whose derivation is completely new. The size of northern/southern boundary layers is much larger than the one of western boundary layers (ν1/4 vs. ν1/3). We explain in detail how the superposition takes place, depending on the geometry of the boundary. Moreover, when the domain Ω is not connected in the x direction, ψ0 is not continuous in Ω, and singular layers appear in order to correct its discontinuities. These singular layer are concentrated in the vicinity of horizontal lines, and therefore penetrate the interior of the domain Ω. Hence we exhibit some kind of boundary layer separation. However, we emphasize that we remain able to prove a convergence theorem, so that the boundary layers somehow remain stable, in spite of the separation. Eventually, the effect of boundary layers is non-local in several aspects. On the first hand, for algebraic reasons, the boundary layer equation is radically different on the west and east parts of the boundary. As a consequence, the Sverdrup equation is endowed with a Dirichlet condition on the East boundary, and no condition on the West boundary. Therefore western and eastern boundary layers have in fact an influence on the whole domain Ω, and not only near the boundary. On the second hand, the northern and southern boundary layer profiles obey a propagation equation, where the space variable x plays the role of time, and are therefore not local.

Authors: Anne-Laure Dalibard (LJLL), Laure Saint-Raymond (DMA)
Category: Research article
Date: 26 mars 2015

In this paper, we want to understand the Proudman resonance, which is a linear resonance in shallow water of a water body to a traveling atmospheric disturbance. We show here that the same kind of resonance exists for landslides tsunamis and we propose a mathematical approach to investigate these phenomena based on the derivation, justification and analysis of relevant asymptotic models. This approach allows us to investigate more complex phenomena that are not dealt with in the physics literature such as the influence of a variable bottom or the generalization of the Proudman resonance in deeper water. First, we prove a local well-posedness of the water waves equations with a moving bottom and a non constant pressure taking into account the dependence of small physical parameters and we show that these equations are a Hamiltonian system (which extend the result of Zakharov). Then, we justify some linear asymptotic models in order to study the Proudman resonance and submarine landslide tsunamis; we study the linear water waves equations and dispersion estimates allow us to investigate the amplitude of the sea level. To complete these asymptotic models, we add some numerical simulations.

We consider the flow of a viscous, incompressible, Newtonian fluid in a perforated domain in the plane. The domain is the exterior of a regular lattice of rigid particles. We study the simultaneous limit of vanishing particle size and distance, and of vanishing viscosity. Under suitable conditions on the particle size, particle distance, and viscosity, we prove that solutions of the Navier-Stokes system in the perforated domain converges to solutions of the Euler system, modeling inviscid, incompressible flow, in the full plane. That is, the flow is not disturbed by the porous medium and becomes inviscid in the limit. Convergence is obtained in the energy norm with explicit rates of convergence.

We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous newtonian flows. We use a model introduced by Ybert and coauthors in which the rough surface is replaced by a flat plane with alternating small areas of slip and no-slip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a non-local system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on works by Allaire, we compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.

Authors: Matthieu Bonnivard (LJLL), Anne-Laure Dalibard (DMA), David Gérard-Varet (IMJ)
Category: Research article
Date: 19 février 2015

The goal of this paper is to describe the formation of Kelvin-Helmholtz instabilities at the interface of two fluids of different densities and the ability of vari-ous shallow water models to reproduce correctly the formation of these instabilities. Working first in the so called rigid lid case, we derive by a simple linear anal-ysis an explicit condition for the stability of the low frequency modes of the inter-face perturbation, an expression for the critical wave number above which Kelvin-Helmholtz instabilities appear, and a condition for the stability of all modes when surface tension is present. Similar conditions are derived for several shallow water asymptotic models and compared with the values obtained for the full Euler equa-tions. Noting the inability of these models to reproduce correctly the scenario of formation of Kelvin-Helmholtz instabilities, we derive new models that provide a perfect matching. A comparisons with experimental data is also provided. Moreover, we briefly discuss the more complex case where the rigid lid is re-placed by a free surface. In this configuration, it appears that some frequency modes are stable when the velocity jump at the interface is large enough; we explain why such stable modes do not appear in the rigid lid case.

Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in H2(ℝ).

Authors: T. Gallay
(Institut Fourier), D. Pelinovsky (U. Toronto)
Category: Research article
Date: 21 décembre 2014

We obtain existence and conormal Sobolev regularity of strong solutions to the 3D com-pressible isentropic Navier-Stokes system on the half-space with a Navier boundary condi-tion, over a time that is uniform with respect to the viscosity parameters when these are small. These solutions then converge globally and strongly in L 2 towards the solution of the compressible isentropic Euler system when the viscosity parameters go to zero.

This paper addresses the global existence problem of so-called κ-entropy solutions of Navier-Stokes equations for viscous compressible and barotropic fluids with degenerate viscosities. Such solutions satisfy in a weak sense the mass and momentum conservation equations and also a generalization of the BD-entropy identity called: κ-entropy. This new entropy involves a mixture parameter κ∈(0,1) between the two velocities u and u+2∇φ(ϱ) (the latter was introduced by the first two authors in [C. R. Acad. Sci. Paris 2004]), where u is the velocity field and φ is a function of the density ϱ defined by φ′(s)=μ′(s)/s. The assumption λ(ϱ)=2(μ′(ϱ)ϱ−μ(ϱ)) is also required as in previous works by the two first authors to establish the κ-entropy identity. As a byproduct of the existence proof, a two-velocity hydrodynamical model (in the spirit of works by S.C. Shugrin fnr istence) is shown to be included in the barotropic compressible flows with degenerate viscosities. It allows for the construction of approximate solutions based on an augmented approximate scheme of the compressible Navier-Stokes equations extending an approach developed for some inviscid compressible systems with capillarity by other authors.

Authors: Didier Bresch, Benoit Desjardins, Ewelina Zatorska
Category: Research article
Date: 20 novembre 2014

These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014. They are entirely devoted to the initial value problem and the long-time behavior of solutions for the two-dimensional incompressible Navier-Stokes equations, in the particular case where the domain occupied by the fluid is the whole plane ℝ2 and the velocity field is only assumed to be bounded. In this context, local well-posedness is not difficult to establish, and a priori estimates on the vorticity distribution imply that all solutions are global and grow at most exponentially in time. Moreover, as was recently shown by S. Zelik, localized energy estimates can be used to obtain a much better control on the uniformly local energy norm of the velocity field. The aim of these notes is to present, in an explanatory and self-contained way, a simplified and optimized version of Zelik's argument which, in combination with a new formulation of the Biot-Savart law for bounded vorticities, allows one to show that the uniform norm of the velocity field grows at most linearly in time. The results do not rely on the viscous dissipation, and remain therefore valid for the so-called "Serfati solutions" of the two-dimensional Euler equations. Finally, a recent work by S. Slijepcevic and the author shows that all solutions remain uniformly bounded in the viscous case if the velocity field and the pressure are periodic in one space direction.

In this paper we prove global in time existence of weak solutions to zero Mach number systems arising in fluid mechanics. Relaxing a certain algebraic constraint between the viscosity and the conductivity introduced in [D. Bresch, E.H. Essoufi, and M. Sy, J. Math. Fluid Mech. 2007] gives a more complete answer to an open question formulated in [P.-L. Lions, Oxford 1998]. A new mathematical entropy shows clearly the existence of two-velocity hydrodynamics with a fixed mixture ratio. As an application of our result we first discuss a model of gaseous mixture extending the results of [P. Embid, Comm. Partial Diff. Eqs. 1987] to the global weak solutions framework. Second, we present the ghost effect system studied by [C.D. Levermore, W. Sun, K. Trivisa, SIAM J. Math. Anal. 2012] and discuss a contribution of the density-dependent heat-conductivity coefficient to the issue of existence of weak solutions.

Authors: Didier Bresch, Vincent Giovangigli, Ewelina Zatorska
Category: Research article
Date: 20 novembre 2014

In this paper we consider the motion of a rigid body immersed in a two dimensional unbounded incompressible perfect fluid with vorticity. We prove that when the body shrinks to a massless pointwise particle with fixed circulation, the "fluid+rigid body" system converges to the vortex-wave system introduced by Marchioro and Pulvirenti in [11]. This extends both the paper [2] where the case of a solid tending to a massive pointwise particle was tackled and the paper [3] where the massless case was considered but in a bounded cavity filled with an irrotational fluid.

Authors: Olivier Glass,
Christophe Lacave, Franck Sueur
Category: Research article
Date: 23 octobre 2014

We approximate a two-phase model by the compressible Navier-Stokes equations with a singular pressure term. Up to a subsequence, these solutions are shown to converge to a global weak solution of the compressible system with the congestion constraint studied for instance by P.L. Lions and N. Masmoudi [Annales I.H.P., 1999]. The paper is an extension of the previous result obtained in one-dimensional setting by D. Bresch et al. [C. R. Acad. Sciences Paris, 2014] to the multi-dimensional case with heterogeneous barrier for the density.

In this paper we study the clockwise simply connected rotating patches for Euler equations. By using the moving plane method we prove that Rankine vortices are the only solutions to this problem in the class of {\it slightly} convex domains. We discuss in the second part of the paper the case where the angular velocity Ω=12 and we show without any geometric condition that the set of the V-states is trivial and reduced to the Rankine vortices.

We prove existence of doubly connected V-states for the planar Euler equations which are not annuli. The proof proceeds by bifurcation from annuli at simple "eigenvalues". The bifurcated V-states we obtain enjoy a m-fold symmetry for some m≥3. The existence of doubly connected V-states of strict 2-fold symmetry remains open.

Authors: Taoufik Hmidi, Francisco de la Hoz, Joan Mateu, Joan Verdera
Category: Research article
Date: 24 septembre 2014

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the Floquet-Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the periodic waves of small amplitudes.

We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $\varepsilon$ separated by distances of order $\varepsilon^\alpha$ for some fixed $\alpha$ and the fluid fills the exterior. In [4], we have obtained a value $\alpha_1$ such that if $\alpha < \alpha_1$, then the limit motion is not perturbed by the porous medium, namely we recover Euler in the whole space. Here, we establish that there exists $\alpha_2$ such that if $\alpha > \alpha_2$, an impermeable wall appears.

We study here Green-Naghdi type equations (also called fully nonlinear Boussinesq, or Serre equations) modeling the propagation of large amplitude waves in shallow water. The novelty here is that we allow for a general vorticity, hereby allowing complex interactions between surface waves and currents. We show that the a priori 2+1-dimensional dynamics of the vorticity can be reduced to a finite cascade of two-dimensional equations: with a mechanism reminiscent of turbulence theory, vorticity effects contribute to the averaged momentum equation through a Reynolds-like tensor that can be determined by a cascade of equations. Closure is obtained at the precision of the model at the second order of this cascade. We also show how to reconstruct the velocity field in the 2 + 1 dimensional fluid domain from this set of 2-dimensional equations and exhibit transfer mechanisms between the horizontal and vertical components of the vorticity, thus opening perspectives for the study of rip currents for instance.

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point (t0,x0) is the identically zero solution. Such conclusion holds provided the physical parameter γ of the model (related to the finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa--Holm equation, corresponding to γ=1. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincar\'e inequalities.

In this paper we consider the nonlinear dispersive wave equation on the real line, ut−utxx+[f(u)]x−[f(u)]xxx+[g(u)+f″(u)2u2x]x=0, that for appropriate choices of the functions f and g includes well known models, such as Dai's equation for the study of vibrations inside elastic rods or the Camassa--Holm equation modelling water wave propagation in shallow water. We establish a local-in-space blowup criterion (i.e., a criterion involving only the properties of the data u0 in a neighbourhood of a single point) simplifying and extending earlier blowup criteria for this equation. Our arguments apply both to the finite and infinite energy case, yielding the finite time blowup of strong solutions with possibly different behavior as x→+∞ and x→−∞.

The aim of this article is to provide a mathematical framework giving access to a better understanding of whistler-mode chorus emissions in space plasmas. There is presently a general agreement that the emissions of whistler waves involve a mechanism of wave-particle interaction which can be decribed in the framework of the relativistic Vlasov-Maxwell equations, with a penalized skew-symmetric term where the inhomogeneity of the magnetic eld plays an essential part. The description of the related phenomena is achieved in two stages. The first provides a new approach allowing to extend in longer times the classical insights on fast rotating fluids. The second is based on a study of oscillatory integrals implying special phases.

We deal with the local well-posedness theory for the two-dimensional inviscid Boussinesq system with rough initial data of Yudovich type. The problem is in some sense critical due to some terms involving Riesz transforms in the vorticity-density formulation. We give a positive answer for a special sub-class of Yudovich data including smooth and singular vortex patches. For the latter case we assume in addition that the initial density is constant around the singular part of the patch boundary.

We study the incompressible Navier-Stokes equations in the two-dimensional strip ℝ×[0,L], with periodic boundary conditions and no exterior forcing. If the initial velocity is bounded, we prove that the solution remains uniformly bounded for all times, and that the vorticity distribution converges to zero as t→∞. We deduce that, after a transient period, a laminar regime emerges in which the solution rapidly converges to a shear flow governed by the one-dimensional heat equation. Our approach is constructive and gives explicit estimates on the size of the solution and the lifetime of the turbulent period in terms of the initial Reynolds number.

We consider the motion of a rigid body immersed in a two-dimensional perfect fluid. The fluid is assumed to be irrotational and confined in a bounded domain. We prove that when the body shrinks to a point wise massless particle with fixed circulation, its dynamics in the limit is given by the point vortex equation. As a byproduct of our analysis we also prove that when the body shrinks with a fixed mass the limit equation is a second-order differential equation involving a Kutta-Joukowski-type lift force, which extends the result of [Glass O., Lacave C., Sueur F., On the motion of a small body immersed in a two dimensional incompressible perfect fluid. {Preprint 2011}, to appear in Bull. Soc. Math. France. {\tt arXiv:1104.5404}] to the case where the domain occupied by the solid-fluid system is bounded.

Authors: Olivier Glass, Alexandre Munnier, Franck Sueur
Category: Research article
Date: 21 février 2014

The rigid-lid approximation is a commonly used simplification in the study of density-stratified fluids in oceanography. Roughly speaking, one assumes that the displacements of the surface are negligible compared with interface displacements. In this paper, we offer a rigorous justification of this approximation in the case of two shallow layers of immiscible fluids with constant and quasi-equal mass density. More precisely, we control the difference between the solutions of the Cauchy problem predicted by the shallow-water (Saint-Venant) system in the rigid-lid and free-surface configuration. We show that in the limit of small density contrast, the flow may be accurately described as the superposition of a baroclinic (or slow) mode, which is well predicted by the rigid-lid approximation; and a barotropic (or fast) mode, whose initial smallness persists for large time. We also describe explicitly the first-order behavior of the deformation of the surface, and discuss the case of non-small initial barotropic mode.

This paper concerns spectral instability of shear flows in the incompressible Navier-Stokes equations with sufficiently large Reynolds number: R→∞. It is well-documented in the physical literature, going back to Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, that generic plane shear profiles other than the linear Couette flow are linearly unstable for sufficiently large Reynolds number. In this work, we provide a complete mathematical proof of these physical results. In the case of a symmetric channel flow, our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of et/αR√, where α is the small spatial frequency that remains between lower and upper marginal stability curves: αlow(R)≈R−1/7 and αup(R)≈R−1/11. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

In this paper we derive a new formulation of the water waves equations with vorticity that generalizes the well-known Zalkarov-Craig-Sulem formulation used in the irrotational case. We prove the local well-posedness of this formulation, and show that it is formally Hamiltonian. This new for- mulation is cast in Eulerian variable, and in finite depth; we show that it can be used to provide uniform bounds on the lifespan and on the norms of the solutions in the singular shallow water regime. As an application to these re- sults, we derive and provide the first rigorous justification of a shallow water model for water waves in presence of vorticity; we show in particular that a third equation must be added to the standard model to recover the velocity at the surface from the averaged velocity.

This paper is devoted to the well-posedness of the stationary 3d Stokes-Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of G\'erard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes-Coriolis operator at low tangential frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes-Coriolis system with data in the Kato space H1/2uloc.

Author: Anne-Laure Dalibard (DMA, CIMS), Christophe Prange
Category: Research article
Date: 30 janvier 2014

In this paper, we address the problem of weak solutions of Yudovich type for the inviscid MHD equations in two dimensions. The local-in-time existence and uniqueness of these solutions sound to be hard to achieve due to some terms involving Riesz transforms in the vorticity-current formulation. We shall prove that the vortex patches with smooth boundary offer a suitable class of initial data for which the problem can be solved. However this is only done under a geometric constraint by assuming the boundary of the initial vorticity to be frozen in a magnetic field line. We shall also discuss the stationary patches for the incompressible Euler system (E) and the MHD system. For example, we prove that a stationary simply connected patch with rectifiable boundary for the system (E) is necessarily the characteristic function of a disc.

We introduce a new class of two-dimensional fully nonlinear and weakly dispersive Green-Naghdi equations over varying topography. These new Green-Naghdi systems share the same order of precision as the standard one but have a mathematical structure which makes them much more suitable for the numerical resolution, in particular in the demanding case of two dimensional surfaces. For these new models, we develop a high order, well balanced, and robust numerical code relying on an hybrid finite volume and finite difference splitting approach. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a finite difference approach. Higher order accuracy in space and time is achieved through WENO reconstruction methods and through a SSP-RK time stepping. Particular effort is made to ensure positivity of the water depth. Numerical validations are then performed, involving one and two dimensional cases and showing the ability of the resulting numerical model to handle waves propagation and transformation, wetting and drying; some simple treatments of wave breaking are also included. The resulting numerical code is particularly efficient from a computational point of view and very robust; it can therefore be used to handle complex two dimensional configurations.