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Franck Sueur

Publications and source records attributed to Franck Sueur.

50 records · Page 3Linked to original sources
arXiv2012

Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations

It was recently proven by De Lellis, Kappeler, and Topalov that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space Lip (T) endowed with the topology of H^1 (T). We prove here that the Lagrangian flow of these solutions are analytic with respect to time and smooth with respect to the initial data. These results can be adapted to the higher-order Camassa-Holm equations describing the exponential curves of the manifold of orientation preserving diffeomorphisms of T using the Riemannian structure induced by the Sobolev inner product H^l (T), for l in N, l > 1 (the classical Camassa-Holm equation corresponds to the case l=1): the periodic Cauchy problem is locally well-posed in the space W^{2l-1,infty} (T) endowed with the topology of H^{2l-1} (T) and the Lagrangian flows of these solutions are analytic with respect to time with values in W^{2l-1,infty} (T) and smooth with respect to the initial data. These results extend some earlier results which dealt with more regular solutions. In particular our results cover the case of peakons, up to the first collision.

math.AP↗
arXiv2011

A 2d spray model with gyroscopic effects

In this paper we introduce a PDE system which aims at describing the dynamics of a dispersed phase of particles moving into an incompressible perfect fluid, in two space dimensions. The system couples a Vlasov-type equation and an Euler-type equation: the fluid acts on the dispersed phase through a gyroscopic force whereas the latter contributes to the vorticity of the former. First we give a Dobrushin type derivation of the system as a mean-field limit of a PDE system which describes the dynamics of a finite number of massive pointwise particles moving into an incompressible perfect fluid. This last system is itself inferred from a joint work of the second author with O. Glass and C. Lacave, where the system for one massive pointwise particle was derived as the limit of the motion of a solid body when the body shrinks to a point with fixed mass and circulation. Then we deal with the well-posedness issues including the existence of weak solutions. Next we exhibit the Hamiltonian structure of the system and finally, we study the behavior of the system in the limit where the mass of the particles vanishes.

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arXiv2011

On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data

A famous result by Delort about the two-dimensional incompressible Euler equations is the existence of weak solutions when the initial vorticity is a diffuse bounded Radon measure with distinguished sign. In this paper we are interested in the case where there is a rigid body immersed in the fluid moving under the action of the fluid pressure. We succeed to prove the existence of solutions à la Delort in a particular case. These solutions satisfy the energy inequality and the body acceleration is bounded.

math.AP↗
arXiv2011

A Kato type Theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body

The issue of the inviscid limit for the incompressible Navier-Stokes equations when a no-slip condition is prescribed on the boundary is a famous open problem. A result by Tosio Kato says that convergence to the Euler equations holds true in the energy space if and only if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes. Of course, if one considers the motion of a solid body in an incompressible fluid, with a no-slip condition at the interface, the issue of the inviscid limit is as least as difficult. However it is not clear if the additional difficulties linked to the body's dynamic make this issue more difficult or not. In this paper we consider the motion of a rigid body in an incompressible fluid occupying the complementary set in the space and we prove that a Kato type condition implies the convergence of the fluid velocity and of the body velocity as well, what seems to indicate that an answer in the case of a fixed boundary could also bring an answer to the case where there is a moving body in the fluid.

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arXiv2011

Boundary layers interactions in the plane parallel incompressible flows

We study the inviscid limit problem of the incompressible flows in the presence of both impermeable regular boundaries and a hypersurface transversal to the boundary across which the inviscid flow has a discontinuity jump. In the former case, boundary layers have been introduced by Prandtl as correctors near the boundary between the inviscid and viscous flows. In the latter case, the viscosity smoothes out the discontinuity jump by creating a transition layer which has the same amplitude and thickness as the Prandtl layer. In the neighborhood of the intersection of the impermeable boundary and of the hypersurface, interactions between the boundary and the transition layers must then be considered. In this paper, we initiate a mathematical study of this interaction and carry out a strong convergence in the inviscid limit for the case of the plane parallel flows introduced by Di Perna and Majda in \cite{DM}.

math.AP↗
arXiv2011

On the motion of a rigid body in a two-dimensional irregular ideal flow

We consider the motion of a rigid body immersed in an ideal flow occupying the plane, with bounded initial vorticity. In that case there exists a unique corresponding solution which is global in time, in the spirit of the famous work by Yudovich for the fluid alone. We prove that if the body's boundary is Gevrey then the body's trajectory is Gevrey. This extends the previous work [Glass-Sueur-Takahashi, 2011] to a case where the flow is irregular.

math.AP↗
arXiv2011

On the motion of a small body immersed in a two dimensional incompressible perfect fluid

In this paper we prove that the motion of a solid body in a two dimensional incompressible perfect fluid converges, when the body shrinks to a point with fixed mass and circulation, to a variant of the vortex-wave system where the vortex, placed in the point occupied by the shrunk body, is accelerated by a lift force similar to the Kutta-Joukowski force of the irrotational theory.

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arXiv2011

The movement of a solid in an incompressible perfect fluid as a geodesic flow

The motion of a rigid body immersed in an incompressible perfect fluid which occupies a three- dimensional bounded domain have been recently studied under its PDE formulation. In particular classical solutions have been shown to exist locally in time. In this note, following the celebrated result of Arnold concerning the case of a perfect incompressible fluid alone, we prove that these classical solutions are the geodesics of a Riemannian manifold of infinite dimension, in the sense that they are the critical points of an action, which is the integral over time of the total kinetic energy of the fluid-rigid body system.

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arXiv2010

Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain of R^2, for flows with bounded vorticity, for which Yudovich proved, in 1963, global existence and uniqueness of the solution. We prove that if the boundary of the domain is C^infty (respectively Gevrey of order M > 1) then the trajectories of the fluid particles are C^infty (resp. Gevrey of order M + 2). Our results also cover the case of "slightly unbounded" vorticities for which Yudovich extended his analysis in 1995. Moreover if in addition the initial vorticity is Holder continuous on a part of the domain then this Holder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.

math.AP↗
arXiv2010

Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the Holder space C^{1,r}. In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic (till the classical solution exists and till the solid does not hit the boundary). Moreover in this case this motion depends smoothly on the initial data.

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arXiv2008

Vorticity internal transition layers for the Navier-Stokes equations

We deal with the incompressible Navier-Stokes equations, in two and three dimensions, when some vortex patches are prescribed as initial data i.e. when there is an internal boundary across which the vorticity is discontinuous. We show -thanks to an asymptotic expansion- that there is a sharp but smooth variation of the fluid vorticity into a internal layer moving with the flow of the Euler equations; as long as this later exists and as t * nu is small, where nu is the viscosity coefficient.

math.AP↗
arXiv2006

On the ferromagnetism equations with large variations solutions

We exhibit some large variations solutions of the Landau-Lifschitz equations as the exchange coefficient ε^2 tends to zero. These solutions are described by some asymptotic expansions which involve some internals layers by means of some large amplitude fluctuations in a neighborhood of width of order ε of an hypersurface contained in the domain. Despite the nonlinear behaviour of these layers we manage to justify locally in time these asymptotic expansions.

math.AP↗
arXiv2005

Entropy boundary layers

We consider the Euler system of compressible and entropic gaz dynamics in a bounded open domain with wall boundary condition. We prove the existence and the stability of families of solutions which correspond to a ground state plus a large entropy boundary layer. The ground state is a solution of the Euler system which satisfies some explicit additional conditions on the boundary. These conditions are used in a reduction of the system. We construct BKW expansions at all order. The profile problems are linear thanks to a transparency property. We prove the stability of these expansions by proving epsilon-conormal estimates for a characteristic boundary value problem.

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