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Daniel Coutand

Publications and source records attributed to Daniel Coutand.

18 recordsLinked to original sources
arXiv2024

Global existence and convergence near equilibrium for the moving interface problem between Navier-Stokes and the linear wave equation

We first establish existence for all positive time near equilibrium for the moving interface problem between the Navier-Stokes equations for the evolving fluid phase (moved by the fluid velocity) and an elastic body modelled by the linear wave equation. This problem has an infinite number of simple solutions with a flat interface (with zero velocity in the fluid, and zero horizontal velocity in the solid), that we call flat interface solutions. We then show that if the initial data is close enough to the canonical equilibrium, the solution converges towards a flat interface solution in large time, showing that these flat interface solutions capture the long time behaviour of this fluid-structure problem near the canonical equilibrium. This result is established with gravity (which can be set to zero or not). It is established for the case where the solid has initial volume close to the volume of its reference configuration (where the linear wave equation is naturally written).

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arXiv2017

Finite time singularity formation for moving interface Euler equations

This paper proposes a new general methodology for finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. The first problem considered is the two-phase Euler vortex sheets problem with surface tension for which is proved the finite time singularity of the natural norm of the problem for suitable initial data. This is in striking contrast with the case of finite time splash and splat singularity formation for the one phase Euler equations introduced in [4] and studied in a more general context in [8], for which the natural norm (in the one phase fluid) stays finite all the way until contact. The second problem considered involves the presence of a heavier rigid body moving in the perfect fluid. We first establish that the rigid body will hit the bottom of the fluid domain in finite time (in a more general context and very different methodology than first done for this problem in [19]). We next establish that a surface energy blows up, and characterize the acceleration of the rigid body at contact, depending on the nature of the contact zone: Acceleration opposes the motion at contact, and is either positive finite if the contact zone contains a curve, or infinite otherwise.

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arXiv2016

On the impossibility of finite-time splash singularities for vortex sheets

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, i.e. for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false.

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arXiv2015

Regularity of the velocity field for Euler vortex patch evolution

We consider the vortex patch problem for both the 2-D and 3-D incompressible Euler equations. In 2-D, we prove that for vortex patches with $H^{k-0.5}$ Sobolev-class contour regularity, $k \ge 4$, the velocity field on both sides of the vortex patch boundary has $H^k$ regularity for all time. In 3-D, we establish existence of solutions to the vortex patch problem on a finite-time interval $[0,T]$, and we simultaneously establish the $H^{k-0.5}$ regularity of the two-dimensional vortex patch boundary, as well as the $H^k$ regularity of the velocity fields on both sides of vortex patch boundary, for $k \ge 3$.

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arXiv2015

On the splash singularity for the free-surface of a Navier-Stokes fluid

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for $d$-dimensional flows, $d=2$ or $3$, the free-surface of a viscous water wave, modeled by the incompressible Navier-Stokes equations with moving free-boundary, has a finite-time splash singularity. In particular, we prove that given a sufficiently smooth initial boundary and divergence-free velocity field, the interface will self-intersect in finite time.

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arXiv2013

On the finite-time splash and splat singularities for the 3-D free-surface Euler equations

We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time "splash" (or "splat") singularity first introduced in [9], wherein the evolving 2-D hypersurface, the moving boundary of the fluid domain, self-intersects at a point (or on surface). Such singularities can occur when the crest of a breaking wave falls unto its trough, or in the study of drop impact upon liquid surfaces. Our approach is founded upon the Lagrangian description of the free-boundary problem, combined with a novel approximation scheme of a finite collection of local coordinate charts; as such we are able to analyze a rather general set of geometries for the evolving 2-D free-surface of the fluid. We do not assume the fluid is irrotational, and as such, our method can be used for a number of other fluid interface problems, including compressible flows, plasmas, as well as the inclusion of surface tension effects.

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arXiv2012

Global existence and decay for solutions of the Hele-Shaw flow with injection

We study the global existence and decay to spherical equilibrium of Hele-Shaw flows with surface tension. We prove that without injection of fluid, perturbations of the sphere decay to zero exponentially fast. On the other hand, with a time-dependent rate of fluid injection into the Hele-Shaw cell, the distance from the moving boundary to an expanding sphere (with time-dependent radius) also decays to zero but with an algebraic rate, which depends on the injection rate of the fluid.

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arXiv2012

Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit

We prove that the 3-D compressible Euler equations with surface tension along the moving free-boundary are well-posed. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and Friedrichs as $p(ρ) = αρ^ γ- β$ for consants $γ>1$ and $ α, β> 0$. The analysis is made difficult by two competing nonlinearities associated with the potential energy: compression in the bulk, and surface area dynamics on the free-boundary. Unlike the analysis of the incompressible Euler equations, wherein boundary regularity controls regularity in the interior, the compressible Euler equation require the additional analysis of nonlinear wave equations generating sound waves. An existence theory is developed by a specially chosen parabolic regularization together with the vanishing viscosity method. The artificial parabolic term is chosen so as to be asymptotically consistent with the Euler equations in the limit of zero viscosity. Having solutions for the positive surface tension problem, we proceed to obtain a priori estimates which are independent of the surface tension parameter. This requires choosing initial data which satisfy the Taylor sign condition. By passing to the limit of zero surface tension, we prove the well-posedness of the compressible Euler system without surface on the free-boundary, and without derivative loss.

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arXiv2010

Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum

We prove well-posedness for the 3-D compressible Euler equations with moving physical vacuum boundary, with an equation of state given by the so-called gamma gas-law for gamma > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss--Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a new higher-order Hardy-type inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.

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arXiv2009

Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum

The free-boundary compressible 1-D Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws which are both characteristic and degenerate. The physical vacuum singularity (or rate-of-degeneracy) requires the sound speed $c= γρ^{γ-1}$ to scale as the square-root of the distance to the vacuum boundary, and has attracted a great deal of attention in recent years. We establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary. The proof is founded on a new higher-order Hardy-type inequality in conjunction with an approximation of the Euler equations consisting of a particular degenerate parabolic regularization. Our regular solutions can be viewed as degenerate viscosity solutions.

math.AP↗
arXiv2009

A priori estimates for the free-boundary 3-D compressible Euler equations in physical vacuum

We prove a priori estimates for the three-dimensional compressible Euler equations with moving {\it physical} vacuum boundary, with an equation of state given by $p(ρ) = C_γρ^γ$ for $γ>1$. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic {\it free-boundary} system to which standard methods of symmetrizable hyperbolic equations cannot be applied.

math.AP↗
arXiv2006

Navier-Stokes equations interacting with a nonlinear elastic fluid shell

We study a moving boundary value problem consisting of a viscous incompressible fluid moving and interacting with a nonlinear elastic fluid shell. The fluid motion is governed by the Navier-Stokes equations, while the fluid shell is modeled by a bending energy which extremizes the Willmore functional and a membrane energy that extremizes the surface area of the shell. The fluid flow and shell deformation are coupled together by continuity of displacements and tractions (stresses) along the moving material interface. We prove existence and uniqueness of solutions in Sobolev spaces.

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arXiv2006

Navier-Stokes equations interacting with a nonlinear elastic solid shell

We study a moving boundary value problem consisting of a viscous incompressible fluid moving and interacting with a nonlinear elastic solid shell. The fluid motion is governed by the Navier-Stokes equations, while the shell is modeled by the nonlinear Koiter shell model, consisting of both bending and membrane tractions. The fluid is coupled to the solid shell through continuity of displacements and tractions (stresses) along the moving material interface. We prove existence and uniqueness of solutions in Sobolev spaces.

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arXiv2006

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order.

math.AP↗
arXiv2005

On the interaction between quasilinear elastodynamics and the Navier-Stokes equations

The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems.

math.AP↗
arXiv2004

On the motion of an elastic solid inside of an incompressible viscous fluid

The motion of an elastic solid inside of an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic and hyperbolic phases, the latter inducing a loss of regularity which has left the basic question of existence open until now. In this paper, we prove the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. Our functional framework is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.

math.AP↗
arXiv2003

Unique solvability of the free-boundary Navier-Stokes equations with surface tension

We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the nonlinear iteration process.

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