As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to have characteristic-function densities. The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: Any two shapes of equal volume can be approximately connected by an Euler spray---a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays. Each such Wasserstein geodesic is also the unique minimizer of a relaxed least-action principle for a two-fluid mixture theory corresponding to incompressible fluid mixed with vacuum.
We consider $L^2$ minimizing geodesics along the group of volume preserving maps $SDiff(D)$ of a given 3-dimensional domain $D$. The corresponding curves describe the motion of an ideal incompressible fluid inside $D$ and are (formally) solutions of the Euler equations. It is known that there is a unique possible pressure gradient for these curves whenever their end points are fixed. In addition, this pressure field has a limited but unconditional (internal) regularity. The present paper completes these results by showing: 1) the uniqueness property can be viewed as an infinite dimensional phe…
In this paper we show that the incompressible Euler equation on the Sobolev space $H^s(\R^n)$, $s > n/2+1$, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.
In this thesis we prove that the homogeneous incompressible Euler equation of hydrodynamics on the Sobolev spaces $H^s(\R^n)$, $n \geq 2$ and $s > n/2+1$, can be expressed as a geodesic equation on an infinite dimensional manifold. As an application of this geometric formulation we prove that the solution map of the incompressible Euler equation, associating intial data in $H^s(\R^n)$ to the corresponding solution at time $t > 0$, is nowhere locally uniformly continuous and nowhere differentiable.
In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists in embedding the incompressible Euler equation with a potential term coming from classical mechanics into incompressible Euler of a manifold and seeing the CH2 equation as a particular case of such fluid dynamic equation.
It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler's equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy …
We study Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field. Moreover, adapting some ideas of Shnirelman, we show that, even for non-deterministic final c…
The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in ${\rm SL}(2,{\mathbb R})$. In the case of perfect fluids, the motion is given by geodesic flow in ${\rm SL}(2,{\mathbb R})$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in ${\rm SL}(2,{\mathbb R})$. A complete classification of the dynamics is given including rigid motions, rotating edd…
There are not too many known explicit solutions to the 2-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the 19th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently -- in the 1980s -- obtained new explicit solutions with a similar feature. We present a more general method for constru…
A new proof is given of the fact that the particle trajectories of the ideal incompressible fluid are analytic curves, though the solutions of the Euler equations may have a finite regularity. This is a consequence of a general fact that the geodesic exponential map on the group of volume preserving diffeomorphisms belonging to the Sobolev space is real-analytic. The proof is based on the general properties of holomorphic maps in complex Banach spaces.
We consider the motion of several rigid bodies immersed in a two-dimensional incompress-ible perfect fluid, the whole system being bounded by an external impermeable fixed boundary. The fluid motion is described by the incompressible Euler equations and the motion of the rigid bodies is given by Newton's laws with forces due to the fluid pressure. We prove that, for smooth solutions, Newton's equations can be recast as a second-order ODE for the degrees of freedom of the rigid bodies with coefficients depending on the fluid vorticity and on the circulations around the bodies, but not anymore o…
We formulate a stochastic least-action principle for solutions of the incompressible Navier-Stokes equation, which formally reduces to Hamilton's principle for the incompressible Euler solutions in the case of zero viscosity. We use this principle to give a new derivation of a stochastic Kelvin Theorem for the Navier-Stokes equation, recently established by Constantin and Iyer, which shows that this stochastic conservation law arises from particle-relabelling symmetry of the action. We discuss issues of irreversibility, energy dissipation, and the inviscid limit of Navier-Stokes solutions in t…
It was proved by Karch and Pilarzyc that Landau solutions are asymptotically stable under any $L^2$-perturbation. In our earlier work with L. Li, we have classified all $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any $L^2$-perturbation.
We discuss general incompressible inviscid models, including the Euler equations, the surface quasi-geostrophic equation, incompressible porous medium equation, and Boussinesq equations. All these models have classical unique solutions, at least for short time. We show that they have real analytic Lagrangian paths. More precisely, we show that as long as a solution of any of these equations is in a class of regularity that assures Hölder continuous gradients of velocity, the corresponding Lagrangian paths are real analytic functions of time. The method of proof is conceptually straightforward …
We consider the Euler equations of incompressible fluids and attempt to solve the initial value problem with the help of a concave maximization problem.We show that this problem, which shares a similar structure with the optimal transport problemwith quadratic cost, in its "Benamou-Brenier" formulation,always admits a relaxed solution that can be interpretedin terms of $sub-solution$ of the Euler equations in the sense of convex integration theory.Moreover, any smooth solution of the Euler equations can be recovered from this maximization problem, at least for short times.
In this article, we investigate the geometry of a central extension $\widehat{\mathcal D}_μ(S^{2})$ of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with the $L^{2}$-metric, whose geodesics correspond solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misiolek curvature of this group. This value is related to the existence of a conjugate point and its positivity directly implies the positivity of the sectional curvature.
In this paper we discuss the incompressible limit for multicomponent fluids in the isothermal ideal case. Both a direct limit-passage in the equation of state and the low Mach-number limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocity-field under the condition that the incompressible model possesses a sufficiently smooth solution, which is granted at least for a short time. Moreover, in comparison to single-component flows, uniform estimates and the convergence of the pressure are needed in the multicom…
The group of diffeomorphisms of a compact manifold endowed with the L^2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein-Fisher-Rao distance is a natural extension of the classical L^2-Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the Hdiv right-invariant metric on the group of diffeomorphisms, wh…
We show that the widely used model governing the motion of two incompressible immiscible fluids in a possibly heterogeneous porous medium has a formal gradient flow structure. More precisely, the fluid composition is governed by the gradient flow of some non-smooth energy. Starting from this energy together with a dissipation potential, we recover the celebrated Darcy-Muskat law and the capillary pressure law governing the flow thanks to the principle of least action. Our interpretation does not require the introduction of any algebraic transformation like, e.g., the global pressure or the Kir…
For the $2D$ Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $L^\infty \cap H^1$ but escapes $H^1$ immediately for $t>0$. Our main observation is that a localized chunk of vorticity bounded in $L^\infty \cap H^1$ with odd-odd symmetry is able to generate a hyperbolic flow with large velocity gradient at least for a short period of time, which stretches the vorticity gradient.
We study in this paper the low Mach number limit for the 2d isentropic Euler system with ill-prepared initial data belonging to the critical Besov space $B_{2,1}^2$. By combining Strichartz estimates with the special structure of the vorticity we prove that the lifespan goes to infinity as the Mach number goes to zero. We also prove the strong convergence in the space of the initial data $B_{2,1}^2$ of the incompressible parts to the solution of the incompressible Euler system. There are at least two main difficulties: the first one concerns the Beale-Kato-Majda criterion which is not known to…
We consider a general class of parametrized displacement boundary value problems in incompressible nonlinear elasticity. We prove the existence of an unbounded solution branch of classical injective solutions emanating from the unforced stress-free reference configuration, which constitutes a sharp global implicit function theorem. The ramifications of this are: There exists at least one solution for any given applied loading, and or there exists solutions of arbitrarily large norm for some finite loading. We refer to this as solutions in the large. The nonlinear constraint equation enforcing …
This paper is devoted to the geometric analysis of the incompressible averaged Euler equations on compact Riemannian manifolds with boundary. The equation also coincides with the model for a second-grade non-Newtonian fluid. We study the analytical and geometrical properties of the Lagrangian flow map. We prove existence and uniqueness of smooth-in-time solutions for initial data in $H^s$, $s > n/2 +1$ by establishing the existence of smooth geodesics of a new weak right invariant metric on new subgroups of the volume-preserving diffeomorphism group. We establish smooth limits of zero viscosit…
We study the geodesic problem on the group of diffeomorphism of a domain M$\subset$Rd, equipped with the H(div) metric. The geodesic equations coincide with the Camassa-Holm equation when d=1, and represent one of its possible multi-dimensional generalizations when d>1. We propose a relaxation {à} la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M. We use this relaxation to prove that smooth H(div) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure fi…
We explore the existence of global weak solutions to the Hookean dumbbell model, a system of nonlinear partial differential equations that arises from the kinetic theory of dilute polymers, involving the unsteady incompressible Navier--Stokes equations in a bounded domain in two or three space dimensions, coupled to a Fokker--Planck-type parabolic equation. We prove the existence of large-data global weak solutions in the case of two space dimensions. Indirectly, our proof also rigorously demonstrates that, in two space dimensions at least, the Oldroyd-B model is the macroscopic closure of the…
Consider a diffusion-free passive scalar $θ$ being mixed by an incompressible flow $u$ on the torus $\mathbb{T}^d$. Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field.Our main result shows that the mix-norm ($||θ(t)||_{H^{-1}}$) is bounded below by an exponential function of time. The exponential decay rate we obtain is not universal and depends on the size of the support of the initial data. We also perform numerical simulations and confirm that the numerically observed decay rate scales similarly to the rigorous lower bound, at…
Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain D are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving diffeomorphisms of D. This problem admits a well-established convex relaxation which generates a set of "relaxed", "multi-stream", version of the Euler equations. However, it is unclear that such relaxed equations are appropriate for the initial value problem and the theory of turbulence, due to their lack of well-posedness for most initial data. As an atte…
In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic, group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface. We also develop a general framework for Euler-Arnold equations for …
We propose a spectral viscosity method (SVM) to approximate the incompressible Euler equations driven by a multiplicative noise. We show that SVM solution converges to a dissipative measure-valued martingale solution. These solutions are weak in the probabilistic sense i.e. the probability space and the driving Wiener process are an integral part of the solution. We also exhibit weak (measure-valued)-strong uniqueness principle. Moreover, we establish strong convergence of approximate solutions to the regular solution of the limit system at least on the lifespan of the latter, thanks to the we…
We investigate the influence of a perforated domain on the 2D Euler equations. Small inclusions of size $\varepsilon$ are uniformly distributed on the unit segment or a rectangle, and the fluid fills the exterior. These inclusions are at least separated by a distance $\varepsilon^α$ and we prove that for $α$ small enough (namely, less than 2 in the case of the segment, and less than 1 in the case of the square), the limit behavior of the ideal fluid does not feel the effect of the perforated domain at leading order when $\varepsilon\to 0$.
