We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in $\Bbb R^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.
In this brief note we show that the author's previous result in \cite{cha} on the nonexistence of self-similar singularities for the 3D incompressible Euler equations implies actually the nonexistence of `locally self-similar' singular solution, which has been sought by many physicists.
We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region which shrinks to a point dynamically as the time, $t$, approaches the singularity time, $T$. The solution outside the inner core region is assumed to be regular. Under the assumption that the local self-similar velocity profile converges to a limiting profile as $t \to T$ in $L^p$ for some $p \in (3,\infty)$,…
A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompresibility of the limit for the full Euler flow is from a combinatio…
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.
In this paper we introduce (I,J) similar method for incompressible two and three dimensional Euler equations and Navier-Stokes equations, obtain a series of explicit (I,J) similar solutions to the incompressible two dimensional Euler equations, they include all of twin wave solutions, some new singularity solutions, and some global smooth solutions with finite energy. We also discover that twin wave solution and affine solution to two dimensional incompressible Euler equations are respectively plane wave and constant vector. Finally, we supply some explicit piecewise smooth solutions to incomp…
They are possible other cases at first done not contemplate. This paper has been withdrawn by the author due to a crucial sign error in equation 1 page 4
We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for $s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length parameter introduced by P. Constantin in \cite{co1}. In particular, we derive lower bounds on the blowup rate of such solutions.
Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such effects to decrease with distance from the localized change, giving the solutions a type of stability. Indeed, this is the case for solutions having spatial decay, as can be easily shown. We consider the more difficult case of solutions lacking spatial decay, and show that such stability still holds, albeit in a somewhat weaker form.
We study the asymptotic regime of strong electric fields that leads from the Vlasov-Poisson system to the Incompressible Euler equations. We also deal with the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external field. In turn, the limiting equation is set in a bounded domain, the shape of which is determined by the external confining potential. The analysis extends to the situation where the limiting density is non-homogeneous and where the Euler equation is replaced…
In this article, I will report a Lax pair structure, a Backlund-Darboux transformation, and the investigation of homoclinic structures for 2D Euler equations of incompressible inviscid fluids.
Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given through the lens of infinite-dimensional dynamical systems, and various open problems are listed and discussed.
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.
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 $\varepsilon\to 0$.
In this paper we prove nonexistence of stationary weak solutions to the Euler-Poisson equations and the Navier-Stokes-Poisson equations in $\Bbb R^N$, $N\geq 2$, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler-Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier-Stokes-Poisson equations this asymptotic behavior of the density can be proved if we assume the standard energy inequal…
This paper studies the pressureless Euler-Poisson system and its fully non-linear counterpart, the Euler-Monge-Ampère system, where the fully non-linear Monge-Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.
For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x,y,z. In this paper, the Clarkson-Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.
A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open question in the affirmative when the initial data of fluid are well-prepared in a real analytic space, in 3D half space. As a key of this advance we capture the Navier-Stokes equations of $$\textit{viscosity} \sim \frac{\textit{Knudsen number}}{\textit{Mach number}}$$ satisfying the no-slip boundary condition, as an $\textit{intermediary}$ approximation of the Eu…
A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli arXiv:2006.14924 showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of $N$ particles interacting in $\mathbb{T}^d$, $d\geq 2$, via Newton's second law through a supercritical mean-field limit. Namely, the coupling constant $λ$ in front of the pair potential, which is Coulombic, scales like $N^{-θ}$ for some $θ\in (0,1)$, in contrast to the usual mean-field scaling $λ\sim N^{-1}$. As…
In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $\dot{B}^0_{1, \infty}(\Bbb R^3)$. For the Navier-Stokes equations the convergence of the velocity to the self-simila…
Incompressible flows of an ideal two-dimensional fluid on a closed orientable surface of positive genus are considered. Linear stability of harmonic, i.e. irrotational and incompressible, solutions to the Euler equations is shown using the Hodge-Helmholtz decomposition. We also demonstrate that any surface Euler flow is stable with respect to harmonic velocity perturbations.
We introduce a new concept of dissipative measure-valued martingale solutions to the stochastic compressible Euler equations. 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 derive the relative energy inequality for the stochastic compressible Euler equations and, as a corollary, we exhibit pathwise weak-strong uniqueness principle. Moreover, making use of the relative energy inequality, we investigate the low Mach limit (incompressible limit) of the underlying system of equations. As a main…
We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^α$ vorticities not necessarily vanishing on the boundary with any $0<α<1$, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of $\mathbb{R}^3$ via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in $\…
By DNS of Navier-Stokes turbulence, Goto-Saito-Kawahara (2017) showed that turbulence consists of a self-similar hierarchy of anti-parallel pairs of vortex tubes, in particular, stretching in larger-scale strain fields creates smaller-scale vortices. Inspired by their numerical result, we examine the Goto-Saito-Kawahara type of vortex-tubes behavior using the 3D incompressible Euler equations, and show that such behavior induces energy cascade in the absence of nonlinear scale-interaction. From this energy cascade, we prove a modified version of the zeroth-law. In the Appendix, we derive Kolmo…
This paper construct a family of explicit self-similar blowup axisymmetric solutions for the 3D incompressible Euler equations in R^3. Those singular solutions admit infinite energy.
For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem. In this paper we consider the physical dimensions $d=2,3$ and show that if we perturb any given smooth initial data in $H^{s_c}$ norm, then the corresponding solution can have infinite $H^{s_c}$ norm instantaneously at $t>0$. The constructed solutions are unique and even $C^{\infty}$-smooth in some cases. To prove these results we introduce a new strategy: la…
We consider self-similar solutions of the 2d incompressible Euler equations. We construct a class of solutions with vorticity forming algebraic spirals near the origin, in analogy to vortex sheets rolling up into algebraic spirals.
We consider various questions about the 2d incompressible Navier-Stokes and Euler equations on a torus when dissipation is removed from or added to some of the Fourier modes.
We give a geometric formulation of 3D incompressible Euler that contains the Eulerian and Lagrangian gauges as special cases. In the Lagrangian gauge, incompressible Euler is a real analytic ODE in Banach space; a short proof of this known result is given. We then describe (in a self-contained section) some basic properties of a hypothetical self-similar solution to 3D incompressible Euler.
In this paper, we show the existence of a family of compactly supported smooth vorticities, which are solutions of the 2D incompressible Euler equation and rotate uniformly in time and space.
In this paper, we prove propagation of chaos in the context of wave turbulence for a generic quasisolution. We then apply the result to full solutions to the incompressible Euler equation.
This article is a survey concerning the state-of-the-art mathematical theory of the Euler equations of incompressible homogenous ideal fluid. Emphasis is put on the different types of emerging instability, and how they may be related to the description of turbulence.
Singular vorticty solutions of the incompressible 3D-Euler equation are constructed which satisfy the BKM criterion (cf. [2]). The construction is done by inviscid limits of vorticity solutions of transformed incompressible Navier Stokes type equations with a damping potential term, where the latter equations admit a global regular solution for positive viscosity. The inviscid limit vorticity solution of the incompressible Euler vorticity equation becomes singular at a point of the boundary of a finite domain.
We give a new sufficient criteria to prove the uniqueness of the incompressible Euler equation in dimension $N\geq2$. In their celebrated works by V. Scheffer [18], A. Shnirelman [19], C. De Lellis and L. Székelyhidi Jr. [7] they have obtained the nonuniqeness of weak solutions of incompressible Euler equation. Here we obtain uniqueness criteria for the same equation under some mild regularity condition on weak solutions. Our proof is simple and can be employed to other equations like inhomogeneous incompressible Euler and Euler-Boussinesq equations. One of the key ingredients in our proof is …
Non-uniqueness of Regular Solutions for Incompressible Static Euler Equations with Given Boundary Conditions and Turbulent Global Solutions of Incompressible Navier-Stokes Equations
We establish energy-balance for weak solutions of the stochastically forced incompressible Euler equations, enjoying Hölder regularity $C^α$, $α>1/3$. It is well known as the Onsager's conjecture for the deterministic incompressible Euler equations, which describes the energy conservation of weak solutions having Hölder regularity $C^α$, $α>1/3$. Additionally, we obtain energy-balance for the inhomogeneous incompressible Euler system driven by a cylindrical Wiener process.
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.
The paper is devoted to the study of a stabilization problem for the 2D incompressible Euler system in an infinite strip with boundary controls. We show that for any stationary solution (c, 0) of the Euler system there is a control which is supported in a given bounded part of the boundary of the strip and stabilizes the system to (c, 0).
In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal magnitude and opposite signs. The results are obtained by using an improved vorticity method.
We prove the existence of time-quasi-periodic solutions of the incompressible Euler equation on the three-dimensional torus $\T^3$, with a small time-quasi-periodic external force. The solutions are perturbations of constant (Diophantine) vector fields, and they are constructed by means of normal forms and KAM techniques for reversible quasilinear PDEs.
In Part II of the paper, we prove linear instability of a certain class of radially symmetric flows of an ideal incompressible fluid in dimension two used in Part I
We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad-Shafranov equations and is inspired by the recent result \cite{gav}.
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.
We consider the three-dimensional incompressible free-boundary Euler equations in a bounded domain and with surface tension. Using Lagrangian coordinates, we establish a priori estimates for solutions with minimal regularity assumptions on the initial data.
We present a local existence result for the three dimensional incompressible Euler equations. The solution is constructed using a formulation of the equations as an active vector system in Eulerian coordinates. The formulation employs the inverse of the Lagrangian path map (the "back-to-labels" map) and its Eulerian gradient. We express sufficient conditions for regularity in terms of this gradient.
This work gathers new results concerning the semi-geostrophic equations: existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the incompressible Euler equations. Meanwhile, a general technique to prove uniqueness of sufficiently smooth solutions to non-linearly coupled system is introduced, using optimal transportation.
In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is locally integrable with values in the same Lebesgue space.
We are interested in the hydrodynamic limit of the Boltzmann equation for a gas of fermions in the incompressible Euler regime. We use the relative entropy method as improved by Saint-Raymond. Our result is analogous to what is obtained in the classical case, but the treatment is slightly complicated by the cubic nonlinearity of the collision operator.
In this paper we consider the incompressible Euler equation on the Sobolev space $H^s(\R^n)$, $s > n/2+1$, and show that for any $T > 0$ its solution map $u_0 \mapsto u(T)$, mapping the initial value to the value at time $T$, is nowhere locally uniformly continuous and nowhere differentiable.
We provide a simple proof that the Cauchy problem for the incompressible Euler equations in $\mathbb{R}^{d}$ with any $d\ge3$ is ill-posed in critical Sobolev spaces, extending an earlier work of Bourgain and Li in the case $d = 3$. The ill-posedness is shown for certain critical Lorentz spaces as well.
We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as $t\to \infty$. To our knowledge, t…
In the present paper, we study the fast rotation limit for the density-dependent incompressible Euler equations in two space dimensions with the presence of the Coriolis force. In the case when the initial densities are small perturbation of a constant profile, we show the convergence of solutions towards the solutions of a quasi-homogeneous incompressible Euler system. The proof relies on a combination of uniform estimates in high regularity norms with a compensated compactness argument for passing to the limit. This technique allows us to treat the case of ill-prepared initial data.
