Claude Bardos, Edriss S. Titi
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.
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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.
We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal 1D Euler equations.
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.
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.
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
研究不可压静态 Euler 方程在给定边界条件下的无穷多正则解与随机解，并证明其作为高 Reynolds 数 Navier-Stokes 流长时路径极限可刻画湍流。
Strong Ill-posedness of the 2d Incompressible Euler Equation in Critical Besov Spaces
本文证明二维不可压缩欧拉方程在临界Besov空间$B^{1}_{\infty, q}$（$1<q<\infty$）中速度场的强不适定性。
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.
Cluster of vortex helices in the incompressible $3$d Euler equations
该研究针对三维无粘不可压缩 Euler 方程，证明了多个螺旋丝状涡旋结构可相互坍缩，形成涡旋螺旋簇。
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.
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.
The paper has been withdrawn by the author due to a gap in Proof of Theorem 1.1.
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.
We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.
We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or Hölder continuous for any exponent $θ<\frac{1}{16}$. Using the techniques introduced in \cite{DS12} and \cite{DS12H}, we prove the existence of infinitely many (Hölder) continuous initial vector fields starting from which there exist infinitely many (Hölder) continuous solutions with preassigned total kinetic energy.
Using a recent result of C. De Lellis and L. Székelyhidi Jr. we show that, in the case of periodic boundary conditions and for dimension greater or equal 2, there exist infinitely many global weak solutions to the incompressible Euler equations with initial data $v_0$, where $v_0$ may be any solenoidal $L^2$-vectorfield. In addition, the energy of these solutions is bounded in time.
We construct an initial data for two-dimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known to be sharp - the double exponential growth is the fastest possible growth rate.
This paper has been withdrawn by the author due to a crucial definition error of Triebel space.
On local well-posedness of the stochastic incompressible density-dependent Euler equations
本文研究三维空间中随机非齐次不可压缩欧拉方程，通过将问题转化为随机问题并利用输运方程估计，证明了加性和乘性噪声下局部解的存在性与路径唯一性。
Regular and Singular Steady States of 2D incompressible Euler equations near the Bahouri-Chemin Patch
该研究针对二维环面上不可压缩Euler方程的稳态问题，围绕Bahouri-Chemin涡斑这一特殊奇异稳态，构造出收敛于该涡斑的光滑及奇异稳态解族，丰富了相关方程的稳态解类型。
We investigate multidimensional model for incompressible non-Newtonian fluids. Using method of energy estimates we prove the property of finite speed of propagations of the solution support for this problem. We find sharp bounds of the propagations by $L^2$--norm and $L^1$--norm of initial data.
Derivation of Yudovich solutions of Incompressible Euler from the Vlasov-Poisson system
本文通过调制能量方法，从Vlasov-Poisson方程推导出二维不可压缩欧拉方程，并提出一种能处理速度梯度仅为BMO的解的策略，与Yudovich定理相符。
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.
A Lagrangian Approach to the Inhomogeneous Incompressible Euler Equation
本文通过拉格朗日方法研究不均匀不可压缩欧拉方程的测地描述及其几何结构，并推导了其拉格朗日形式。
Local wellposedness for the free boundary incompressible Euler equations with interfaces that exhibit cusps and corners of nonconstant angle
研究自由边界不可压缩欧拉方程在界面含尖点和角点时的局部适定性，证明其解存在且唯一，并允许波峰角度随时间变化。
We consider weak stationary solutions to the incompressible Euler equations and show that the analogue of the h-principle obtained in [5, 7] for time-dependent weak solutions continues to hold. The key difference arises in dimension d = 2, where it turns out that the relaxation is strictly smaller than what one obtains in the time-dependent case.
Weak solutions to the steady incompressible Euler equations with source terms
本文基于凸积分格式将带源项的定常不可压缩Euler方程转化为微分包含，通过构造高频扰动平面波解，结合迭代与Baire纲论证证明了其定常弱解的非唯一性，且存在大量满足给定能量剖面的弱解。
In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted $L^2$- estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.
Energy conservations are studied for inhomogeneous incompressible and compressible Euler equations with general pressure law in a torus or a bounded domain. We provide sufficient conditions for a weak solution to conserve the energy. By exploiting a suitable test function, the spatial regularity for the density is only required to be of order $2/3$ in the incompressible case, and of order $1/3$ in the compressible case. When the density is constant, we recover the existing results for classical incompressible Euler equation.
For any positive regularity parameter $β< \frac 12$, we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in $H^β$ uniformly in time. In particular, we construct solutions which have an $L^2$-based regularity index \emph{strictly larger} than $\frac 13$, thus deviating from the $H^{\frac{1}{3}}$-regularity corresponding to the Kolmogorov-Obhukov $\frac 53$ power spectrum in the inertial range.
We utilize undetermined coefficient method and an iterative method to construct the series solutions of the 3D Cauchy problem for a class of incompressible Navier-Stokes and Euler Equations. Then we can turn the Navier-Stokes Equations (Euler Equations) into the Cauchy problem for finitely (infinitely) many ordinary differential equations. We get the finite series solution of the Navier-Stokes Equations. By using some combinatorial identities techniques, we prove that the sum of the solutions to these ordinary differential equations is an infinite series solution of the Euler Equations in some…
A class of singular 3D-velocity vector fields is constructed which satisfy the incompressible 3D-Euler equation. It is shown that such a solution scheme does not exist in dimension 2. The solutions constructed are bounded and smooth up to finite time where they become singular. Although the solution is smooth and bounded there seems to be no bound in L2 of the velocity field.
We consider the full Navier-Stokes-Fourier system in the singular regime of small Mach and large Reynolds and Peclet numbers, with ill prepared initial data on an unbounded domain in the three dimensional Euclidean space with a compact boundary. We perform the singular limit in the framework of weak solutions and identify the Euler-Boussinesq system as the target problem.
We prove that the 2D Euler equations are not locally well-posed in $C^1$. Our approach relies on the technique of Lagrangian deformations and norm inflation of Bourgain and Li. We show that the assumption that the data-to-solution map is continuous in $C^1$ leads to a contradiction with a well-posedness result in $W^{1,p}$ of Kato and Ponce.
In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschtiz function and the free surface belongs to $C^{\f32+\varepsilon}$. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.
Self-similar algebraic spiral vortex sheets of 2-D incompressible Euler equations
本文首次严格构造了二维不可压缩欧拉方程的自相似代数螺旋涡面解，并通过处理超出经典奇异积分理论的柯西积分，刻画了涡面曲率奇异性形成后的典型卷起模式。
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 paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with initial data in the Schwartz class develop non-trivial spatial asymptotic expansions of the type considered here.
We study the multiphasic formulation of the incompressible Euler equation introduced by Brenier: infinitely many phases evolve according to the compressible Euler equation and are coupled through a global in-compressibility constraint. We are able to prove that the entropy, when averaged over all phases, is a convex function of time, a result that was conjectured by Brenier. The novelty in our approach consists in introducing a time-discretization that allows us to import a flow interchange inequality previously used by Matthes, McCann and Savar{é} to study first order in time PDE, namely the …
Stability of a class of exact solutions of the incompressible Euler equation in a disk
针对圆盘内二维不可压缩欧拉方程中以第一类贝塞尔函数表示的精确定常解，通过守恒量变分特征与紧性论证，证明其严格轨道稳定性，涵盖截断兰姆偶极子特例。
The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \\ \mathrm{div}_g u &= 0, \end{align*} where $u: [0,T] \to Γ(T M)$ is the velocity field and $p: [0,T] \to C^\infty(M)$ is the pressure field. In this paper we show that if one is permitted to extend the base manifold $M$ by taking an arbitrary warped product with a torus, then the space of solutions to this equation becomes "non-rigid'"in the sense that a non-empty open set of smooth incompressible flows $u: [0,T] \to Γ(T M)$ can be approxim…
Euler-Leray data functions of first and second order are defined by first and second order derivatives of the nonlinear spatial part of the incompressible Euler equation operator in Leray projection form applied to Cauchy data. The Lipschitz continuity of these functions for a certain class of strongly regular Cauchy data is sufficient for the existence of global regular upper bounds of incompressible Navier Stokes equation solutions. Global regular upper bounds of global solution branches of the incompressible Euler equation can be obtained for a class of strongly regular Cauchy data, if the …
Recently, De Lellis and Székelyhidi constructed Hölder continuous, dissipative (weak) solutions to the incompressible Euler equations in the torus $\mathbb T^3$. The construction consists in adding fast oscillations to the trivial solution. We extend this result by establishing optimal h-principles in two and three space dimensions. Specifically, we identify all subsolutions (defined in a suitable sense) which can be approximated in the $H^{-1}$-norm by exact solutions. Furthermore, we prove that the flows thus constructed on $\mathbb T^3$ are genuinely three-dimensional and are not trivially …
In this paper we consider the locally backward self-similar solutions for the Euler system, and focus on the case that the possible nontrivial velocity profiles have non-decaying asymptotics. We derive the meaningful representation formula of the pressure profile in terms of velocity profiles in this case, and by using it and the local energy inequality of profiles, we prove some nonexistence results and show the energy behavior concerning the possible velocity profiles.
We study behaviors of scalar quantities near the possible blow-up time, which is made of smooth solutions of the Euler equations, Navier-Stokes equations and the surface quasi-geostrophic equations. Integrating the dynamical equations of the scaling invariant norms, we derive the possible blow-up behaviors of the above quantities, from which we obtain new type of blow-up criteria and some necessary conditions for the blow-up.
On concentrated vortices of 3D incompressible Euler equations under helical symmetry: with swirl
本文通过变分方法构造了带旋度的三维不可压缩欧拉方程在螺旋对称下的集中螺旋涡旋，并证明其渐近收敛于涡丝。
We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions, and study the potential formation of a finite-time singularity under certain convexity conditions for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent numerical s…
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