Diego Cordoba, Charles Fefferman
We study the rate of growth of sharp fronts of the Quasi-geostrophic equation and 2D incompressible Euler equations.. The development of sharp fronts are due to a mechanism that piles up level sets very fast. Under a semi-uniform collapse, we obtain a lower bound on the minimum distance between the level sets.
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We introduce a model for the two-dimensional Euler equations which is designed to study whether or not double exponential growth can be achieved at an interior point of the flow.
Support growth of vorticity for bi-rotational Euler flows in high dimensions
该研究在高维空间中探讨了双旋转对称性下欧拉方程的涡量支持增长，证明了特定初始条件下的涡量支持直径无限增长。
Maximal Double-Exponential Growth for the Euler Equation on the Half-Plane
本文研究半平面 Euler 方程光滑解的涡量梯度增长，证明其可达双指数级，确定最大增长率并构造饱和解。
This paper has been withdrawn by the author due to a crucial definition error of Triebel space.
For any $A > 2$, we construct solutions to the two-dimensional incompressible Euler equations on the torus $\mathbb{T}^2$ whose vorticity gradient $\nablaω$ grows exponentially in time: $$\|\nablaω(t, \cdot)\|_{L^\infty} \gtrsim e^{At},\quad \forall\ t \geq 0.$$
Growth of vorticity gradient for the Euler equation on the sphere
研究欧拉方程在球面上涡度梯度的最大双指数增长，并通过构造具有奇对称性的显式解证明该上界是紧致的，同时将结果扩展到旋转球体的情况。
We prove that there are solutions to the Euler equation on the torus with $C^{1,α}$ vorticity and smooth except at one point such that the vorticity gradient grows in $L^\infty$ at least exponentially as $t\to\infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Sverak.
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.
We consider the incompressible 2D Euler equation in an infinite cylinder $\mathbb{R}\times \mathbb{T}$ in the case when the initial vorticity is non-negative, bounded, and compactly supported. We study $d(t)$, the diameter of the support of vorticity, and prove that it allows the following bound: $d(t)\leq Ct^{1/3}\log^2 t$ when $t\rightarrow\infty$.
We consider the 3D axisymmetric Euler equations without swirl on some bounded axial symmetric domains. In this setting, well-posedness is well known due to the essentially 2D geometry. The quantity $ω^θ/r$ plays the role of vorticity in 2D. First, we prove that the gradient of $ω^θ/r$ can grow at most double exponentially with improving a priori bound close to the axis of symmetry. Next, on the unit ball, we show that at the boundary, one can achieve double exponential growth of the gradient of $ω^θ/r$.
By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li. Under some additional assumption on the vorti…
We consider smooth, double-odd solutions of the two-dimensional Euler equation in $[-1, 1)^2$ with periodic boundary conditions. It is tempting to think that the symmetry in the flow induces possible double-exponential growth in time of the vorticity gradient at the origin, in particular when conditions are such that the flow is "hyperbolic". This is because examples of solutions with $C^{1, γ}$-regularity were already constructed with exponential gradient growth by A. Zlatos. We analyze the flow in a small box around the origin in a strongly hyperbolic regime and prove that the compression of…
Growth rates for anti-parallel vortex tube Euler flows in three and higher dimensions
研究三维及更高维度中无旋对称欧拉方程解的增长下界，改进了Choi和Jeong在三维中的结果。
We consider the Euler equations on the two-dimensional torus and construct invariant measures for the dynamics of these equations, concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist. The proof follows the method of Kuksin and we obtain in particular that these measures do not have atoms, excluding trivial invariant measures. Then we prove that almost every initial data with respect to the constructed measures give rise to global solutions for which the growth of the Sobolev norms are at most polynomial. To do this, we rely on an argument of Bou…
Euler-Lagrange equations for variable-growth total variation
针对Musielak-Orlicz型变增长积分泛函（含p(x)幂型、p=1双相型），通过Young不等式定义Anzellotti乘积变体确定其L²次微分并给出局部表征，进而得到含变增长正则项的ROF图像去噪变体的欧拉-拉格朗日方程。
Remarks on Linear Growth of Vorticity Gradients and Support Diameters for 2D Euler Flow in Half-Plane
本文通过奇对称设定下的任意背景扰动原理，证明对于半平面中任意紧支撑非负函数，存在任意小的光滑非负扰动，使得其解在象限内经历时间线性纤维化，从而支持了二维欧拉方程一般光滑解中涡度梯度线性增长的猜想。
We consider the two-dimensional Euler equations in non-smooth domains with corners. It is shown that if the angle of the corner $θ$ is strictly less than $π/2$, the Lipschitz estimate of the vorticity at the corner is at most single exponential growth and the upper bound is sharp. %near the stagnation point. For the corner with the larger angle $π/2 < θ<2π$, $θ\neq π$, we construct an example of the vorticity which loses continuity instantaneously. For the case $θ\le π/2$, the vorticity remains continuous inside the domain. We thus identify the threshold of the angle for the vorticity maintain…
Superlinear gradient growth for 2D Euler equation without boundary
本文在环面和全平面中研究二维欧拉方程涡量梯度的超线性增长，通过构造接近轨道稳定鞍点稳态的初值及利用Lamb-Chaplygin偶极子扰动，首次在开集光滑初值和平面紧支撑涡量情形下证明超线性增长。
On the upper bound for the vorticity growth of bi-rotational Euler flows without swirl
针对d≥4的无旋双旋转欧拉流，研究得到Yudovich型解的局部适定性及d≤6时的全局适定性，明确其涡量增长速率与无旋轴对称流一致。
A famous theorem of Bers and Finn states that isolated singularities of solutions to the non-parametric minimal surface equation are removable. We show that this result remains valid, if the area functional is replaced by a general functional of linear growth depending on the modulus of the gradient. We emphasize that Serrin ([1]) in fact proved the removability of singularities on sets of $(n-1)$-dimensional Hausdorff measure zero in an even more general setting. Our main interest is to generalize the comparison principles as outlined, for instance, in Section 10 of [2] without having the par…
It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. We study here the Euler vortex patch in a disk. We prove global in time regularity by providing the upper bound of the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that such upper bound is qualitatively sharp.
For two-dimensional Euler equation on the torus, we prove that the uniform norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of the gradient for the finite time.
In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of Kull-Anisimov [14]. This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by L0 an associated characteristic length. The system of equations is linearized around a stationary solution, and each perturbed quantity is written using the normal modes method. The resultin…
It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this paper, we study Euler vortex patches in a general smooth bounded domain. We prove global in time regularity by providing an upper bound on the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.
Growth estimates for axisymmetric Euler equations without swirl
本文研究了无旋轴对称欧拉方程中解的增长行为，证明了径向矩和涡度范数的幂律增长，并首次建立了光滑初始涡度的$L^p$-范数增长结果。
The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_ΩF(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(Ω, \mathbb R^N )\cap K$, where $K$ is a closed convex subset of the Dirichlet class $\mathrm W^{k,1}_{g}(Ω, \mathbb R^N ),$ are characterised as the energy solutions to the Euler-Lagrange inequality for $\mathscr F$. We assume that the essentially smooth integrand $F\colon \mathbb R^{N} \otimes \odot^{k}\mathbb R^{n} \to \mathbb R\cup\{+\infty\}$ is convex, lower semi-continuous, proper and at least super-linear at infinity…
For the two-dimensional Euler equation on the torus, we prove that the uniform norm of the vorticity gradient can grow as double exponential over arbitrarily long but finite time provided that at time zero it is already sufficiently large. Our result is equivalent to the statement that the Euler evolutions is linearly unbounded in Lipschitz norm for any time t>0.
In this note we study locally self-similar blow up for the Euler equation. The main result states that under a mild $L^p$-growth assumption on the profile $v$, namely, $\int_{|y| \sim L} |v|^p dy \lesssim L^{\g}$ for some $\g <p-2$, the self-similar solution carries a positive amount of energy up to the time of blow-up $T$, namely, $\int_{|y| \sim L} |v|^2 dy \sim L^{N-2\a}$. The result implies and extends several previously known exclusion criteria. It also supports a general conjecture relating fractal local dimensions of the energy measure with the rate of velocity growth at the time of pos…
We study the Euler-Poincaré equations that are the regularized Euler equations derived from the Euler-Poincaré framework. It is noteworthy to remark that the Euler-Poincaré equations are a generalization of two well-known regularizations, the vortex blob method and the Euler-$α$ equations. We show the global existence of a unique weak solution for the two-dimensional (2D) Euler-Poincaré equations with the initial vorticity in the space of Radon measure. This is a remarkable feature of these equations since the existence of weak solutions with the Radon measure initial vorticity has not been es…
We consider the limit $α\to0$ for the $α$-Euler equations in a two-dimensional bounded domain with Dirichlet boundary conditions. Assuming that the vorticity is bounded in $L^p$, we prove the existence of a global solution and we show the convergence towards a solution of the incompressible Euler equation with $L^p$ vorticity. The domain can be multiply-connected. We also discuss the case of the second grade fluid when both $α$ and $ν$ go to 0.
We consider the incompressible Euler equations on ${\mathbb R}^d$, where $d \in \{ 2,3 \}$. We prove that: (a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius). (b) In Lagrangian coordinates the equations are well-posed in highly anisotropic spaces, e.g.~Gevrey-class regularity in the label $a_1$ and Sobolev regularity in the labels $a_2,...,a_d$. (c) In Eulerian coordinates both results (a) and (b) above are false.
We give a determination of the equivalence group of Euler-Bernoulli equation and of one of its generalizations, and thus derive some symmetry properties of this equation.
By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the …
In this paper we prove that solutions of the 2D Euler equations in vorticity formulation obtained via vanishing viscosity approximation are renormalized.
After reformulate the incompressible Euler-$α$ equations in 3D smooth domain with Drichlet data, we obtain the unique classical solutions to Euler-$α$ equations exist in uniform time interval independent of $α$. We also show the solution of the Euler-$α$ converge to the corresponding solution of Euler equation in $L^2$ in space, uniformly in time. In the sequel, it follows that the $H^s$ $(s>\frac{n}{2}+1)$ solutions of Euler-$α$ equations exist in any fixed sub-interval of the maximum existent interval for the Euler equations provided that initial is regular enough and $α$ is small sufficient…
In this article, we will report the recent developments on Lax pairs and Darboux transformations for Euler equations of inviscid fluids.
This notes gives several criteria which exclude the existence of discretely self-similar solutions of the three dimensional incompressible Euler equations.
In this paper we study the weighted Gevrey class regularity of Euler equation in the whole space R 3. We first establish the local existence of Euler equation in weighted Sobolev space, then obtain the weighted Gevrey regularity of Euler equation. We will use the weighted Sobolev-Gevrey space method to obtain the results of Gevrey regularity of Euler equation, and the use of the property of singular operator in the estimate of the pressure term is the improvement of our work.
The aim of this manuscript is to study the influence of the vorticity on the existence time in fluid systems for which global smoothness and decay is known in the case of small irrotational data. We focus on two examples: the Euler-Korteweg system and the two-fluid Euler Maxwell system. We prove that the lower bound of the lifespan of these systems is no less than the inverse of the $H^s$ $(s>5/2)$ norm of the rotational part of the initial velocity. Our approach is based on energy estimates and the fast time decay results of global solutions to these systems with small irrotational initial da…
In this paper we study the uniqueness property of solutions to the steady incompressible Euler equations with perturbations in $\Bbb R^N$. Our perturbations include as special cases the Euler equations with a `single signed' nonlinear term, the self-similar Euler equations, and the steady Navier-Stokes equations. For these equations show that suitable decay assumptions at infinity on the solution or its derivatives, imposed by the $L^q$ conditions imply that the only possible solution is zero.
We consider the 2D Euler equation of incompressible fluids on a strip and prove the stability of the rectangular stationary state.
This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in some critical Besov spaces
We give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible Euler and the compressible isentropic Euler systems in order for the energy to be conserved. Our strategy relies on commutator estimates similar to those employed by P. Constantin et al. for the homogeneous incompressible Euler equations.
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.
The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we construct rigorously self-similar solutions to the Riemann problem of the spherically symmetric Euler equations for general equations of state. We used the assumption of self-similarity to reduce the spherically symmetric Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence.
The article `Global regularity for the 3D Navier-Stokes and the 3D Euler equations'(arXiv:0711.2453) is withdrawn due to a serious error in the proof.
We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal 1D Euler equations.
We prove well-posedness of vortex sheets with surface tension in the 3D incompressible Euler equations with vorticity.
We prove a localized non blow-up theorem of the Beale-Kato-Majda type for the solution of the 3D incompressible Euler equations.
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 prove new \emph{a priori} estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.
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 paper, we develop an abstract framework to establish ill-posedness in the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable spectra. We apply it to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system.
Cahn-Hillard and Keller-Segel systems as high-friction limits of Euler-Korteweg and Euler-Poisson equations
本文研究了带摩擦的Euler-Korteweg和Euler-Poisson方程联合系统，证明了耗散测度值解的存在性，并证明在γ≥2时Cahn-Hillard-Keller-Segel系统的强解是该系统的高摩擦极限。
Semi-classical limit for Klein-Gordon equation toward relativistic Euler equations via an adapted modulated energy method
通过适配调制能量方法，研究Klein-Gordon方程在半经典极限下向相对论欧拉方程的收敛性，并证明其动量和密度在Lebesgue范数下的收敛性。
We construct spherically symmetric solutions to the Einstein-Euler equations, which contains a positive cosmological constant, say, the Einstein-Euler-de Sitter equations. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler-de Sitter equations are given by the Tolman-Oppenheimer-Volkoff-de Sitter equation. We can construct solutions near time periodic linearized solutions around the equilibrium. The Cauchy problem around the equilibrium can be solved. This work can be considered as a trial of the generalization of the previous work on the …
In this note, we construct solutions to the 2D Euler equations which belong to the Yudovich class but lose $W^{1,p}$ regularity continuously with time.
This paper deals with the global existence and uniqueness results for the three-dimensional incompressible Euler equations with a particular structure for initial data lying in critical spaces. In this case the BKM criterion is not known.
A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in S…
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.
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