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This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

This paper is devoted to show a couple of typicality results for weak solutions $v\in C^\theta$ of the Euler equations, in the case $\theta0}W^{\frac{2\theta}{1-\theta} + \varepsilon,p}(I)$ for any open $I \subset [0,T]$, are a residual set in $X_\theta$. This, in particular, partially solves [9, Conjecture 1]. We also show that smooth solutions form a nowhere dense set in the space of all the $C^\theta$ weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one., Comment: 21 pages. arXiv admin note: substantial text overlap with arXiv:1701.08678 by other authors

In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity.

We consider a family of nonlocal problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.

We prove a localized non blow-up theorem of the Beale–Kato–Majda type for the solution of the 3D incompressible Euler equations.

In this paper we prove local in time well-posedness for the incompressible Euler equations in R n for the initial data in L 1 ( 1 ) 1 ( R n ) , which corresponds to a critical case of the generalized Campanato spaces L q ( N ) s ( R n ) . The space is studied extensively in our companion paper [9] , and in the critical case we have embeddings B ∞ , 1 1 ( R n ) ↪ L 1 ( 1 ) 1 ( R n ) ↪ C 0 , 1 ( R n ) , where B ∞ , 1 1 ( R n ) and C 0 , 1 ( R n ) are the Besov space and the Lipschitz space respectively. In particular L 1 ( 1 ) 1 ( R n ) contains non- C 1 ( R n ) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L 1 ( 1 ) 1 ( R n ) , for which the solution to the Euler equations blows up in finite time.

We prove the non-uniform continuity of the data-to-solution map of the incompressible Euler equations in Besov spaces, B p , q s , where the parameters p , q and s considered here are such that the local existence and uniqueness result holds.

We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose time-evolution is determined from the underlying Euler equations. We prove partial well-posedness results for dissipative statistical solutions and propose a Monte Carlo type algorithm, based on spectral viscosity spatial discretizations, to approximate them. Under verifiable hypotheses on the computations, we prove that the approximations converge to a statistical solution in a suitable topology. In particular, multi-point statistical quantities of interest converge on increasing resolution. We present several numerical experiments to illustrate the theory., Mathematical Models and Methods in Applied Sciences, 31 (02), ISSN:0218-2025, ISSN:1793-6314

In this paper, we use a surprised system to construct some exact solutions of compressible Euler equations with two and three dimension. Furthermore, we also give other exact solutions of three dimension incompressible Euler equations.

In this paper, we study the desingularization of multiscale rotating vortex patches with fixed angular velocity to the 2D Euler equations in a disk. We prove the existence of two-parameter multiscale V-states concentrating near two points. These two points are determined by the Robin function to the Green’s function, the Green’s function and angular velocity, and not symmetric about the origin, which is new to the former results. The existence of multiscale V-states with N-folds for $$ N\ge 2 $$ is also proved.

In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of traveling vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal intensities but opposite signs. The results are obtained by using an improved vorticity method.

We prove the global well-posedness of the free interface problem for the two-phase incompressible Euler Equations with damping for the small initial data, where the effect of surface tension is included on the free surfaces. Moreover, the solution decays exponentially to the equilibrium.

In this paper, we study the asymptotic behavior of solutions to second grade fluid equations, a model for viscoelastic fluids, in an expanding domain. We prove that, the solutions converge to a solution of the incompressible Euler equations in the whole plane, as the elastic response α and the viscosity ν vanish, and the radius of domain becomes infinite. Meanwhile, we also establish precise convergence rates in terms of ν , α and the radius of the family of spatial domains.

We exclude Type I blow-up, which occurs in the form of atomic concentrations of the $$L^2$$ norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale.

Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin-Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler-Poincar\'{e} and stochastic Navier-Stokes-Poincar\'{e} equations respectively. The stochastic Euler-Poincar\'{e} equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems., Comment: 26 pages

By using an idea of localized Galilean boost, we show that the data-to-solution map for incompressible Euler equations is not uniformly continuous in $${H^s({\mathbb{R}}^d)}$$, $${s \ge 0}$$. This settles the end-point case (s = 0) left open in Himonas–Misiolek (Commun Math Phys 296(1):285–301, 2010) and gives a unified treatment for all Hs. We also show the solution map is nowhere uniformly continuous.

The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in Dinvay (2018). It is numerically shown to be stable and a good approximation to the incompressible Euler equations. Here we prove local in time well-posedness. Our proof relies on an energy method and a compactness argument. In addition some numerical experiments, supporting the validity of the system as an asymptotic model for water waves, are carried out.

Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question, by first describing a class of potentially singular solutions to the Euler equations numerically discovered in axisymmetric geometries, and then by presenting evidence from rigorous analysis that strongly supports the existence of such singular solutions. The initial data leading to these singular solutions possess certain special symmetry and monotonicity properties, and the subsequent flows are assumed to satisfy a periodic boundary condition along the axial direction and a no-flow, free-slip boundary condition on the solid wall. The numerical study employs a hybrid 6th-order Galerkin/finite difference discretization of the governing equations in space and a 4th-order Runge--Kutta discretization in time, where the emerging singularity is captured on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10¹²)² near the point of the singularity, the simulations are able to advance the solution to a point that is asymptotically close to the predicted singularity time, while achieving a pointwise relative error of O(10⁻⁴) in the vorticity vector and obtaining a 3 x 10⁸-fold increase in the maximum vorticity. The numerical data are checked against all major blowup/nonblowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A close scrutiny of the data near the point of the singularity also reveals a self-similar structure in the blowup, as well as a one-dimensional model which is seen to capture the essential features of the singular solutions along the solid wall, and for which existence of finite-time singularities can be established rigorously.

In this note we extend a 2018 result of Bardos and Titi (Arch Ration Mech Anal 228(1):197–207, 2018) to a new class of functional spaces $$C^{0,\alpha }_{\lambda }(\bar{\Omega })$$ . It is shown that weak solutions $$\,u\,$$ satisfy the energy equality provided that $$u\in L^3((0,T);C^{0,\alpha }_{\lambda }(\bar{\Omega }))$$ with $$\alpha \ge \frac{1}{3}$$ and $$\lambda >0$$ . The result is new for $$\,\alpha =\,\frac{1}{3}.$$ Actually, a quite stronger result holds. For convenience we start by a similar extension of a 1994 result of Constantin and Titi (Commun Math Phys 165:207–209, 1994), in the space periodic case. The proofs follow step by step those of the above authors. For the readers convenience, and completeness, proofs are presented in a quite complete form.

We consider the initial value problem of the 3D inviscid Boussinesq equations for stably stratified fluids. We prove the long time existence of classical solutions for large initial data when the buoyancy frequency is sufficiently high. Furthermore, we consider the singular limit of the strong stratification, and show that the long time classical solution converges to that of 2D incompressible Euler equations in some space-time Strichartz norms.

We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical limit as $c\rightarrow \infty$. As our main result, we prove that the fluid pressure of solutions of these incompressible "relativistic" Euler equations satisfies an elliptic equation on each of the hypersurfaces orthogonal to the fluid four-velocity, which indicates infinite speed of propagation., Comment: 7 pages. Version 2: Improved wording and presentation

The combined quasi-neutral and inviscid limit of the Navier-Stokes-Poisson-Korteweg system with density-dependent viscosity and cold pressure in the torus T3 is studied. It is shown that, for the well-prepared initial data, the global weak solution of the Navier-Stokes-Poisson-Korteweg system converges strongly to the strong solution of the incompressible Euler equations when the Debye length and the viscosity coefficient go to zero simultaneously. Furthermore, the rate of convergence is also obtained.

We consider the free boundary compressible and incompressible Euler equations with surface tension. In both cases, we provide a priori estimates for the local existence with the initial velocity in H 3 , with the H 3 condition on the density in the compressible case. An additional condition is required on the free boundary. Compared to the existing literature, both results lower the regularity of initial data for the Lagrangian Euler equation with surface tension.

The quasineutral limit of the isothermal Navier–Stokes–Poisson system is rigorously proved when the combined quasineutral and vanishing viscosity limit is considered in a domain with boundary. The convergence of the global weak solution for Navier–Stokes–Poisson system to the strong solution for incompressible Euler equations is obtained.

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers (Dinvay, 2019; Dinvay et al., 2019) the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approach and a concentration-compactness argument. The main difficulties stem from the fact that in the considered Euler–Lagrange equation we have a non-local operator of positive order appearing both in the linear and non-linear parts. Our approach allows us to obtain solitary waves for a particular Boussinesq system as well.

We consider the Cauchy problem for the incompressible Euler equations on R d for d ≥ 3 . Then we demonstrate the local-in-time solvability of classical solutions with the nondecaying derivatives and finite kinetic energy. Moreover, we establish the blowup criterion of such solutions in terms of the vorticity.

In this paper, we generalize an h-adaptive Runge–Kutta discontinuous Galerkin scheme developed earlier in Zhu et al. (J Sci Comput 69:1346–1365, 2016) for the 1D Vlasov–Poisson system to the guiding center Vlasov model and the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation. The main difficulty of this generalization lies in solving the 2D Poisson equation due to the irregular adaptive mesh with hanging nodes. We adopt a local discontinuous Galerkin method to solve the Poisson equation. The full adaptive algorithm and the related numerical implementation details are included. Extensive numerical tests have been performed to showcase the effectiveness of the adaptive scheme and its advantage over the fixed-mesh scheme in saving computational cost and improving solution quality.

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., 14 pages

El estudio de las funciones univalentes (holomorfas e inyectivas) en el disco unidad D, considerado piedra angular en la teor´ıa geom´etrica de funciones, es un ´area de investigaci´on ya cl´asica pero que sigue muy viva a d´ıa de hoy. Sin duda, la famosa conjetura de Bieberbach de 1916 (demostrada por de de Branges en 1984), impuls´o en gran medida el estudio de este tipo de funciones. En este trabajo, se revisan las principales propiedades que deben satisfacer las funciones en la c´elebre clase S de aplicaciones univalentes y normalizadas (de manera est´andar) en D y como muchas de ellas se deducen del teorema de Bieberbach. Tambi´en se estudia la teor´ıa de Loewner, de gran relevancia en esta ´area y parte fundamental en la demostraci´on de de Branges de la conjetura de Bieberbach. Por ´ultimo, y como muestra de la relaci´on entre la teor´ıa de funciones anal´ıticas y otras ´areas de las Matem´aticas (concretamente, la Mec´anica de Fluidos), se recogen algunos resultados recientes sobre las soluciones expl´ıcitas de la ecuaci´on bidimensional de Euler incompresible. The study of univalent (holomorphic and injective) functions in the unit disk D, considered as a milestone in geometric function theory, is a classical subject of research which continues active nowadays. With no doubt, the famous Bieberbach conjecture from 1916 (proved by de Branges in 1984) was one of the main sources for the development of the analysis of the properties of this type of functions. In this Bachelor Thesis, we review the main properties that a function in the class S of univalent mappings in the unit disk (normalized in the standard way) must satisfy. Most of them are derived from the classical Bieberbach theorem. An indepth analysis of Loewner’s theory, of great relevance in the area (and a fundamental tool in de Branges’ proof of the Bieberbach conjecture) is developed as well. Finally, and as a proof of the relationship between Complex Analysis and other areas of Mathematics (in particular, Fluid Mechanics), we review some recent results on explicit solutions of the incompressible Euler equation.

We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on $T^2$ and $R^2$ whose perimeter grows with time. More precisely, for any constant $M > 0$, we construct a vortex patch in $T^2$ whose smooth boundary has length of order 1 at the initial time such that the perimeter grows up to the given constant $M$ within finite time. The construction is done by cutting a thin stick out of an almost square patch. A similar result holds for an almost round patch with a thin handle in $R^2$., 6 pages, 2 figures

In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disk. By choosing a suitable vector field to deform the patch, we show that each simply-connected rotating vortex patch D with angular velocity Ω, Ω ≥ max ⁡ { 1 / 2 , ( 2 l 2 ) / ( 1 − l 2 ) 2 } or Ω ≤ − ( 2 l 2 ) / ( 1 − l 2 ) 2 , where l = sup x ∈ D ⁡ | x | , must be a disk. The main idea of the proof, which has a variational flavor, comes from a recent paper of Gomez-Serrano–Park–Shi–Yao, arXiv:1908.01722 , where radial symmetry of rotating vortex patches in the whole plane was studied.

In this note we would like to present "an analysts' point of view" on the Nash-Kuiper theorem and in particular highlight the very close connection to some aspects of turbulence -- a paradigm example of a high-dimensional phenomenon., Comment: To appear in the Bulletin of the AMS

In this paper, we first show the existence of local-in-time solutions the inhomogeneous incompressible Euler equations in R 3 . We then derive a refined blow-up criterion of these solutions.

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 (Gavrilov in A steady Euler flow with compact support. Geom Funct Anal 29(1):90–197, [Gav19]).

This paper mainly concerns the mathematical justification of the asymptotic limit of the Gross-Pitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the Gross-Pitaevskii equation to the strong solution of the incompressible Euler equations. Furthermore, we also obtain the rates of the convergence.

In this paper, we are concerned with the rigorous proof of the convergence of the quantum Navier–Stokes-Poisson system to the incompressible Euler equations via the combined quasi-neutral, vanishing damping coefficient and inviscid limits in the three-dimensional torus for general initial data. Furthermore, the convergence rates are obtained.

This paper introduces a well-balanced second-order finite volume scheme, based on the Q-scheme of Roe, for simulating granular type flows. The proposed method is applied to solve the incompressible Euler equations under Savage–Hutter assumptions. The model is derived in a local coordinate system along a non-erodible bed to take its curvature into account. Moreover, simultaneous appearance of flowing/static regions is simulated by considering a basal friction resistance which keeps the granular flow from moving when the angle of granular flow is less than the angle of repose. The proposed scheme preserves stationary solutions up to second order and deals with different situations of wet/dry transitions by a modified nonlinear wet/dry treatment. Numerical results indicate the improved properties and robustness of the proposed finite volume structure. In addition, the granular flow properties are estimated with a computational error of less than 5%. These errors are consistently less than those obtained by using similar existing finite volume schemes without the proposed modifications, which can result in up to 30% overestimation.

We show that the 2D Euler equations are not locally well-posed in the sense of Hadamard in the \(C^1\) space and in the Besov space \(B^1_{\infty ,1}\). Our approach relies on the technique of Lagrangian deformations of Bourgain and Li (Strong ill-posedness of the incompressible Euler equations in borderline Sobolev spaces. arXiv:1307.7090). We show that the assumption that the data-to-solution map is continuous in either \(C^1\) or \(B^1_{\infty ,1}\) leads to a contradiction with a result in \(W^{1,p}\) of Kato and Ponce (Rev Mat Iberoam 2:73–88, 1986).

This paper investigates temporal regularity of solutions for the incompressible Euler equations in a critical Besov space B d p +1 p,1 (R ) for 1 ≤ p ≤ d.

In this work we investigate the helicity regularity for weak solutions of the incompressible Euler equations. To prove regularity and conservation of the helicity we will treat the velocity u u and its curl ⁡ u \operatorname {curl} u as two independent functions and we mainly show that the helicity is a constant of motion assuming u ∈ L t 2 r ( C x θ ) u \in L^{2r}_t(C^\theta _x) and curl ⁡ u ∈ L t κ ( W x α , 1 ) \operatorname {curl} u \in L^{\kappa }_t(W^{\alpha ,1}_x) , where r , κ r,\kappa are conjugate Hölder exponents and 2 θ + α ≥ 1 2\theta +\alpha \geq 1 . Using the same techniques we also show that the helicity has a suitable Hölder regularity even in the range where it is not necessarily constant.

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., Comment: 18 pages

The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity problem concerning the 2D inviscid Boussinesq equations remains widely open. In an attempt to understand this problem, we examine the damped 2D Boussinesq equations and study how damping affects the regularity of solutions. Since the damping effect is insufficient in overcoming the difficulty due to the “vortex stretching”, we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption. By positioning the solutions in a suitable functional setting (more precisely, the homogeneous Besov space B ˚ ∞ , 1 1 ), we are able to obtain a unique global solution under a minimal smallness assumption.

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $\Bbb R^N$. If we assume "single signedness condition" on the force, then we can show that a $C^1 (\Bbb R^N)$ solution $(v,p)$ with $|v|^2+ |p|\in L^{\frac{q}{2}}(\Bbb R^N)$, $q\in (\frac{3N}{N-1}, \infty)$ is trivial, $v=0$. For the solution of of the steady Navier-Stokes equations, satisfying $v(x)\to 0$ as $|x|\to \infty$, the condition $\int_{\Bbb R^3} |\Delta v|^{\frac65} dx, Comment: 15 pages(to appear in Comm. Math. Phys.). arXiv admin note: substantial text overlap with arXiv:1105.3639

For any \theta, Comment: 40 pages

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.

Methods originally developed to study the finite time blow-up problem of the regular solutions of the three dimensional incompressible Euler equations are used to investigate the regular solutions of the Camassa–Holm equation. We obtain results on the relative behaviors of the momentum density, the deformation tensor and the nonlocal term along the trajectories. In terms of these behaviors, we get new types of asymptotic properties of global solutions, blow-up criterion and blow-up time estimate for local solutions. More precisely, certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions. Finite time blow-up of the accumulated momentum density is necessary and sufficient for the finite time blow-up of the solution. An upper estimate of the blow-up time and a blow-up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term.

The 3D incompressible Euler equations with a passive scalar $\theta$ are considered in a smooth domain $\Omega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\bu\cdot\bhn|_{\partial\Omega} = 0$. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector $\bB = \nabla q\times\nabla\theta$, provided $\bB$ has no null points initially\,: $\bom = \mbox{curl}\,\bu$ is the vorticity and $q = \bom\cdot\nabla\theta$ is a potential vorticity. The presence of the passive scalar concentration $\theta$ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (2000) on the non-existence of Clebsch potentials in the neighbourhood of null points., Comment: 5 pages, no figures