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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 monograph, we study the notion of measure-valued solution for the compressible and incompressible Euler equations and investigate how such solutions can be generated by singular limits of weak solutions of related fluid models. Of particular interest will be limits of compressible weak solutions, vanishing viscosity, and low Mach number limits. After laying the abstract linear foundations used in this thesis we take a deeper look at the relationship between weak and measure-valued solutions of compressible fluid flows. First, we construct examples of energy admissible compressible measure-valued solutions arising from deterministic and continuous initial data, which cannot be generated by vanishing viscosity limits or limits of weak solutions. This is in sharp contrast to the incompressible situation, where every (classical) measure-valued solution is the limit of weak solutions. In general, any generable compressible measure-valued solution has to satisfy a necessary Jensen-type inequality. On the other hand, we obtain also sufficient conditions involving a slightly different Jensen inequality on the level of potential operators from an L^1-version of Fonseca-Müller’s characterization result for A-free Young measures together with a certain A-free truncation technique. Proving the aforementioned truncation method is a major milestone in our approach. Given a sequence of A-free functions in potential form converging in an L^1-sense to some limit set, the aim is to truncate this sequence in such a way that the new sequence converges uniformly to the limit set while preserving the A-freeness and remaining sufficiently close to the original sequence. We establish such a truncation result for potentials of first and second order, including the potential of the linearly relaxed Euler system. Moreover, we introduce a measure-valued framework in which the low Mach limit can be treated adequately. This will lead to the consideration of the novel concept of augmented measure-valued solutions. We give sufficient conditions on low Mach sequences for generating such an augmented solution. More importantly, we also obtain necessary Jensen-type conditions on augmented solutions to arise from low Mach sequences. Our discussion of the low Mach limit will shed some new light on the role of the pressure in the compressible and incompressible situation. As a consequence of our results, we propose two selection criteria in order to discard some compressible measure-valued solutions and augmented incompressible solutions out of the in general infinitely many as unphysical. To a certain extent, this tackles the question of uniqueness for measure-valued solutions.

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.

In this paper, we study the inviscid and zero Froude number limits of the viscous shallow water system. We prove that the limit system is represented by the incompressible Euler equations on the whole space. Furthermore, the rate of convergence is also obtained.

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.

La dinàmica d'un fluid ideal en un espai 2-dimensional pot ser aproximat mitjançant la evolució de N vortexs singulars a mesura que fem tendir N cap a infinit. Al llarg d'aquest treball estudiem la dinàmica de vortexs singulars, normalment denominats com "point vortices", en una superficie no orientable com és l'espai projectiu real, $\RP^2$. Primer estudiem les equacions de Euler per fluids incompressibles així com la seva generalització a varietats no orientables. L'objectiu es obtenir un nou model de vortexs singulars que sigui compatible amb varietats no orientables. Veurem que la dinamica de N vòretxs en un espai projectiu pot ser visualitzat com la dinàmica de N dipols antipodals en l'esfera $S^2$, la qual és un espai recobridor de dos fulls de l'espai projectiu. En la segona part del treball presentem un estudi dels casos integrables, en el sentit de Liouville, a $\RP^2$ entre els quals trobem tant fenòmens nous com fenòmens similars als que es tenen en el problema estandar a l'esfera. Dins de l'estudi incluim l'estudi de fenòmens de equilibri estàtic, equilibri dinàmic, estabilitat i del fenòmen de col·lapse singular. La dinámica de un fluido ideal en un espacio de dimensión 2 puede ser aproximado mediante la evolución de N vortices singulares a medida que N tiende a infinito. A lo largo de este trabajo estudiamos la dinámica de vortices singulares (point vortices) en una superficie no orientable, el plano proyectivo. Primero estudiamos las equaciones de Euler para fluidos incompressibles asi como su generalización para variedades no orientables. La intencion es obtener un nuevo modelo de vortices singulares que sea compatible con la no-orientabilidad de la variedad. Veremos que la dinámica de N vortices en el espacio proyectivo es equivalente a la dinámica de N dipolos antipodales en la esfera, la cual es un recubrimiento de 2 hojas del espacio proyectivo. En la siguiente parte presentamos una comparación de los casos integrables, en el sentido de Liouville, en RP^2 con dinámicas analogas en S^2. Veremos que la dinámica de N vortices en el espacio proyectivo presenta algunas similitudes con el problema estándar así como algunas diferencias. Dentro del trabajo incluimos el estudio de equilibrios estáticos, relativos, estabilidad y colapso singular. The motion of an ideal two-dimensional fluid can be approximated by the evolution of N point vortices, as N → ∞. Throughout this work, we study the motion of point vortices on a non- orientable surface, the projective plane RP2. First, we study the incompressible Euler equations, as well as their generalization to non-orientable manifolds. The aim is to derive a new model for point vortices that is compatible with non-orientability. We will see that the motion of N vortices on the projective space can be regarded as the motion of N antipodal dipoles on the sphere S2, the orientation cover of RP2. In the next part, we will present a comparison of the integrable cases of the vortex motions on RP2 with analogous motions on S2 describing new phenomena, as well as similarities. Among the cases we study, we include the study of equilibria, relative equilibria, their stability, and collapse. Outgoing

Energy stable and conservative nonlinear weakly imposed interface conditions for the incompressible Euler equations are derived in the continuous setting. By discretely mimicking the continuous analysis using summation-by-parts operators, we prove that the numerical scheme is stable and conservative. The theoretical findings are verified by numerical experiments. Funding: Vetenskapradet [2018-05084, 2021-05484]

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.

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

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.

Can every physical system simulate any Turing machine? This is a classical problem that is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore [C. Moore, Nonlinearity 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently, Tao launched a program based on the Turing completeness of the Euler equations to address the blow-up problem in the Navier¿Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem to quantum-field theories. To the best of our knowledge, the existence of undecidable particle paths of three-dimensional fluid flows has remained an elusive open problem since Moore¿s works in the early 1990s. In this article, we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao¿s approach to the blow-up problem in the Navier¿Stokes equations., Robert Cardona was supported by the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Program for Units of Excellence in R&D (MDM-2014-0445) via an FPI grant. R.C. and E.M. are partially supported by Grants MTM2015-69135-P/FEDER, the Spanish Ministry of Science and Innovation PID2019-103849GB-I00/AEI/10.13039/501100011033, and Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR932. E.M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. D.P.-S. is supported by MICINN Grant MTM PID2019-106715GB-C21 and MCIU Grant Europa Excelencia EUR2019-103821. F.P. is supported by MICINN/FEDER Grants MTM2016-79400-P and PID2019-108936GB-C21. This work was partially supported by ICMAT–Severo Ochoa Grant CEX2019-000904-S.

We consider the initial value problem for the Nernst-Planck equations coupled to the incompressible Euler equations in $\mathbb T^2$. We prove global existence of weak solutions for vorticity in $L^p$. We also obtain global existence and uniqueness of smooth solutions. We show that smooth solutions of the Nernst-Planck-Navier-Stokes equations converge to solutions of the Nernst-Planck-Euler equations as viscosity tends to zero. All the results hold for large data.

Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty., Minor typos corrected

For a nonlinear operator T satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) T is surjective, ii) T is open at zero, and iii) T has a bounded right inverse. The theorem applies to numerous scale-invariant PDEs in regularity regimes where the equations are stable under weak⁎ convergence. Two particular examples we explore are the Jacobian equation and the equations of incompressible fluid flow. For the Jacobian, it is a long standing open problem to decide whether it is onto between the critical Sobolev space and the Hardy space. Towards a negative answer, we show that, if the Jacobian is onto, then it suffices to rule out the existence of surprisingly well-behaved solutions. For the incompressible Euler equations, we show that, for any p, Advances in Mathematics, 415, ISSN:0001-8708, ISSN:1090-2082

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 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

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.