Euler Equations Of Incompressible Fluid Research Articles

Nearly Toroidal, Periodic and Quasi-periodic Motions of Fluid Particles Driven by the Gavrilov Solutions of the Euler Equations

We consider the smooth, compactly supported solutions of the steady 3D Euler equations of incompressible fluids constructed by Gavrilov (Geom Funct Anal (GAFA) 29(1):190–197, 2019), and we study the corresponding fluid particle dynamics. This is an ode analysis, which contributes to the description of Gavrilov’s vector field.

On the Semi-Analytical Solutions in Hydrodynamics of Ideal Fluid Flows Governed by Large-Scale Coherent Structures of Spiral-Type

We have presented here a clearly formulated algorithm or semi-analytical solving procedure for obtaining or tracing approximate hydrodynamical fields of flows (and thus, videlicet, their trajectories) for ideal incompressible fluids governed by external large-scale coherent structures of spiral-type, which can be recognized as special invariant at symmetry reduction. Examples of such structures are widely presented in nature in “wind-water-coastline” interactions during a long-time period. Our suggested mathematical approach has obvious practical meaning as tracing process of formation of the paths or trajectories for material flows of fallout descending near ocean coastlines which are forming its geometry or bottom surface of the ocean. In our presentation, we explore (as first approximation) the case of non-stationary flows of Euler equations for incompressible fluids, which should conserve the Bernoulli-function as being invariant for the aforementioned system. The current research assumes approximated solution (with numerical findings), which stems from presenting the Euler equations in a special form with a partial type of approximated components of vortex field in a fluid. Conditions and restrictions for the existence of the 2D and 3D non-stationary solutions of the aforementioned type have been formulated as well.

Dynamics of nearly parallel vortex filaments for the Gross–Pitaevskii equation

Klein et al. (J Fluid Mech 288:201–248, 1995) have formally derived a simplified asymptotic motion law for the evolution of nearly parallel vortex filaments in the context of the three dimensional Euler equation for incompressible fluids. In the present work, we rigorously derive the corresponding asymptotic motion law in the context of the Gross–Pitaevskii equation.

The Initial Value Problem for the Euler Equations of Incompressible Fluids Viewed as a Concave Maximization Problem

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 Riccati-type solution of 3D Euler equations for incompressible flow

In fluid mechanics, a lot of authors have been reporting analytical solutions of Euler and Navier-Stokes equations. But there is an essential deficiency of non-stationary solutions indeed.In our presentation, we explore the case of non-stationary flows of the Euler equations for incompressible fluids, which should conserve the Bernoulli-function to be invariant for the aforementioned system.We use previously suggested ansatz for solving of the system of Navier-Stokes equations (which is proved to have the analytical way to present its solution in case of conserving the Bernoulli-function to be invariant for such the type of the flows). Conditions for the existence of exact solution of the aforementioned type for the Euler equations are obtained. The restrictions at choosing of the form of the 3D nonstationary solution for the given constant Bernoulli-function B are considered. We should especially note that pressure field should be calculated from the given constant Bernoulli-function B, if all the components of velocity field are obtained.

Leapfrogging Vortex Rings for the Three Dimensional Gross-Pitaevskii Equation

Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation.

Topology-Preserving Diffusion of Divergence-Free Vector Fields and Magnetic Relaxation

The usual heat equation is not suitable to preserve the topology of divergence-free vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, one can find examples of topology-preserving diffusion equations for divergence-free vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of “dissipative solutions”, which shares common features with both P.-L. Lions’s dissipative solutions to the Euler equations and the concept of “curves of maximal slopes”, à la De Giorgi, recently used to study the scalar heat equation in very general metric spaces. We show that the initial value problem admits such global solutions, at least in the two space variable case, and they are unique whenever they are smooth.

A logarithmically improved blow-up criterion of smooth solutions for nematic liquid crystal flows with partial viscosity

In this paper we investigate the Cauchy problem for nematic liquid crystal flows with partial viscosity in R 3 . A logarithmically improved blow-up criterion of smooth solutions is established. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.

A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity

We study the incompressible magneto-micropolar fluid equations with partial viscosity in . A blow-up criterion of smooth solutions is obtained. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.

Energy conservation and Onsager's conjecture for the Euler equations

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space . We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood–Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

On the Euler equations of incompressible fluids

Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

Stabilization of the incompressible 2D Euler equations in a simply connected domain utilizing the Lorentz force

In this paper, the null asymptotic stabilization of the 2D Euler equations of incompressible fluids in a simply connected bounded domain is investigated by utilizing the Lorentz force given by the Maxwell equations with Ohm's law.

Applications of Nambu Mechanics to Systems of Hydrodynamical Type II

In this paper we further investigate some applications of Nambu mechanics in hydrodynamical systems. Using the Euler equations for a rotating rigid body Névir and Blender [J. Phys. A 26 (1993), L1189–L1193] had demonstrated the connection between Nambu mechanics and noncanonical Hamiltonian mechanics. Nambu mechanics is extended to incompressible ideal hydrodynamical fields using energy and helicity in three dimensional (enstrophy in two dimensional). In this paper we discuss the Lax representation of systems of Névir-Blender type. We also formulate the three dimensional Euler equations of incompressible fluid in terms of Nambu-Poisson geometry. We discuss their Lax representation. We also briefly discuss the Lax representation of ideal incompressible magnetohydrodynamics equations.

Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles

Dissipative solutions [12] of the Euler equations of incompressible fluids are obtained as the hydrodynamic limit of a properly scaled BGK equation. This stability result comes from refined entropy and entropy dissipation bounds. It uses in a crucial way the local conservation laws which are known to hold for weak solutions of this simplified model of the Boltzmann equation.

A continuation principle for the $3$-D Euler equations for incompressible fluids in a bounded domain

A continuation principle for the $3$-D Euler equations for incompressible fluids in a bounded domain

Well-Posedness of the Euler and Navier-Stokes Equations in the Lebesgue Spaces $L^p_s(\mathbb R^2)$

In this paper we show that the Euler equation for incompressible fluids in R2 is well posed in the (vector-valued) Lebesgue spaces Lsp = (1 - ?)-s/2 Lp(R2) with s > 1 + 2/p, 1 < p < 8 and that the same is true of the Navier-Stokes equation uniformly in the viscosity ?.

Nonlinear stability of circular vortex patches

This paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the Euler equations of incompressible fluids. This is achieved by establishing a relative variational principle in terms of either energy or angular momentum. Thus, we exploit and extend Arnold's idea in (1965, 1969) to a nonsmooth setting as well.

Nonlinear evolution equations and the Euler flow

A new theorem on abstract nonlinear equations of evolution is proved. As an application, the existence, uniqueness, regularity, and continuous dependence on the data are proved for the solution of the Euler equation for incompressible fluids in a bounded domain in R m .

A Beale-Kato-Madja breakdown criterion for an Oldroyd-B fluid in the creeping flow regime

We derive a criterion for the breakdown of solutions to the Oldroyd-B model in R3 in the limit of zero Reynolds number (creeping flow). If the initial stress field is in the Sobolev space Hm(R3), m > 5/2, then either a unique solution exists within this space indefinitely, or, at the time where the solution breaks down, the time integral of the L 1 -norm of the stress tensor must diverge. This result is analogous to the celebrated Beale-Kato-Majda breakdown criterion for the inviscid Euler equations of incompressible fluids. AMS subject classifications. 35B35, 35Q35