On a Lagrangian Formulation of the Incompressible Euler Equation

In this paper we show that the incompressible Euler equation on the Sobolev space $H^s(\mathbb{R}^n), s › n ⁄ 2+1$, 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.     

On the Free Boundary to the Incompressible Euler Equations

In this paper, we consider an inviscid, incompressible, irrotational fluid in a region of R^3 with free boundary. Motivated by [1], we find that in this particular case, we do not need the complicated energy functional in [1], instead we can use a simpler replacement and get the a priori energy estimate for a positive time, which depends only on the initial data.     

A Priori Estimates for the Compressible Euler Equations for a Liquid with Free Surface Boundary and the Incompressible Limit

In this paper, we prove a new type of energy estimate for the compressible Euler equations with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed k. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin to when you have a free boundary. In the incompressible case our energies reduce to those in Christodoulou and Lindblad, and our proof in particular gives a simplified proof of their estimates with improved error estimates. Since for an incompressible irrotational liquid with free surface there are small data global existence results, our result leaves open the possibility of long‐time existence also for slightly compressible liquids with a free surface.© 2017 Wiley Periodicals, Inc.     

Regularity Criterion for a Nonhomogeneous Incompressible Ginzburg-Landau-Navier-Stokes System

Applications of Mathematics - We prove a regularity criterion for a nonhomogeneous incompressible Ginzburg-Landau-Navier-Stokes system with the Coulomb gauge in ?3. It is proved that if the...     

A Regularity Result of Nikolskii Type for an Evolution Equation on Compact Homogeneous Spaces

In the previous paper [2], we proposed a definition of a class of function spaces of Nikolskii type on a special class of riemannian manifolds, namely the manifolds M admitting a transitive compact connected Lie group of isometries. In this paper we continue the subject and we complete the proof that the spaces we have introduced coincide with the classical ones, defined through (finitely many) local charts of M. Moreover, our equivalent definition of these spaces allows us to prove here a regularity result for the solution of a Stefan-like problem on M, in which the initial datum is assumed to be of Nikolskii type.     

A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D.     

三维不可压缩Euler方程的H1(R~3)轴对称解

在适当的初值条件下，对三维不可压缩Ｅｕｌｅｒ方程组，我们证明了Ｈ１（Ｒ３）轴对称解的存在性．     

Regularity criteria for the Euler equations with nondecaying data

The local-in-time existence and uniqueness of strong solutions to the Euler equations in the whole space with nondecaying and certainly regular initial velocity are concerned. It is obtained that the spatial regularity of solutions coincides with that of initial velocity under the suitable setting of external forcing terms. Regularity criteria focusing into the vorticity are also discussed due to the similar arguments of Beale-Kato-Majda.     

A Regularity Criterion for the 3D Incompressible Density-dependent Navier-Stokes-Allen-Cahn Equations

In this paper, we study the initial boundary problem for 3D incompressible density-dependent Navier-Stokes-Allen-Cahn equations, and give a regularity criterion for local strong solutions. Our result refines the blow-up criterion in [1].     

A Threshold Result for a Non-local Parabolic Equation

In this paper, we consider the Cauchy problem: (ECP) u_t−Δu+p(x)u=u(x,t)∫u²(y,t)/∣x−y∣dy; x∈ℝ³, t>0, u(x, 0)=u₀(x)⩾0 x∈ℝ³, (0.2) The stationary problem for (ECP) is the famous Choquard–Pekar problem, and it has a unique positive solution ū(x) as long as p(x) is radial, continuous in ℝ³, p(x)⩾ā>0, and lim_{∣x∣→∞}p(x)=p¯>0. In this paper, we prove that if the initial data 0⩽u₀(x)⩽(≢)ū(x), then the corresponding solution u(x, t) exists globally and it tends to the zero steady-state solution as t→∞, if u₀(x)⩾(≢)ū(x), then the solution u(x,t) blows up in finite time. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

截断Euler方程的两流体模型的正则性准则

该文研究具有部分粘性的截断Euler方程的两流体模型.在齐次Besov空间中给出强解的一些正则性条件.     

A Blow-Up Result in a Nonlinear Wave Equation with Delay

In this paper, we consider a nonlinear wave equation with delay. We show that under suitable conditions on the initial data, the weights of the damping, the delay term and the nonlinear source, the energy of solutions blows up in a finite time.     

A Five-Dimensional System of Navier-Stokes Equation for a Two-Dimensional Incompressible Fluid on a Torus

A five-mode truncation of Navier-Stokes equation for a two-dimensional incompressible fluid on a torus is studied. Its stationary solutions and stability are presented, the existence of attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through an involved sequence of bifurcations with the changing of Reynolds number, is simulated numerically. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system are revealed.     

On Regularity of Incompressible Fluid with Shear Dependent Viscosity

The authors consider a non-Newtonian fluid governed by equations with p-structure in a cubic domain.A fluid is said to be shear thinning(or pseudo-plastic) if 1 < p < 2,and shear thickening(or dilatant) if p > 2.The case p > 2 is considered in this paper.To improve the regularity results obtained by Crispo,it is shown that the secondorder derivatives of the velocity and the first-order derivative of the pressure belong to suitable spaces,by appealing to anisotropic Sobolev embeddings.     

The Incompressible Limit and the Initial Layer of the Compressible Euler Equation in ℝn+

We consider the incompressible limit of the compressible Euler equation in the half-space ℝⁿ₊. It is proved that the solutions of the non-dimensionalized compressible Euler equation converge to the solution of the incompressible Euler equation when the Mach number tends to zero. If the initial data v^∞₀ do not satisfy the condition ‘∇⋅v^∞₀=0’, then the initial layer will appear. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Existence of Solutions for Three Dimensional Stationary Incompressible Euler Equations with Nonvanishing Vorticity

In this paper, solutions with nonvanishing vorticity are established for the three dimensional stationary incompressible Euler equations on simply connected bounded three dimensional domains with smooth boundary. A class of additional boundary conditions for the vorticities are identified so that the solution is unique and stable.     

Persistence of Besov Regularity for a Generalized Drift-Diffusion Equation with Pressure

In this paper, we prove that Besov regularity of the initial data can persist for a generalized drift-diffusion equation with pressure under a very weak condition on the drift velocity. In particular, the solution is Hölder continuous.     

Regularity Results for a Strongly Degenerate Parabolic Equation

M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.