Existence and Regularity of the Pressure for the Stochastic Navier–Stokes Equations

We prove, on one hand, that for a convenient body force with values in the distribution space (H ^-1(D)) ^d , where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V of the divergence free subspace V of (H ¹ ₀(D)) ^d , in general it is not possible to solve the stochastic Navier–Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier–Stokes equations could be meaningful for them.     

Decay properties of solutions to the incompressible magnetohydrodynamics equations in a half space

We consider the asymptotic behavior of the strong solution to the incompressible magnetohydrodynamics (MHD) equations in a half space. The L^r‐decay rates of the strong solution and its derivatives with respect to space variables and time variable, including the L¹ and L ^∞ decay rates of its first order derivatives with respect to space variables, are derived by using L^q ? L^r estimates of the Stokes semigroup and employing a decomposition for the nonlinear terms in MHD equations. In addition, if the given initial data lie in a suitable weighted space, we obtain more rapid decay rates than observed in general. Similar results are known for incompressible Navier–Stokes equations in a half space under same assumption. Copyright © 2012 John Wiley & Sons, Ltd.     

Periodic solutions of the Navier–Stokes equations in a perturbed half‐space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half‐space and an aperture domain of the dimension n?3. Our proof is based on L^p–L^q estimates of the Stokes semigroup. We apply L^p–L^q estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

The Cauchy Problem for the Kadomtsev-Petviashvili (KPII) Equation in Three Space Dimensions

We prove the existence and uniqueness of local solutions in time for the Kadomtsev-Petviashvili equation (KPII) in ?³ for initial data in a suitable anisotropic Sobolev space of functions which have essentially 1⁺ derivative in the variable x and 0⁺ derivatives in the variables y and z.     

Averaging of a 3D primitive equations with oscillating external forces

In this article, we consider a non-autonomous three-dimensional primitive equations of the ocean with a singularly oscillating external force g ^ε?=?g ₀(t)?+?ε^?ρ g ₁(t/ε) depending on a small parameter ε?>?0 and ρ?∈?[0,?1) together with the averaged system with the external force g ₀(t), formally corresponding to the case ε?=?0. Under suitable assumptions on the external force, we prove as in [V.V. Chepyzhov, V. Pata, and M.M.I. Vishik, Averaging of 2D Navier–Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), pp. 351–370] the boundness of the uniform global attractor 𝒜^ε as well as the convergence of the attractors 𝒜^ε of the singular systems to the attractor 𝒜⁰ of the averaged system as ε?→?0⁺. When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by Kε^(1?ρ). Let us note that the main difference between this work and that of Chepyzhov et al. (2009) is that the non-linearity involved in the three-dimensional primitive equation is stronger than the one in the two-dimensional Navier–Stokes equations considered in Chepyzhov et al. (2009), which makes the analysis of the problem studied in this article more involved.     

Global solutions of the evolutionary Faddeev model with small initial data

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space ℝ¹⁺ⁿ to the unit sphere $ \mathbb{S} $ \mathbb{S} ², which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.     

Compatible wavelet-Galerkin method to solve stokes interior problem with circular boundary

This article addresses Neumann boundary value interior problem of Stokes equations with circular boundary. By using natural boundary element method, the Stokes interior problem is reduced into an equivalent natural integral equation with a hyper-singular kernel, which is viewed as Hadamard finite part. Based on trigonometric wavelet functions, the compatible wavelet space is constructed so that it can serve as Galerkin trial function space. In proposed compatible wavelet-Galerkin method, the simple and accurate computational formulae of the entries in stiffness matrix are obtained by singularity removal technique. It is also proved that the stiffness matrix is almost a block diagonal matrix, and its diagonal sub-blocks all are both symmetric and circulant submatrices. These good properties indicate that a 2 ^J+3 × 2 ^J+3 stiffness matrix can be determined only by its 2 ^J + 3J + 1 entries. It greatly decreases the computational complexity. Some error estimates for the compatible wavelet-Galerkin projection solutions are established. Finally, several numerical examples are given to demonstrate the validity of the proposed approach.     

Global periodicity and complete integrability of discrete dynamical systems

Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F ^p (x)=x for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.     

The initial boundary value problem for Navier-Stokes equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC ³ boundary, under the assumption that |a| _L ²(Θ) + |f| _L ¹(0,∞;L ²(Θ)) or |∇a| _L ²(Θ) + |f| _L ²(0,∞;L ²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed. This work is supported by foundation of Institute of Mathematics, Academia Sinica     

Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L∞（0, T; L^n（Ω））

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞（0, T; L^n（Ω）. We prove that if the velocity u belongs to the critical space L∞（0, T; L^n（Ω）, the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.     

BMO Functions on Compact Sets

Let P be an elliptic differential operator of order p with real analytic coefficients on in open set Q ? ?ⁿ. Given a compact set K ? Ω, we describe the closure in BMO(K) of the space of mentions of Pf = 0 on neighborhoods of K.     

On a decay rate for 1D-viscous compressible barotropic fluid equations

The Navier-Stokes equations of a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H ¹ as well as the mass force such that the stationary density is positive. The uniform lower bound for the density is proved. By constructing suitable Lyapunov functionals, decay rate estimates in L ²-norm and H ¹-norm are given. The decay rate is exponential if so the decay rate of the nonstationary part of the mass force is. The results are proved in the Eulerian coordinates for a wide class of increasing state functions including with any γ > 0 as well as functions of arbitrarily fast growth. We also extend the results for equations of a multicomponent compressible barotropic mixture (in the absence of chemical reactions). Received December 20, 2000; accepted February 27, 2001.     

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

We propose and analyze a fully discrete H ¹-Galerkin method with quadrature for nonlinear parabolic advection–diffusion–reaction equations that requires only linear algebraic solvers. Our scheme applied to the special case heat equation is a fully discrete quadrature version of the least-squares method. We prove second order convergence in time and optimal H ¹ convergence in space for the computer implementable method. The results of numerical computations demonstrate optimal order convergence of scheme in H ^k for k = 0, 1, 2. Support of the Australian Research Council is gratefully acknowledged.     

Two-level linearized and local uncoupled difference schemes for the two-component evolutionary Korteweg-de Vries system

The two-level linearized and local uncoupled spatial second order and compact difference schemes are derived for the two-component evolutionary system of nonhomogeneous Korteweg-de Vries equations. It is shown by the mathematical induction that these two schemes are uniquely solvable and convergent in a discrete L_∞ norm with the convergence order of O(τ² + h²) and O(τ² + h⁴), respectively, where τ and h are the step sizes in time and space. Three numerical examples are given to confirm the theoretical results.     

On the Nonexistence of Global Weak Solutions to the Navier–Stokes–Poisson Equations in ℝ N

In this paper we prove nonexistence of stationary weak solutions to the Euler–Poisson equations and the Navier–Stokes–Poisson equations in ? ^N , N ≥ 2, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler–Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier–Stokes–Poisson equations in ? ^N this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case.     

Decay of global solutions of two nonlinear evolution equations

1.IntroductionTherehavebeenconsiderableliteratuxeonthedecayofsolutionstothebestialvalueproblemsforsomenonlinearevolutionequations[3,4,6,7,161.Undercertainassumptions,LZdecayandLoodecayofsolutionstotheseproblemswereestablished.Thereadersinterestedcanfindsuchworksinourreferences.OurillterestisfocusedonthedecayofsolutionsoftheinitialvalueproblemsfornonlinearBenjamin--OnthBurgers(BOB)l"'19--21]andSchlodinger-Burgers(SB)equationwhereHisHilberttransform,definedbyWewallttoshowthattheLZandLoon…     

Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions

In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space L^p( R ⁿ). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in H^s(s≥0) and ill posed in H^s(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in L^p( R ²) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit p_c=1. In one‐dimensional case, we show that the problem is locally well posed in L¹( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.     

Weak Dispersive Estimates for Schrödinger Equations with Long Range Potentials

We prove local smoothing estimates for the Schrödinger initial value problem with data in the energy space L ²(? ^d ), d ≥ 2 and a general class of potentials. In the repulsive setting we have to assume just a power like decay (1 + |x|)^?γ for some γ > 0. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.     

Sharp regularity estimates for solutions of the wave equation and their traces with prescribed neumann data

In this paper the regularity properties of second-order hyperbolic equations defined over a rectangular domain Θ with boundary Γ under the action of a Neumann boundary forcing term inL ² (0,T;H ^1/4 (Γ)) are investigated. With this given boundary input, we prove by a cosine operator/functional analytical approach that not only is the solution of the wave equation and its derivatives continuous in time, with their pointwise values in a basic energy space (in the interior of Ω), but also that a trace regularity thereof can be assigned for the solution’s time derivative in an appropriate (negative) Sobolev space. This new-found information on the solution and its traces is crucial in handling a mathematical model derived for a particular fluid/structure interaction system.     

Boundary Value Problems with Compatible Boundary Conditions

If Y is a subset of the space ℝⁿ × ℝⁿ, we call a pair of continuous functions U, V Y-compatible, if they map the space ℝⁿ into itself and satisfy Ux · Vy ≥ 0, for all (x, y) ∈ Y with x · y ≥ 0. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential n-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer's fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points P _δ is obtained. Then passing to the limits as δ tends to zero the so-obtained accumulation points are solutions of the problem.