Singular initial data and uniform global bounds for the hyper-viscous Navier-Stokes equation with periodic boundary conditions

In the hyper-viscous Navier-Stokes equations of incompressible flow, the operator A=−Δ is replaced by A_α,a,b≡aA^α+bA for real numbers α,a,b with α?1 and b?0. We treat here the case a>0 and equip A (and hence A_α,a,b) with periodic boundary conditions over a rectangular solid Ω⊂Rn. For initial data in Lp(Ω) with α?n/(2p)+1/2 we establish local existence and uniqueness of strong solutions, generalizing a result of Giga/Miyakawa for α=1 and b=0. Specializing to the case p=2, which holds a particular physical relevance in terms of the total energy of the system, it is somewhat interesting to note that the condition α?n/4+1/2 is sufficient also to establish global existence of these unique regular solutions and uniform higher-order bounds. For the borderline case α=n/4+1/2 we generalize standard existing (for n=3) “folklore” results and use energy techniques and Gronwall's inequality to obtain first a time-dependent H^α-bound, and then convert to a time-independent global exponential H^α-bound. This is to be expected, given that uniform bounds already exist for n=2,α=1 ([6, pp. 78-79]), and the folklore bounds already suggest that the α?n/4+1/2 cases for n?3 should behave as well as the n=2 case. What is slightly less expected is that the n?3 cases are easier to prove and give better bounds, e.g. the uniform bound for n?3 depends on the square of the data in the exponential rather than the fourth power for n=2. More significantly, for α>n/4+1/2 we use our own entirely semigroup techniques to obtain uniform global bounds which bootstrap directly from the uniform L²-estimate and are algebraic in terms of the uniform L²-bounds on the initial and forcing data. The integer powers on the square of the data increase without bound as α↓n/4+1/2, thus “anticipating” the exponential bound in the borderline case α=n/4+1/2. We prove our results for the case a=1 and b=0; the general case with a>0 and b?0 can be recovered by using norm-equivalence. We note that the hyperviscous Navier-Stokes equations have both physical and numerical application.     

Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

Existence of global solutions of multicomponent reactive transport problems with mass action kinetics in porous media

We prove the existence and uniqueness of time-global solutions for multi-species multi-reaction advection-diffusion-dispersion problems with mass action kinetics in the space \(W_p^{2,1}([0,T]\!\times\!\Omega)\). The reaction terms of mass action kinetics may contain polynomial expressions of arbitrarily high order. The difficulty to obtain an a~priori estimate for the semilinar system of PDEs is tackled with a special Lyapunov function.     

On the Cauchy Problem of the Camassa-Holm Equation

The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method proposed and studied by T. Kato and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa-Holm equation in L ^p frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.     

The Cauchy problem for non-autonomous nonlinear Schrödinger equations

In this paper we study the Cauchy problem for cubic nonlinear Schrödinger equation with space- and time-dependent coefficients on ∝^m and \(\mathbb{T}^m \). By an approximation argument we prove that for suitable initial values, the Cauchy problem admits unique local solutions. Global existence is discussed in the cases of m = 1, 2.     

Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third ＋α the second partial dervative of u to x ,the second the third ＋β the third partial dervative of u to x ,the second the thire ＋γ the fifth partial dervative of u to x = 0,（x,t）∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo（x） ∈ H^s（R） with s ≥ 1/4, and a global solution exists if s ≥ 2.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

A doubly critical degenerate parabolic problem

It is shown that the Dirichlet problem for where Ω??ⁿ is critical in that it has first eigenvalue one, is globally solvable for any continuous positive initial datum vanishing at ?Ω. Moreover, for p<3 all solutions are bounded and tend to some nonnegative eigenfunction of the Laplacian as t→∞, while if p?3 then there are both bounded and unbounded solutions. Finally, it is shown that unlike the case p∈[0,1), all steady states are unstable if p?1. Copyright © 2004 John Wiley & Sons, Ltd.     

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.     

七阶非线性色散方程初值问题解的局部和整体存在性

该文研究七阶非线性弱色散方程:∂u/∂t + au(∂u/∂x) +β(∂^3 u/∂x^3) +γ(∂^5 u/∂x^5) + μ(∂^7 u/∂x^7)=0, (x,t)∈R^2的初值问题,通过运用震荡积分衰减估计的最近结果, 首先对相应线性方程的基本解建立了几类Strichartz型估计. 其次, 应用这些估计证明了七阶非线性弱色散方程初值问题解的局部与整体存在性和唯一性. 结果表明, 当初值u_0(x)∈H^s(R), s≥2/13 时, 存在局部解; 当s≥1时, 存在整体解.     

Global existence of three dimensional incompressible MHD flows

We investigate the initial value problem for the three‐dimensional incompressible magnetohydrodynamics flows. Global existence and uniqueness of flows are established in the function space , provided that the norm of the initial data is less than the minimal value of the viscosity coefficients of the flows. Copyright © 2016 John Wiley & Sons, Ltd.     

一类偏差依赖状态自身的泛函微分方程

本文证明了一类偏差依赖状态自身的泛函微分方程ｘ′（ｔ）＝ａ（ｔ）ｆ（ｘ（ｔ），ｘ（ｘ（ｔ）））之所有强弱解都必单调，并且首次详尽地研究了方程ｘ′（ｔ）＝ａｘ（ｔ）＋ｂｘ（ｘ（ｔ）），ａ·ｂ≠０强解的存在性及渐近性态。     

Existence of solution to Korteweg‐de Vries equation in domains that can be transformed into rectangles

We study the Korteweg‐de Vries equation subject to boundary condition in nonrectangular domain where , with some assumptions on functions (φ_i(t))_1≤i≤2 and the coefficients of equation. The right‐hand side and its derivative with respect to t are in the Lebesgue space L²(Ω). Our goal is to establish the existence, the uniqueness, and the regularity of the solution.     

Global solutions for nonlinear Klein-Gordon equations in infinite homogeneous waveguides

In this paper we prove a global existence result for nonlinear Klein-Gordon equations in infinite homogeneous waveguides, R×M, with smooth small data, where M=(M,g) is a Zoll manifold, or a compact revolution hypersurface. The method is based on normal forms, eigenfunction expansion and the special distribution of eigenvalues of the Laplace-Beltrami on such manifolds.     

Boundedness in a chemotaxis model with oxygen consumption by bacteria

This paper deals with the Keller-Segel model

     

Absence of collapse in a parabolic chemotaxis system with signal‐dependent sensitivity

We consider the chemotaxis system under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ? ?ⁿ. The chemotactic sensitivity function is assumed to generalize the prototype It is proved that no chemotactic collapse occurs in the sense that for any choice of nonnegative initial data (with some r > n), the corresponding initial‐boundary value problem possesses a unique global solution that is uniformly bounded (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.