Blow-up for doubly degenerate nonlinear parabolic equations

In this paper, we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem of some doubly degenerate nonlinear parabolic equations. The project is supported by the Natural Science Foundation of Fujian Province of China (No. Z0511048)     

广义指数二分性与微分方程的不变流形

用 Hadamar和 Bogoliubov的方法,在常微分方程、泛函微分方程以及半线性抛物型方程所能满足的条件下对Banach或Hilbert空间上的非自治抽象微分方程建立了不变流形理论。首先,对相应的线性方程提出了“广义指数二分性”概念并讨论了它和线性方程谱的关系,然后,我们给出了不变流形存在性结论以及强稳定(不稳定)流形、弱稳定(不稳定)流形和弱双曲流形的分类。进而,我们对这些不变流形给出了C~k光滑性、周期性、概周期性和吸斥性的结论。     

PERSISTENCE AND TRANSVERSALITY OF HOMOCLINIC ORBITS FOR DIFFEOMORPHISMS

A generalized method combining the exponential dichotomy and the theory of trans-versality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeomorphisms.     

Ergodic behaviour of stochastic parabolic equations

The ergodic behaviour of homogeneous strong Feller irreducible Markov processes in Banach spaces is studied; in particular, existence and uniqueness of finite and -finite invariant measures are considered. The results obtained are applied to solutions of stochastic parabolic equations.     

Invariant algebraic curves and rational first integrals of holomorphic foliations in CP(2)

In this paper, using the tools of algebraic geometry we provide sufficient conditions for a holomor-phic foliation in CP(2) to have a rational first integral. Moreover, we obtain an upper bound of the degrees of invariant algebraic curves of a holomorphic foliation in CP(2). Then we use these results to prove that any holomorphic foliation of degree 2 does not have cubic limit cycles.     

A space decomposition method for parabolic equations

A convergence proof is given for an abstract parabolic equation using general space decomposition techniques. The space decomposition technique may be a domain decomposition method, a multilevel method, or a multigrid method. It is shown that if the Euler or Crank–Nicolson scheme is used for the parabolic equation, then by suitably choosing the space decomposition, only O(| log τ |) steps of iteration at each time level are needed, where τ is the time-step size. Applications to overlapping domain decomposition and to a two-level method are given for a second-order parabolic equation. The analysis shows that only a one-element overlap is needed. Discussions about iterative and noniterative methods for parabolic equations are presented. A method that combines the two approaches and utilizes some of the good properties of the two approaches is tested numerically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 27–46, 1998     

Transversal infinitesimal automorphisms of harmonic foliations on complete manifolds

Summary In this paper we study infinitesimal automorphisms of finiteL ²-norm for harmonic Riemannian and Kähler foliations admitting a complete bundle-like metric. The results generalize facts established recently in the compact case.1980 Mathematics Subject Classification: Primary 57 R 30, Secondary 58 E 20.Work supported in part by a grant from the National Science Foundation.     

Boundedness and blowup for nonlinear degenerate parabolic equations

The author deals with the quasilinear parabolic equation _ut=[u^α+g(u)]Δu+bu^α+1+f(u,∇u)$u_t=[u^{\alpha }+g\left(u\right)]\Delta u+b{u}^{\alpha +1}+f(u,\nabla u)$ with Dirichlet boundary conditions in a bounded domain Ω$\Omega$, where f$f$ and g$g$ are lower-order terms. He shows that, under suitable conditions on f$f$ and g$g$, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −Δ$-\Delta$ in Ω$\Omega$ with Dirichlet boundary condition. For some special cases, the result is sharp.     

Uniform exponential dichotomy of parabolic operators with rapidly oscillating coefficients

The paper deals with averaging problems for parabolic equations. We prove that exponential dichotomy is preserved without any assumption concerning the almost-periodicity of the coefficients.     

A second-order ADI scheme for three-dimensional parabolic differential equations

A second-order unconditionally stable ADI scheme has been developed for solving three-dimensional parabolic equations. This scheme reduces three-dimensional problems to a succession of one-dimensional problems. Further, the scheme is suitable for simulating fast transient phenomena. Numerical examples show that the scheme gives an accurate solution for the parabolic equation and converges rapidly to the steady state solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:159–168, 1998     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Degenerate parabolic stochastic partial differential equations

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.     

Error estimates for discontinuous Galerkin method for nonlinear parabolic equations

We consider the nonlinear parabolic partial differential equations. We construct a discontinuous Galerkin approximation using a penalty term and obtain an optimal L^∞(L²) error estimate.     

Stability of solutions of quasilinear parabolic equations

We bound the difference between solutions u and v of ut=aΔu+divxf+h and vt=bΔv+divxg+k with initial data φ and ψ, respectively, by

     

On the maximum principle for linear parabolic equations

We prove extensions of our previous estimates for linear elliptic equations with inhomogeneous terms in L ^p spaces, p ≤ n to linear parabolic equations with inhomogeneous terms in L ^p , p ≤ n + 1. As with the elliptic case, our results depend on restrictions on parabolicity determined by certain subcones of the positive cone . They also extend the maximum principle of Krylov for the case p = n + 1, corresponding to the usual parabolicity.     

Time periodic solutions of a class of degenerate parabolic equations

1.IntroductionManypapershavebeendevotedtotheexistenceoftimeperiodicsolutionsforsemilinearparabolicequations,see[1--8].Atthesametime,thestudyofquasilinearperiodic-parabolicequationsalsoattractedmanyauthors,seealso[9--141.Inparticular,recentlyHess,PozioandTesei[13]usedthemonotonicitymethodstodealwiththeequationsonot=aam a(x,t)u,wherem>1andaisafunctionperiodicint,andMizoguchi[lllappliedtheLeray-Schauderdegreetheorytoinvestigatetheequationswithsuperlinearforcingtermwherem>1,hisapositiveperiodicf…     

Dynamic adaptation for parabolic equations

A dynamic adaptation method is presented that is based on the idea of using an arbitrary time-dependent system of coordinates that moves at a velocity determined by the unknown solution. Using some model problems as examples, the generation of grids that adapt to the solution is considered for parabolic equations. Among these problems are the nonlinear heat transfer problem concerning the formation of stationary and moving temperature fronts and the convection-diffusion problems described by the nonlinear Burgers and Buckley-Leverette equations. A detailed analysis of differential approximations and numerical results shows that the idea of using an arbitrary time-dependent system of coordinates for adapted grid generation in combination with the principle of quasi-stationarity makes the dynamic adaptation method universal, effective, and algorithmically simple. The universality is achieved due to the use of an arbitrary time-dependent system of coordinates that moves at a velocity determined by the unknown solution. This universal approach makes it possible to generate adapted grids for time-dependent problems of mathematical physics with various mathematical features. Among these features are large gradients, propagation of weak and strong discontinuities in nonlinear transport and heat transfer problems, and moving contact and free boundaries in fluid dynamics. The efficiency is determined by automatically fitting the velocity of the moving nodes to the dynamics of the solution. The close relationship between the adaptation mechanism and the structure of the parabolic equations allows one to automatically control the nodes’ motion so that their trajectories do not intersect. This mechanism can be applied to all parabolic equations in contrast to the hyperbolic equations, which do not include repulsive components. The simplicity of the algorithm is achieved due to the general approach to the adaptive grid generation, which is independent of the form and type of the differential equations.