Semigroup Estimates Research Articles

An application of weighted Hardy spaces to the Navier–Stokes equations

In this article, we consider the mapping properties of convolution operators with smooth functions on weighted Hardy spaces Hp(w) with w belonging to Muckenhoupt class A∞. As a corollary, one obtains decay estimates of heat semigroup on weighted Hardy spaces. After a weighted version of the div–curl lemma is established, these estimates on weighted Hardy spaces are applied to the investigation of the decay property of global mild solutions to Navier–Stokes equations with the initial data belonging to weighted Hardy spaces.

Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the L 1-estimate between the entropy solution and the geometric optics expansion function is bounded by O(ϵ2), independent of the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.

ALMOST PERIODIC SOLUTIONS TO STOCHASTIC EVOLUTION EQUATIONS ON BANACH SPACES

We prove the existence of almost periodic solutions to a class of abstract stochastic evolution equations on a Banach space E, [Formula: see text] Both autonomous (A is a C0-semigroup generator) and non-autonomous (A(t) satisfies conditions of Acquistapace–Terreni and generates a strongly continuous evolution family) cases are studied. Results are based on the theory of stochastic integration on Banach spaces of van Neerven and Weis and R-boundedness estimates for semigroups and evolution families due to Hytönen and Veraar. An example is given for a non-autonomous second order boundary value problem on a domain in ℝd.

Regularity of global solution to atmospheric circulation equations with humidity effect

In this article, the regularity of the global solutions to atmospheric circulation equations with humidity effect is considered. Firstly, the formula of the global solutions is obtained by using the theory of linear operator semigroups. Secondly, the regularity of the global solutions to atmospheric circulation equations is presented by using mathematical induction and regularity estimates for the linear semigroups. MSC:35D35, 35K20, 35Q35.

Existence and Regularity of Solutions to Model for Liquid Mixture of 3He-4He

Existence and regularity of solutions to model for liquid mixture of 3He-4He is considered in this paper. First, it is proved that this system possesses a unique global weak solution in H1(ω,C×R) by using Galerkin method. Secondly, by using an iteration procedure, regularity estimates for the linear semigroups, it is proved that the model for liquid mixture of 3He-4He has a unique solution in Hk(ω,C×R) for all k ≥ 1.

Hardy Inequality and Heat Semigroup Estimates for Riemannian Manifolds with Singular Data

Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on ∂D, and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in L 2(D) satisfies a strong Hardy inequality with weight δ2, (ii) the initial temperature distribution, and the specific heat of D are given by δ−α and δ−β respectively, where δ is the distance to ∂D, and 1 < α <2, 1 < β <2.

Regularity of global attractor for the fourth-order reaction–diffusion equation

In this paper, it is proved that the fourth-order reaction–diffusion equation possesses a global attractor in Sobolev space Hk for all k>0, which attracts any bounded subset of Hk(Ω) in the Hk-norm by using an iteration procedure, regularity estimates for the linear semigroups and a classical existence theorem of global attractor.

Regularity of Global Attractor for the Reaction‐Diffusion Equation

By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction‐diffusion equation possesses a global attractor in Sobolev space Hk for all k &gt; 0, which attracts any bounded subset of Hk(Ω) in the Hk‐norm.

Estimate of an Hypoelliptic Heat-Kernel outside the Cut-Locus in Semi-Group Theory

We give a proof in semi-group theory based on the Malliavin Calculus of Bismut type in semi-group theory and Wentzel-Freidlin estimates in semi-group of our result giving an expansion of an hypoelliptic heat-kernel outside the cut-locus.

Global existence and optimal decay rate of the compressible Navier–Stokes–Korteweg equations with external force

In this paper, we consider the three-dimensional compressible fluid models of Korteweg type. The equation takes the form of the compressible Navier–Stokes equation, with the Cauchy stress tensor in the momentum equations. Under the smallness assumption of the external force in some Sobolev space, we first show the existence of stationary solution by standard iterative arguments. Next, we present that the global-in-time solution of the initial value problem for the three-dimensional compressible Navier–Stokes–Korteweg equations exists uniquely and approaches the stationary solution as t→∞, provided the prescribed initial data and the external force are sufficiently small. Finally, based on the elaborate energy estimates of the solution for the nonlinear system and L2-decay estimates of semigroup generated by the corresponding linearized equation, we show the optimal L2-convergence rates of the solution towards the stationary solution. As far as we know, this is the first result about the global existence and L2-decay rate of smooth solutions for the compressible Navier–Stokes–Korteweg equations with the external force.

Global attractor of the extended Fisher-Kolmogorov equation in H k spaces

The long-time behavior of solution to extended Fisher-Kolmogorov equation is considered in this article. Using an iteration procedure, regularity estimates for the linear semigroups and a classical existence theorem of global attractor, we prove that the extended Fisher-Kolmogorov equation possesses a global attractor in Sobolev space H k for all k > 0, which attracts any bounded subset of H k (Ω) in the H k -norm. 2000 Mathematics Subject Classification: 35B40; 35B41; 35K25; 35K30.

Blow-up profiles of solutions for the exponential reaction-diffusion equation

We consider the blow-up of solutions for a semilinear reaction diffusion equation with exponential reaction term. It is know that certain solutions that can be continued beyond the blow-up time possess a nonconstant selfsimilar blow-up profile. Our aim is to find the final time blow-up profile for such solutions. The proof is based on general ideas using semigroup estimates. The same approach works also for the power nonlinearity.

EXPONENTIAL MIXING FOR SOME SPDEs WITH LÉVY NOISE

We show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in infinite dimensions. We concentrate on semilinear systems driven by cylindrical α-stable noises introduced in [18], α ∈ (0,2). We first prove that if the nonlinearity is bounded, then the system is ergodic and strong mixing. Then we show that the system is exponentially mixing provided that the nonlinearity, or its Lipschitz constant, are sufficiently small.

Semigroup estimates and stability/instability results for the linearized three waves interaction equations

We consider the three waves interaction system and its linearization (the “pump-wave approximation”). We give some estimates on the semigroup as well as stability or instability results for the linearized problem in suitable norms. We work in the whole space and with periodic boundary condition, and our analysis relies on energy estimates and not on the complete integrability of the system.

Global attractor of a modified Swift–Hohenberg equation in [formula omitted] spaces

In this paper, by using an iteration procedure, regularity estimates for the linear semigroups and a classical existence theorem of global attractor, we prove that a modified Swift–Hohenberg equation u t = − Δ 2 u + g ( u ) possesses a global attractor in Sobolev space H k for all k ≥ 0 , which attracts any bounded subset of H k ( Ω ) in the H k -norm.

Global strong solution to the semi-linear Keller–Segel system of parabolic–parabolic type with small data in scale invariant spaces

We shall show existence of global strong solution to the semi-linear Keller–Segel system in R n , n ⩾ 3 , of parabolic–parabolic type with small initial data u 0 ∈ H n r − 2 , r ( R n ) and v 0 ∈ H n r , r ( R n ) for max { 1 , n / 4 } < r < n / 2 . Our method is based on the perturbation of linealization together with the L p – L q estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall prove the decay property of solutions as the time goes to infinity.

Global attractor of the Cahn–Hilliard equation in Hk spaces

In this paper, by using an iteration procedure, regularity estimates for the linear semigroups and a classical existence theorem of global attractor we prove that the Cahn–Hilliard equation u t = − Δ 2 u + Δ g ( u ) possesses a global attractor in Sobolev space H k for all k ⩾ 0 , which attracts any bounded subset of H k ( Ω ) in the H k -norm.

Difference estimates for continuous and discrete operator semigroups

For suitable bounded operator semigroups (etA)t≥0 in a Banach space, we characterize the estimate ‖AetA‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖Tn−Tn+1‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n1/2 for large n.

The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(\mathbb{R}^n)$ and its application to self-similar solutions

We shall show the existence of a global strong solution to the semilinear Keller-Segel system in Rn , n=3 of parabolic-parabolic type with small initial data u0?Lwn/2(Rn) and v0?BMO . Our method is based on the perturbation of linearization together with the Lp-Lq -estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall construct a self-similar solution and prove the smoothing effect. Furthermore, the stability problem on our strong solutions will be also discussed.

Stability of travelling waves with noncritical speeds for double degenerate Fisher-Type equations

This paper is concerned with the asymptotic stability of travel- ling wave solutions for double degenerate Fisher-type equations. By spectral analysis, each travelling front solution with non-critical speed is proved to be linearly exponentially stable in some exponentially weighted spaces. Further by Evans function method and detailed semigroup estimates each travelling front solution with non-critical speed is proved to be locally algebraically stable to perturbations in some polynomially weighted spaces, and it is also locally exponentially stable to perturbations in some polynomially and exponentially weighted spaces.