Global Mild Solution Research Articles

On the initial value problem of fractional stochastic evolution equations in Hilbert spaces

In this article, we are concerned with the initial value problem of fractional stochastic evolution equations in real separable Hilbert spaces. The existence of saturated mild solutions and global mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using $\alpha$-order fractional resolvent operator theory, the Schauder fixed point theorem and piecewise extension method. Furthermore, the continuous dependence of mild solutions on initial values and orders as well as the asymptotical stability in $p$-th moment of mild solutions for the studied problem have also been discussed. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract results.

Local and global existence results for the Navier-Stokes equations in the rotational framework

Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided only the horizontal components of the initial data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in [1,\infty)$.

Stochastic modified Boussinesq approximate equation driven by fractional Brownian motion

Abstract The current paper is devoted to the dynamics of a stochastic modified Boussinesq approximate equation driven by fractional Brownian motion with H ∈ ( 1 2 , 1 ) . Based on the different diffusion operators △ 2 and −△ in the stochastic system, we combine two types of operators Φ 1 = I and a Hilbert-Schmidt operator Φ 2 to guarantee the convergence of the corresponding Wiener-type stochastic integrals. Then the existence and regularity of the stochastic convolution for the corresponding additive linear stochastic equation can be shown. By the Banach modified fixed point theorem in the selected intersection space, the existence and uniqueness of the global mild solution are obtained. Finally, the existence of a random attractor for the random dynamical system generated by the mild solution for the modified Boussinesq approximation equation is also established. MSC:35B40, 35Q35, 76D05.

Asymptotic behavior for second order stochastic evolution equations with memory

In this paper, a general second order integro-differential evolution equation with memory driven by multiplicative noise is considered. We prove the existence of global mild solution and asymptotic stability of the zero solution using Lyapunov function techniques. Moreover, we discuss three examples to show that the asymptotic stability results can be applied to various partial differential equations.

Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients

In this paper we study the long--time dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. For this purpose, we begin by showing the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of non-stochastic integrals and the noisy path by using an integration by parts. This latter term causes that at first, only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the Holder norm of the noisy path to be sufficiently small. Subsequently, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, has an associated random attractor.

Stochastic nonlinear wave equation with memory driven by compensated Poisson random measures

In this paper, we study a class of stochastic nonlinear wave equation with memory driven by Lévy noise. We first show the existence and uniqueness of global mild solutions using a suitable energy function. Second, under some additional assumptions we prove the exponential stability of the solutions.

On existence and scattering theory for the Klein–Gordon–Schrödinger system in an infinite $$L^{2}$$ L 2 -norm setting

This paper is concerned with the initial value problem for the nonlinear Klein–Gordon–Schrodinger (KGS) system in $$\mathbb {R}^{n}\times \mathbb {R},\, n\ge 1$$ . We consider general polynomial nonlinearities that include in particular the classical Yukawa–KGS model. We show existence of local and global mild solutions for the KGS system with initial data in weak $$L^{r}$$ -spaces, which is an infinite $$L^{2}$$ -norm setting. Moreover, we obtain a persistence result in $$H^{s}$$ when the initial data belong to this class, which shows that the constructed data-solution map in weak- $$L^{r}$$ recovers $$H^{s} $$ -regularity. We also prove results of scattering and wave operators in that singular framework.

On the initial value problem of fractional evolution equations with noncompact semigroup

In this paper, we study the initial value problem of fractional semilinear evolution equations with noncompact semigroup in Banach spaces. The existence of saturated mild solutions and global mild solutions is obtained under weaker conditions. Particularly, we find that the results obtained are also valid for the initial value problem of fractional ordinary differential equations in Banach spaces. The theorems proved in this paper improve and extend some related conclusions on this topic. An example is given to illustrate that our results are valuable.

Results of local and global mild solution for impulsive fractional differential equation with state dependent delay

In this paper, we establish the existence of local and global mild solution for an impul- sive fractional integro-differential equation with state dependent delay subject to nonlocal initial condition. The existence results for local mild solution are proved by applying the Schauder, nonlinear Larey Schauder alternative and Banach fixed point theorems. Then, we prove global existence result. An example is presented to demonstrate the application of the established re- sults.

On a Fractional SPDE Driven by Fractional Noise and a Pure Jump Lévy Noise inℝd

We study a stochastic partial differential equation in the whole spacex∈ℝd, with arbitrary dimensiond≥1, driven by fractional noise and a pure jump Lévy space-time white noise. Our equation involves a fractional derivative operator. Under some suitable assumptions, we establish the existence and uniqueness of the global mild solution via fixed point principle.

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.

Well-posedness of fractional Ginzburg–Landau equation in Sobolev spaces

This article is concerned with real fractional Ginzburg–Landau equation. Existence and uniqueness of local and global mild solution for both whole space case and flat torus case are obtained by contraction semigroup method, and Gevrey regularity of mild solution for flat torus case is discussed.

The Fractional Ginzburg-Landau equation with distributional initial data

The paper is concerned with real fractional Ginzburg-Landau equation. Existence and uniqueness of local and global mild solution with distributional initial data are obtained by contraction mapping principle and carefully choosing the working space, and Gevrey regularity of mild solution for flat torus case is discussed.

Global mild solutions of the micropolar fluid system in critical spaces

We prove the global in time existence of a small solution for the 3D micropolar fluid system in critical Fourier–Herz spaces by using the Fourier localization method and Littlewood–Paley theory.

Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures

We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The so-called “method of the moving frame” allows us to reduce the SPDE problems to SDE problems.

On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1 driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.

Existence of local and global solutions of fractional-order differential equations

We study the existence of local and global mild solutions of the fractional-order differential equations in an arbitrary Banach space by using the semigroup theory and the Schauder fixed-point theorem. We also give some examples to illustrate the applications of abstract results.

Global Mild Solutions and Attractors for Stochastic Viscous Cahn‐Hilliard Equation

This paper is devoted to the study of mild solutions for the initial and boundary value problem of stochastic viscous Cahn‐Hilliard equation driven by white noise. Under reasonable assumptions we first prove the existence and uniqueness result. Then, we show that the existence of a stochastic global attractor which pullback attracts each bounded set in appropriate phase spaces.

Global Small Solutions for the Navier–Stokes–Maxwell System

We consider a full system of magnetohydrodynamic equations. The system does not enjoy any property of scaling invariance and, at least formally, has an energy estimate. Nevertheless, the existence of a global weak solution seems to remain an interesting open problem in both two and three space dimensions. In three dimensions, we show the existence of global small mild solutions. In two dimensions, we prove the same result in a space “close” to the energy space.

Existence of mild solutions for abstract mixed type semilinear evolution equations

This paper is concerned with the existence of global mild solutions and positive mild solutions to initial value problem for a class of mixed type semilinear evolution equations with noncompact semigroup in Banach spaces. The main method is based on a new fixed point theorem with respect to convex-power condensing operator.