Global Existence Of Mild Solutions Research Articles

Mild solutions for some nonautonomous partial functional differential equations with infinite delay

The aim of this work is to establish several results on the existence of mild solutionse nondensely nonautonomous partial functional differential equations with infinite delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. We show the local existence of the mild solutions which may blow up at the finite time. Secondly, we get the existence of global mild solutions. For illustration, we propose applications for distributed delay and discrete delay.

Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of \begin{document}$\mathbb{R}^N,$\end{document} \begin{document}$N = 2, 3,$\end{document} for small initial data in \begin{document}$L^p$\end{document} -spaces.

Fractional Measure-dependent Nonlinear Second-order Stochastic Evolution Equations with Poisson Jumps

Abstract This paper focuses on a nonlinear second-order stochastic evolution equations driven by a fractional Brownian motion (fBm) with Poisson jumps and which is dependent upon a family of probability measures. The global existence of mild solutions is established under various growth conditions, and a related stability result is discussed. An example is presented to illustrate the applicability of the theory.

Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations

Abstract In this paper, we study a class of fractional semilinear integro-differential equations of order β ∈ (1,2] with nonlocal conditions. By using the solution operator, measure of noncompactness and some fixed point theorems, we obtain the existence of local and global mild solutions for the problem. The results presented in this paper improve and generalize many classical results. An example about fractional partial differential equations is given to show the application of our theory.

Global well-posedness of solutions for magnetohydrodynamics-α model in R3

In this paper, by using Strauss’ inequality, we prove the existence of global mild solutions to the 3D magnetohydrodynamics- α equations with small initial data.

Abstract functional second-order stochastic evolution equations with applications

We investigate a class of abstract second-order damped functional stochastic evolution equations driven by a fractional Brownian motion in a separable Hilbert space. The global existence of mild solutions is established under various growth and compactness conditions. The case of a nonlocal initial condition is addressed. A related convergence result is discussed, and the theory is applied to stochastic wave and beam equations, as well as a spring-mass system, for illustrative purposes.

Global existence for the ellipsoidal BGK model with initial large oscillations

The ellipsoidal BGK model was introduced to fit the correct Prandtl number in the Navier-Stokes approximation of the classical BGK model. In this paper, we establish the global existence of mild solutions to the Cauchy problem on the model for a class of initial data allowed to have large oscillations. The proof is motivated by a recent study of the same topic on the Boltzmann equation.

Existence and analyticity of mild solutions for the 3D rotating Navier–Stokes equations

We study the three-dimensional Navier–Stokes equations in the rotational framework. By using the Littlewood–Paley analysis technique and the dispersive estimates for the Coriolis linear group \(\{e^{\pm i\Omega t\frac{D_3}{|D|}}\}_{t\in \mathbb {R}}\), we prove unique existence of global mild solutions to initial value problem and mild solutions to time-periodic problem of the rotating Navier–Stokes equations under some precise conditions, respectively, which permit the initial velocity and the time-periodic external force to be arbitrarily large provided that the speed of the rotation is fast enough. These results improve the related ones obtained in Iwabuchi and Takada (Math Ann 357:727–741, 2013) and Koh et al. (Adv diff Equ 19:857–878, 2014), and the result on the time-periodic problem can also be regarded as an enhancement and complement of that in Iwabuchi and Takada (J Evol Equ 12:985–1000, 2012). Meanwhile, based on the so-called Gevrey estimates, which are motivated by the work of Foias and Temam (J Funct Anal 87:359–369, 1989), we particularly verify that the obtained mild solutions are analytic in the spatial variables.

Local and global existence of mild solutions for a class of nonlinear fractional reaction–diffusion equations with delay

In this paper, we investigate the local and global existence of mild solutions to a class of nonlinear fractional reaction–diffusion equations with delay via the theory of β -resolvent family and fixed point theorems.

Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup

In this paper, we study local and global existence of mild solution for an impulsive fractional functional integro differential equation with non-compact semi-group in Banach spaces. We establish a general framework to find the mild solutions for impulsive fractional integro-differential equations, which will provide an effective way to deal with such problems. The theorems proved in this paper improve and extend some related conclusions on this topic. Finally, two applications are given to illustrate that our results are valuable.

Global well-posedness and analyticity results to 3-D generalized magnetohydrodynamic equations

This paper is devoted to studying the Cauchy problem for the incompressible generalized magnetohydrodynamic equations in three space dimensions. By using the Lei and Lin (2011) argument in a slightly different fashion, we obtain the global existence of mild solutions, which enables us to prove analyticity of the solution.

Existence and uniqueness theorem on mild solutions to the Keller–Segel system coupled with the Navier–Stokes fluid

We consider the Keller–Segel system coupled with the Navier–Stokes fluid in the whole space, and prove the existence of global mild solutions with the small initial data in the scaling invariant space. Our method is based on the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data. As a byproduct, we show the asymptotic stability of solutions as the time goes to infinity. Since we may deal with the initial data in the weak L p -spaces, the existence of self-similar solutions provided the initial data are small homogeneous functions.

A unified existence theory for evolution equations and systems under nonlocal conditions

We investigate the effect of nonlocal conditions expressed by linear continuous mappings over the hypotheses which guarantee the existence of global mild solutions for functional-differential equations in a Banach space. A progressive transition from the Volterra integral operator associated to the Cauchy problem, to Fredholm type operators appears when the support of the nonlocal condition increases from zero to the entire interval of the problem. The results are extended to systems of equations in a such way that the system nonlinearities behave independently as much as possible and the support of the nonlocal condition may differ from one variable to another.

Local and Global Existence of Mild Solution for Impulsive Fractional Stochastic Differential Equations

In this paper, the local and global existence of mild solutions are studied for impulsive fractional semilinear stochastic differential equation with nonlocal condition in a Hilbert space. The results are obtained by employing fixed-point technique and solution operator. In many existence results for stochastic fractional differential systems, the value of \(\alpha \) is restricted to \(\frac{1}{2} < \alpha \le 1;\) the aim of this manuscript is to extend the results which are valid for all values of \(\alpha \in (0,\,1).\) An example is provided to illustrate the obtained theoretical results.

Global well-posedness of the three dimensional magnetohydrodynamics equations

In this paper we investigate the Cauchy problem for the incompressible MHD equations in three space dimensions. If the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients, global existence of mild solutions is obtained.

Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition

This paper is concerned with the local and global existence of mild solution for an impulsive fractional functional integro differential equations with nonlocal condition. We establish a general framework to find the mild solutions for impulsive fractional integro-differential equations, which will provide an effective way to deal with such problems. The results are obtained by using the fixed point technique and solution operator on a complex Banach space.

The inelastic Enskog equation with external force

In this paper, the Cauchy problem for the inelastic Enskog equation with external force is considered for near vacuum data. Under the assumptions on the bi-characteristic, we proved the global existence of mild solution.

Nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition

The aim of the present paper is to investigate the global existence of mild solutions of nonlinear mixed Volterra-Fredholm integrodifferential equations, with nonlocal condition. Our analysis is based on an application of the Leray-Schauder alternative and rely on a priori bounds of solutions.

The global existence of mild solutions for semilinear fractional Cauchy problems in the α-norm

The global existence of mild solutions for semilinear fractional Cauchy problems in the α-norm

On the Cauchy Problem for the Inelastic Boltzmann Equation with External Force

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered for near vacuum data. Under the assumptions on the bicharacteristic generated by external force which can be arbitrarily large, we prove the global existence of mild solution for initial data small enough with respect to the sup norm with exponential weight by using the contraction mapping theorem. Furthermore, we prove the uniform L1 stability of the mild solution following from the exponential decay estimate and the Gronwall’s inequality for the case of soft potentials.