Cung The Anh

In this paper, we study the Cauchy problem for a coupled system of damped wave equations with mixed product-type nonlinearities. We have succeeded in determining the critical curve, formulated in terms of the power exponents of nonlinearities, for the global (in time) existence of small data mild solutions and the nonexistence of global mild solutions. Moreover, the global existence result is established by means of Schauder's fixed point theorem and Banach's contraction principle, whereas the nonexistence analysis relies on a new iteration scheme adapted to the product-type structure of nonlinearities. To the best of our knowledge, this work provides the first critical threshold results for such a system.

We consider a Voigt-regularization of three-dimensional magnetohydrodynamic (MHD) equations in bounded domains with periodic boundary conditions. Under suitable assumptions on the external forces, we first prove the existence, uniqueness and Sobolev regularity of solutions to the equations. Then we investigate the convergence of solutions of the Voigt-regularization to that of the classical MHD equations as the regularization parameter tends to 0. We also estimate the number of determining nodes for the solutions of the equations. Finally, when the external forces are time-independent, we show the existence of a finite-dimensional global attractor for the associated continuous semigroup.

The aim of this paper is twofold. First, we develop the notion of decay characters and evaluate these characters in important data spaces that often appear in problems related to fluid mechanics. Next, we apply our evaluation for the decay character of the initial datum r^{*}(u_{0}) to study the decay of solution and its derivatives to n -dimensional generalized Navier–Stokes equations with fractional-order Laplace operator (-\Delta)^{\alpha} when 0<\alpha<\frac{n+2}{4} . We establish the optimal uniform decay estimates for the L^{2} -norm of solution u , as well as decay estimates for \|\Lambda^{m} u\|_{L^2} and \|\partial^{k}_{t}\Lambda^{m} u\|_{L^2} for positive integer k and non-negative real number m .

In this paper we study the Cauchy problem for some classes of weakly coupled systems of semilinear damped σ -evolution equations with unbounded coefficients of the form { u tt + a ( x )(– ∆ ) σ u + u t = F 1 ( v , v t ), v tt + a ( x )(– ∆ ) σ u + u t = F 2 ( u , u t ), u ( 0 , x )= u 0 , u t ( 0 , x )= u 1 , v ( 0 , x )= v 0 , v t ( 0 , x )= v 1 , where σ ∈(0,1), a ( x ) is a continuous function satisfying a ( x ) ≈ | x | α a.e. in R n with α ∈[0,2 σ ), the nonlinearities f ( v , v t ) and g ( u , u t ) are assumed to be of the form | v | p or | v t | p , and | u | q or | u t | q , respectively. Our main aim is to prove the global (in time) existence and decay rates of mild solutions to these systems with small initial data belonging to the weighted space L ρ 2 ∩ L ρ m , where ρ : = ρ ( x ) = a ( x ) – 1 . The proof is based on suitable linear estimates and the fixed point method using the Banach contraction mapping principle. A blow-up result in the case of systems with power nonlinearities is also given to illustrate the optimality of the corresponding global existence result.

We study a distributed optimal control problem for Navier-Stokes-Voigt equations in three-dimensional bounded domains with convex control constraints and a general cost functional. We prove the existence of optimal solutions, the first-order necessary optimality conditions, and the second-order sufficient optimality conditions as well as the Lipschitz stability of optimal solutions with respect to the initial data.

This work is a continuation of the previous one in [{\it Optimization} (2023)], where the existence of optimal solutions and first-order necessary optimality conditions in both Pontryagin's maximum principle form and the variational form were proved for a distributed optimal control problem governed by the three-dimensional viscous Camassa-Holm equations in bounded domains with the cost functional of a quite general form and pointwise control constraints. We will establish the second-order sufficient optimality conditions as well as the Lipschitz stability results of the control system with respect to perturbations of the initial data.

We study the three‐dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. First, we study the existence, uniqueness, and regularity of global weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since the system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu which is called evolutionary system to obtain various attractors and their properties.