Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in Lipschitz domains

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in a bounded Lipschitz domain in , with small boundary datum in L²‐based Sobolev spaces. A useful intermediary result is the well‐posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in , with Dirichlet boundary condition and data in L²‐based Sobolev spaces. We obtain this well‐posedness result by showing that the matrix type operator associated with the Poisson problem is an isomorphism. Then, we combine the well‐posedness result from the linear case with a fixed point theorem in order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy–Forchheimer–Brinkman system. Some applications are also included. Copyright © 2014 John Wiley & Sons, Ltd.     

Global well‐posedness for free boundary problem of the Oldroyd‐B model fluid flow

In this paper, we prove the global well‐posedness of non‐Newtonian viscous fluid flow of the Oldroyd‐B model with free surface in a bounded domain of N‐dimensional Euclidean space . The assumption of the problem is that the initial data are small enough and orthogonal to rigid motions. Copyright © 2015 John Wiley & Sons, Ltd.     

Global well‐posedness result for density‐dependent incompressible viscous fluid in with linearly growing initial velocity

In this paper, we consider the density‐dependent incompressible Navier–Stokes equations in with linearly growing initial velocity at infinity. We obtain a blow‐up criterion and global well‐posedness of the two‐dimensional system. It generalized the local well‐posedness results due to the recent work by the first and third authors to the global well‐posedness in . Copyright © 2012 John Wiley & Sons, Ltd.     

Well‐posedness for the Cauchy problem of the modified Hunter–Saxton equation in the Besov spaces

This paper is concerned with the Cauchy problem of the modified Hunter‐Saxton equation. The local well‐posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces H^s) by using Littlewood‐Paley decomposition and transport equation theory. Moreover, the local well‐posedness in critical case (with ) is considered.     

Nonstationary Stokes system in Besov spaces

We examine the solvability in Besov spaces of an initial–boundary value problem for the nonstationary Stokes system with the slip boundary conditions. We prove the existence and uniqueness of solutions to the problem in a bounded domain . The existence is shown by localizing the system to interior and boundary subdomains of Ω. The localized Stokes system is transformed by the Helmholtz–Weyl decomposition to the heat and the Poisson equations, which are solved in the Besov spaces. Next, by the properties of the partition of unity and a perturbation argument, the existence is proved in domain Ω. Copyright © 2013 John Wiley & Sons, Ltd.     

Well‐posedness for fractional Navier–Stokes equations in the largest critical spaces

This note studies the well‐posedness of the fractional Navier–Stokes equations in some supercritical Besov spaces as well as in the largest critical spaces for β ∈ (1/2,1). Meanwhile, the well‐posedness for fractional magnetohydrodynamics equations in these Besov spaces is also studied. Copyright © 2012 John Wiley & Sons, Ltd.     

Global existence and asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

This paper considers the 2‐species chemotaxis‐Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a 3‐dimensional bounded domain with smooth boundary. Both chemotaxis‐fluid systems and 2‐species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2‐species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (u·∇)u in the fluid equation were established in the 2‐dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3‐dimensional case when is sufficiently small.     

Multiplicity of nontrivial solutions to a biharmonic equation via Lusternik–Schnirelman theory

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of biharmonic problem where is a bounded domain with smooth boundary. Using the Lusternik–Schnirelman theory, we relate the number of solutions with the topology of Ω. Copyright © 2012 John Wiley & Sons, Ltd.     

Regular attractor for damped KdV‐Burgers equations on R

We study the well‐posedness and dynamic behavior for the KdV‐Burgers equation with a force on R . We establish L ^p ?L ^q estimates of the evolution , as an application we obtain the local well‐posedness. Then the global well‐posedness follows from a uniform estimate for solutions as t goes to infinity. Next, we prove the asymptotical regularity of solutions in space and by the smoothing effect of . The regularity and the asymptotical compactness in L ² yields the asymptotical compactness in by an interpolation arguement. Finally, we conclude the existence of an globalattractor.     

Global attractor for suspension bridge equations with memory

This paper is concerned with a suspension bridge equation with memory effects , defined in a bounded domain of . For the suspension bridge equation without memory, there are many classical results. Existing results mainly devoted to existence and uniqueness of a weak solution, energy decay of solution and existence of global attractors. However the existence of global attractors for the suspension bridge equation with memory was no yet considered. The object of the present paper is to provide some results on the well‐posedness and long‐time behavior to the suspension bridge equation in a more with past history. Copyright © 2015 John Wiley & Sons, Ltd.     

Global well‐posedness and scattering of a generalized nonlinear fourth‐order wave equation

In this paper, we study the global well‐posedness and scattering theory of the solution to the Cauchy problem of a generalized fourth‐order wave equation where if d ?4, and if d ?5. The main strategy we use in this paper is concentration‐compactness argument, which was first introduced by Kenig and Merle to handle the scattering problem vector so as to control the momentum. Copyright © 2017 John Wiley & Sons, Ltd.     

Global well posedness of 3D‐NSE in Fourier–Lei–Lin spaces

In this paper, we prove a global well posedness of the three‐dimensional incompressible Navier–Stokes equation under an initial data, which belong to the non‐homogeneous Fourier–Lei–Lin space for σ ? ? 1 and if the norm of the initial data in the Lei–Lin space is controlled by the viscosity. Copyright © 2016 John Wiley & Sons, Ltd.     

On the well‐posedness for the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals in critical spaces

In recent paper, we prove the well‐posedness for the heat flow of harmonic maps with initial data and for the hydrodynamic flow of nematic liquid crystals with initial data . Copyright © 2011 John Wiley & Sons, Ltd.     

Pullback attractors of 2D incompressible Navier‐Stokes‐Voight equations with delay

In this paper, the 2D Navier‐Stokes‐Voight equations with 3 delays in is considered. By using the Faedo‐Galerkin method, Lions‐Aubin lemma, and Arzelà‐Ascoli theorem, we establish the global well‐posedness of solutions and the existence of pullback attractors in H¹.     

Bilinear estimate on tent‐type spaces with application to the well‐posedness of fluid equations

We introduce a class of tent‐type spaces and establish a Poisson extension result of Triebel–Lizorkin spaces . As an application, we get the well‐posedness of Navier–Stokes equations and magnetohydrodynamic equations with initial data in critical Triebel–Lizorkin spaces , . Copyright © 2016 John Wiley & Sons, Ltd.     

On the mixed problem for the semilinear Darcy‐Forchheimer‐Brinkman PDE system in Besov spaces on creased Lipschitz domains

The purpose of this paper is to study the mixed Dirichlet‐Neumann boundary value problem for the semilinear Darcy‐Forchheimer‐Brinkman system in L _p ‐based Besov spaces on a bounded Lipschitz domain in ${\mathbb{R}}^3$, with p in a neighborhood of 2. This system is obtained by adding the semilinear term | u | u to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well‐posedness results for the Dirichlet and Neumann problems in L _p ‐based Besov spaces on bounded Lipschitz domains in ${\mathbb{R}}^n$(n ≥3) are also presented. Then, using integral potential operators, we show the well‐posedness in L ₂‐based Sobolev spaces for the mixed problem of Dirichlet‐Neumann type for the linear Brinkman system on a bounded Lipschitz domain in ${\mathbb{R}}^n$(n ≥3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann‐to‐Dirichlet operator, we extend the well‐posedness property to some L _p ‐based Sobolev spaces. Next, we use the well‐posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy‐Forchheimer‐Brinkman system in L _p ‐based Besov spaces, with p ∈(2?ε ,2+ε ) and some parameter ε >0.     

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

This paper is focused on following time‐harmonic Maxwell equation: where is a bounded Lipschitz domain, is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as , we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor and permittivity tensor , ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.     

Global well‐posedness for the 2D dissipative quasi‐geostrophic equations in modulation spaces

We prove the existence of global weak solution of the two‐dimensional dissipative quasi‐geostrophic equations with small initial data in and local well‐posedness with the large initial data in the same space. Our proof is based on constructing a commutator related to the problem, as well as its estimate. Copyright © 2017 John Wiley & Sons, Ltd.     

Dynamic boundary systems with boundary feedback and population system with unbounded birth process

This paper is concerned with a class of dynamic boundary systems with boundary feedback. The well‐posedness of the considered systems is proved under some regularity conditions. Moreover, some spectral properties are derived. As an application, the well‐posedness and the asymptotic behavior of population dynamical systems with unbounded birth process ‘ ’ are solved. Such population dynamical systems were pointed out in [S. Piazzera, Math. Methods Appl. Sci., 27 (2004), 427‐439] to be a current research topic in semigroup theory and still an open problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Well‐posedness of a 1D transport equation with nonlocal velocity in the Lei–Lin space

In this paper, we consider the well‐posedness of a one‐dimensional transport equation with nonlocal velocity in the Lei–Lin space . We first modify the product estimate and then establish the global existence of solutions to the Cauchy problem with small enough initial data. Finally, we discuss the stability of the global solution. Copyright © 2016 John Wiley & Sons, Ltd.