Self-similar Solutions of the Navier–Stokes Equations on Weak Weighted Lorentz Spaces

In the present paper, we prove the existence of global solutions for the Navier–Stokes equations in Rnwhen the initial velocity belongs to the weighted weak Lorentz space Λn,∞(u) with a sufficiently small norm under certain restriction on the weight u. At the same time, self-similar solutions are induced if the initial velocity is, besides, a homogeneous function of degree-1. Also the uniqueness is discussed.     

Existence Uniqueness and Decay of the Global Weak Solutions for a Class of Parabolic Systems with Nonlinear Boundary Conditions

In this paper, a class of parabolic systems with nonlinear boundary conditions is discussed. By introducing a complete metric space with decay of W^s_p-norm, we obiain the existence uniqueness of global weak solutions, our method is simpler than before. A decay estimate of the global weak solutions is obtained simultaneously.     

Remarks on infinite energy solutions of nonlinear wave equations

We first consider the existence of a solution of the critical semilinear wave equation in Besov space which extends the results in [P. Germain, Global infinite energy solutions of critical semilinear wave equation, Revista Matematica Iberoamericana 24 (2) (2008) 463-497] to general dimensions. Next we derive the existence and uniqueness of global solutions for a semilinear wave equation in Marcinkiewicz space.     

Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations

This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method.     

Global small solutions of the cauchy problem for a semilinear system of equations of thermoelasticity

For a semilinear system of equations of thermoelasticity, we establish a theorem on the existence and uniqueness of global solutions in a multidimensional space under the condition that the initial data are sufficiently small. We also obtain estimates for the decrease of solutions as time increases.     

Quasilinear equation in moving domain

In this article we are concerned with the existence and uniqueness of global weak solutions of a mixed problem associated with one-dimensional damped elastic stretched string equation when the supports of the ends have small displacements. In addition, we show that the energy decays exponentially. In previous investigations about string equation in moving domain, local or global solutions for increasing domain with the growth of the time has been shown. Here, thanks to the internal strong damping we eliminate this hypothesis.     

Zero Mach number limit in critical spaces for compressible Navier-Stokes equations

We are concerned with the existence and uniqueness of local or global solutions for slightly compressible viscous fluids in the whole space. In [6] and [7], we proved local and global well-posedness results for initial data in critical spaces very close to the one used by H. Fujita and T. Kato for incompressible flows (see [14]). In the present paper, we address the question of convergence to the incompressible model (for ill-prepared initial data) when the Mach number goes to zero. When the initial data are small in a critical space, we get global existence and convergence. For large initial data and a bit of additional regularity, the slightly compressible solution is shown to exist as long as the corresponding incompressible solution does. As a corollary, we get global existence (and uniqueness) for slightly compressible two-dimensional fluids.     

The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution for a nonlocal Cahn-Hilliard equation with Dirichlet boundary condition on a bounded domain. Under a nondegeneracy assumption the solutions are classical but when this is relaxed, the equation is satisfied in a weak sense. Also we prove that there exists a global attractor in some metric space.     

到齐次空间带位势的P-调和热流的非唯一性

对齐次空间的具有位势的P-调和热流的唯一性问题进行了研究,并且证明了如果初始数据是非稳态的具有位势的P-调和映射,则存在无穷多个全局弱解．     

Global wellposedness for quasi-linear symmetric hyperbolic systems with regularized coefficients

Following the abstract setting of [8] and using the global results of [2], global wellposedness and regularity results are proved for the solutions of quasi-linear symmetric hyperbolic systems with bounded coefficients which are regularized by a convolution in the space variables with a regularizing function. In the case of unbounded regularized coefficients, local existence of classical solutions is proved, as well as uniqueness and regularity of (not necessarily existing) global weak solutions with initial value in a Sobolev space. As the regularizing function tends to Dirac's δ, local-in-time convergence to the classical solution of the non-regularized problem is proved.     

A Lagrangian Approach for the Incompressible Navier‐Stokes Equations with Variable Density

We investigate the Cauchy problem for the inhomogeneous Navier‐Stokes equations in the whole n‐dimensional space. Under some smallness assumption on the data, we show the existence of global‐in‐time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of \input amssym $\dot {B}^{n/p‐1}_{p,1}({\Bbb R}^n)$ . In particular, piecewise‐constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results, as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence. © 2012 Wiley Periodicals, Inc.     

The global weak solution for the shallow water wave model of moderate amplitude

In this paper, we firstly establish the existence theorem of the global weak solutions of the Cauchy problem for the shallow water wave model of moderate amplitude, then following the idea in Xin and Zhang’s work (see Xin and Zhang, 2002), we prove the uniqueness of global weak solutions using the localization analysis in the transport equation theory. Finally, several travelling wave solutions are derived.     

Sewing method for weak solutions of second-order hyperbolic equation with variable domains of discontinuous unbounded operators

We prove the existence and uniqueness of global weak solutions on the entire interval for the Cauchy problem for hyperbolic differential-operator equations with time-discontinuous operators that have variable domains and satisfy certain matching conditions at the points of discontinuity. To this end, we develop a method of successive sewing of existing local weak solutions of Cauchy problems on the smoothness intervals of the operators. The sewing method is based on special energy inequalities, which imply the time continuity of local weak solutions in the main space and of their first derivatives in some negative spaces and hence the existence of the corresponding limit values at the points of discontinuity. These values, with regard for the matching conditions, are taken for the initial data on each successive interval.     

Global Solutions to the Coupled Chemotaxis-Fluid Equations

In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier–Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small.     

Well-posedness of the ferrimagnetic equations

In this paper, we show the existence of global weak solutions of the ferrimagnetic equations on compact Riemannian manifold using the penalty method. We also show the uniqueness of the solution and its well-posedness by the energy estimates method in lower dimensions. In particular, when the space dimension is one, we can prove that the problem is globally well-posed.     

Uniform global attractors for the nonautonomous 3D Navier–Stokes equations

We obtain the existence and the structure of the weak uniform (with respect to the initial time) global attractor and construct a trajectory attractor for the 3D Navier–Stokes equations (NSE) with a fixed time-dependent force satisfying a translation boundedness condition. Moreover, we show that if the force is normal and every complete bounded solution is strongly continuous, then the uniform global attractor is strong, strongly compact, and solutions converge strongly toward the trajectory attractor. Our method is based on taking a closure of the autonomous evolutionary system without uniqueness, whose trajectories are solutions to the nonautonomous 3D NSE. The established framework is general and can also be applied to other nonautonomous dissipative partial differential equations for which the uniqueness of solutions might not hold. It is not known whether previous frameworks can also be applied in such cases as we indicate in open problems related to the question of uniqueness of the Leray–Hopf weak solutions.     

Global attractor for the regularized Bénard problem

In this paper, we study the asymptotic behaviour of weak solutions for the regularized Bénard problem. We establish the global existence and uniqueness of weak solutions of this problem and give a proof for the existence of global attractor in the three-dimensional case for this system.     

Global attractor for thermoelasticity in shape memory alloys without viscosity

We consider a nonlinear system of thermoelasticity in shape memory alloys without viscosity. The existence and uniqueness of strong and weak solutions and the existence of a compact global attractor in an appropriate space are proved. Copyright © 2015 John Wiley & Sons, Ltd.     

On the solvability of a mathematical model for prion proliferation

We show that a model describing the interaction between normal and infectious prion proteins admits global solutions. More precisely, supposing the involved degradation rates to be bounded, we prove global existence and uniqueness of classical solutions. Based on this existence theory, we provide sufficient conditions for the existence of global weak solutions in the case of unbounded splitting rates. Moreover, we prove global stability of the disease-free steady state.     

On the solvability of a boundary value problem for nonlinear wave equations in angular domains

For a one-dimensional wave equation with a weak nonlinearity, we study the Darboux boundary value problem in angular domains, for which we analyze the existence and uniqueness of a global solution and the existence of local solutions as well as the absence of global solutions.