On the existence,uniqueness and regularity of the exterior stokes problem in R3

Conditions, under which a solution exls-ts, and 1s, unique for the statlonarp ftofces, pattern in exterior domains, in R³are dlscussed. The domains, constdered are the complements. In R³ of lounded star shaped-sets and the class of functions, for which existence ta. dlsplayed have "physically reasonable" lehavlor at infinity. 7hls class is. characterized ly, a finite Dieichlet intearal. We use a method that employs, a variational formulation of the p-rollem and avoids, the use of a Helmhoetz decomposltlon of vector fields in unlounded domains     

Existence, regularity and local uniqueness of the solutions to the Maxwell-Landau-Lifshitz system in three dimensions

We study a model of ferromagnetism governed by Maxwell's equations coupled with the non-linear Landau-Lifshitz equation of micromagnetism. We are interested in the cases of space-periodic solutions for 3D domains. Obtaining the regularity of the solution m in space and of the solutions E, H in space we state the existence theorem. Finally, we prove the local uniqueness of the solutions.     

On uniqueness and time regularity of flows of power‐law like non‐Newtonian fluids

Large class of non‐Newtonian fluids can be characterized by index p, which gives the growth of the constitutively determined part of the Cauchy stress tensor. In this paper, the uniqueness and the time regularity of flows of these fluids in an open bounded three‐dimensional domain is established for subcritical ps, i.e. for p>11/5. Our method works for ‘all’ physically relevant boundary conditions, the Cauchy stress need not be potential and it may depend explicitly on spatial and time variable. As a simple consequence of time regularity, pressure can be introduced as an integrable function even for Dirichlet boundary conditions. Moreover, these results allow us to define a dynamical system corresponding to the problem and to establish the existence of an exponential attractor. Copyright © 2010 John Wiley & Sons, Ltd.     

Bem and its convergence for Stationary Stokes problem in three dimensions

This paper discusses BEM for Stationary Stokes problem in three dimensions, studies its convergence and superconvergence, and gives the optimal error estimates as well.     

三维复Ginzburg-Landau方程的整体解的存在惟一性

在三维空间中研究带2σ次非线性项的复值Ginzburg—Landau方程(CGL) ut=ρu (1 iγ)△u-(1 iμ)|u|^2σu,通过先验估计的方法,在适当的σ的假设下,获得该方程周期边值问题整体解的存在性和惟一性．     

Existence and uniqueness of strong–weak solutions for chemically reacting generalized second grade fluids in 2 space dimensions

The present paper is devoted to the analysis of a nonlinear system modeling unsteady flows of an incompressible non‐Newtonian fluid mixed with a reactant. We are interested on generalized second grade fluids, which are chemically reacting and whose viscosity depends both on the shear‐rate and the concentration. We prove existence and uniqueness of strong–weak solution for a flow filling in the plane and subject to space periodic boundary conditions. This result is established under the fulfillment of some assumptions on the viscosity stress tensor and the flux vector of the diffusion–convection equation reflecting the chemical reaction. Copyright © 2016 John Wiley & Sons, Ltd.     

Planar least gradient problem: existence,regularity and anisotropic case

We show existence of solutions to the least gradient problem on the plane for boundary data in \(BV(\partial \varOmega )\). We also provide an example of a function \(f \in L^1(\partial \varOmega ) \backslash \) \((C(\partial \varOmega ) \cup BV(\partial \varOmega ))\), for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.     

On the existence of a solution of a second‐order singular initial value problem

In this paper, we consider a class of second‐order initial value problems. We extend the family of second‐order initial value problems for which the existence and uniqueness of a solution is known. To handle the singularity, we introduce the more general Caratheodory type of restrictions. The solvability results of the Emden–Fowler type of equations is established as a special case.Copyright © 2014 John Wiley & Sons, Ltd.     

Existence,uniqueness, and regularity results for the two-body contact problem

The problem of contact between two elastic bodies is studied under the assumption of nonzero initial gap in the potential contact region. The related variational inequality is stated and existence, uniqueness, and local regularity results are proved for its solution.     

A dynamic thermoviscoelastic problem: An existence and uniqueness result

This work deals with the variational analysis of a dynamic problem which models the temperature evolution in a thermoviscoelastic body. The variational problem is formulated as a coupled system of evolutionary nonlinear variational equations. Then, the existence of a unique weak solution is proved using Banach fixed-point arguments and results on time-dependent families of subgradients.     

On a nonlinear singular diffusion problem:existence and uniqueness

The singular diffusion equation u_t=(u^?1u_x)_x:arises in many areas of application, e.g. in the central limit approximation to Carleman's model of Boltzman equation, or, in the expansion of a thermalized electron cloud in plasma physics. This paper concerns the existence and uniqueness of solution of a mixed boundary value problem of equation u_t=(u^m=1u_x)_x for ?1 < m ≤0.     

Non‐linear bone remodelling: an existence and uniqueness result

Several models have been proposed in order to be able to predict the correct density and stress distributions on long bones. Most of them are of an empiric and/or experimental nature. A few are related to optimization techniques, in the framework of elasticity theory. In this work, we establish an existence and uniqueness result for a modified Cowin–Hegedus model. This models is of great importance not only because it is one of the most sophisticated including a great number of the existing models, but also because it is set in the framework of adaptive elasticity theory giving a sound basis for the development of robust numerical methods. Copyright © 2000 John Wiley & Sons, Ltd.     

The existence of non-tiles and non-facets in three dimensions

Non-tiles are convex polytopes, of which isomorphic copies will not tile the space locally finite and face-to-face (that is, neighbouring tiles meet in a face of each). Non-facets are convex polytopes, of which isomorphic copies will not fit together as the facets of a higher dimensional convex polytope. It is proved that there are infinitely many 3-dimensional convex polytopes which are non-tiles as well as non-facets, thereby answering a question of Danzer.     

The limit behavior of the second‐grade fluid equations

In the paper, the limit behavior of solutions to the second‐grade fluid system with no‐slip boundary conditions is studied as both ν→0 and α→0. More precisely, it is verified that the convergence from second‐grade fluid system to Euler system holds as ν→0 and α→0 independently under the radial symmetry case.     

The existence,uniqueness, and regularity for an incompressible Newtonian flow with intrinsic degree of freedom

We consider the regularity and uniqueness of solution to the Cauchy problem of a mathematical model for an incompressible, homogeneous, Newtonian fluid, taking into account internal degree of freedom. We first show there exist uniquely a local strong solution. Then we show this solution can be extend to the whole interval [0,T] if the velocity u, or its gradient ? u, or the pressure p belongs to some function class, which are similar with that of incompressible Navier–Stokes equations. Our result shows that the solution is unique in these classes, and that velocity field plays a more prominent role in the existence theory of strong solution than the angular velocity field. Finally, if the L^{3 ∕ 2}‐norm of the initial angular velocity vector and some homogeneous Besov norm of initial velocity field are small, then there exists uniquely a global strong solution. Copyright © 2012 John Wiley & Sons, Ltd.     

An analytic approach to existence and uniqueness for martingale problems in infinite dimensions

We prove existence and uniqueness for a class of martingale problems in a Hilbert space. We solve the associated Kolmogorov equation and prove that the corresponding semigroup is determined by a kernel of measures if a Schauder-type regularity is satisfied. Received: 18 May 1998 / Revised version: 27 September 1999 / Published online: 5 September 2000     

Global existence, uniqueness, and continuous dependence for a semilinear initial value problem

In this paper, we consider the semilinear initial value problem associated with an operator A whose spectrum lies in a sector of the complex plane and whose resolvent satisfies (z−A)⁻¹M|z|^γ for some −1<γ<0 and all z outside the sector. The properties of existence and uniqueness of global mild solutions and continuous dependence on the initial data are investigated.     

Existence and uniqueness for a nonlinear inverse reaction‐diffusion problem with a nonlinear source in higher dimensions

This paper analyzes the existence and the uniqueness problem for an n‐dimensional nonlinear inverse reaction‐diffusion problem with a nonlinear source. A transformation is used to obtain a new inverse coefficient problem. Then, a parabolic differential operator L_λ is defined to establish the relation between the solution of L_λ = 0 and the new inverse problem. Following this, it is shown that the inverse problem has at least one solution in the class of admissible coefficients. Furthermore, it is proved that this solution is the unique solution of the undertaken inverse problem. A numerical example is given to illustrate ill‐posedness of the inverse problem. Copyright © 2013 John Wiley & Sons, Ltd.     

Global existence and uniqueness of solution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum

The aims of this paper are to discuss global existence and uniqueness of solution for a class of non-Newtonian fluids with vacuum in one-dimensional bounded interval. The important point in this paper is that we allow the initial vacuum. In particular, these results are used to prove similar results for more general non-Newtonian fluids, and applied to numerical computation. This work is partially supported by the 985 program of Jilin University, China Postdoctoral Sciences Foundation, and NSF Grant ([10571072]; [10601009]).