Global solutions of a dynamical equation in ferrimagnet

We consider global solutions of a dynamical equation in ferrimagnet. We show that it admits a global weak solution by using the penalty method. By the energy estimates method we show there exists a unique global smooth solution. Finally we establish the relationship between this equation and wave maps.     

Well-posedness of higher-order Camassa-Holm equations

We consider higher-order Camassa-Holm equations describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. We establish the existence of global weak solutions. We also present some invariant spaces under the action of the equation. Moreover, we prove a “weak equals strong” uniqueness result.     

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.     

H 1-perturbations of Smooth Solutions for a Weakly Dissipative Hyperelastic-rod Wave Equation

We consider a weakly dissipative hyperelastic-rod wave equation (or weakly dissipative Camassa–Holm equation) describing nonlinear dispersive dissipative waves in compressible hyperelastic rods. We fix a smooth solution and establish the existence of a strongly continuous semigroup of global weak solutions for any initial perturbation from In particular, the supersonic solitary shock waves [8] are included in the analysis. Dedicated to the memory of Professor Aldo Cossu The research of K.H. Karlsen is supported by an Outstanding Young Investigators Award from the Research Council of Norway. The current address of G.M. Coclite is Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy     

Existence and uniqueness of the solution to a free boundary-value problem with thermocapillary convection in an unbounded domain

A free boundary-value problem for a nonisothermal flow of two heavy viscous incompressible capillary liquids descibing a slot coating process is mathematically studied. The problem is stationary and two-dimensional and the Boussinesq approximation for the Navier-Stokes equations is used. The existence and uniqueness of a weak solution is proved in weighted Holder spaces for small data.     

Global energy conservative solutions to a system of variational wave equations

This paper is concerned with a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. The global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy is established by using the method of energy-dependent coordinates and the Young measure theory.     

Global weak solutions for a modified two-component Camassa-Holm equation

We obtain the existence of global-in-time weak solutions for the Cauchy problem of a modified two-component Camassa-Holm equation. The global weak solution is obtained as a limit of viscous approximation. The key elements in our analysis are the Helly theorem and some a priori one-sided supernorm and space-time higher integrability estimates on the first-order derivatives of approximation solutions.     

Generalized Solutions of Semilinear Parabolic Equations

It is well-known that a semilinear parabolic equation has no unique solution in the classical sense. We study such equations from the viewpoint of generalized functions. By using approximations for generalized functions, we obtain results on existence and uniqueness of generalized solutions. Furthermore, we establish the relationship between generalized solutions and classical solutions. Current address: Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan     

A note on the regularity of flows with shear-dependent viscosity

We consider a non-Newtonian fluid governed by stationary, incompressible Navier–Stokes equations with shear-dependent viscosity. Using a fixed point argument in an appropriate functional setting, we establish the existence of a strong solution for small and suitably regular data. Uniqueness results are obtained under similar conditions.     

Weighted estimates for stationary Navier-Stokes equations

We show Morrey-type estimates for the weak solution of the periodic Navier-Stokes equations in dimensionN, 5 <N < 10. ForN < 8, we prove the existence of a maximum solution.     

Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces

The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition. Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate Orlicz-Sobolev space.     

A nonlinear model and viscosity solution for the temperature field of an oil-immersed self-cooled three-phrase transformer

In this paper, on the basis of the characteristics of an oil-immersed self-cooled three-phrase transformer, we establish a mathematical model of the three-dimensional temperature field. But because the specific heat, density, heat sources and coefficient of heat transfer are discontinuous and non-differentiable, the problem has no analytical solution. We decompose the problem into seven subproblems, and prove the existence and uniqueness of a viscosity solution for every subproblem, by combining Perron’s method with the technique of coupled solutions.     

Long-time behavior of solutions to a nonlinear hyperbolic relaxation system

We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel.     

A free boundary problem for the Navier–Stokes equations with measure data

The existence of a weak solution of an initial boundary-value problem for the plane nonstationary Navier–Stokes equations with Radon measure data on the free boundary, is established. The problem may be considered as a model of the blood flow around the heart valves. An inverse problem is studied, it allows us to find the boundary forces acting on the valve from the observed values of the velocity of the fluid in a fixed subregion.     

Existence of weak solutions for steady viscous fluids with general slip boundary conditions

We study the existence of weak solutions for stationary viscous fluids with general slip boundary conditions in this paper. Applying monotone operator theory, we first establish the existence result of weak solutions for an approximation problem. Then using the compactness methods and the point-wise convergence property of velocity gradients, we get the desired results.     

Unique solvability of equations of motion for ferrofluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system is a combination of the Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of the unique strong solution to the system posed in a bounded domain of R³ and equipped with initial and boundary conditions.     

Generalized solutions to partial differential equations of evolution type

This paper deals with a new solution concept for partial differential equations in algebras of generalized functions. Introducing regularized derivatives for generalized functions, we show that the Cauchy problem is wellposed backward and forward in time for every system of linear partial differential equations of evolution type in this sense. We obtain existence and uniqueness of generalized solutions in situations where there is no distributional solution or where even smooth solutions are nonunique. In the case of symmetric hyperbolic systems, the generalized solution has the classical weak solution as macroscopic aspect. Two extensions to nonlinear systems are given: global solutions to quasilinear evolution equations with bounded nonlinearities and local solutions to quasilinear symmetric hyperbolic systems.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

Wasserstein kernels for one-dimensional diffusion problems

We treat the evolution as a gradient flow with respect to the Wasserstein distance on a special manifold and construct the weak solution for the initial-value problem by using a time-discretized implicit scheme. The concept of Wasserstein kernel associated with one-dimensional diffusion problems with Neumann boundary conditions is introduced. On the basis of this, features of the initial data are shown to propagate to the weak solution at almost all time levels, whereas, in a case of interest, these features even help with obtaining the weak solution. Numerical simulations support our theoretical results.