Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

Asymptotic behavior of strong solutions of a simplified energy-transport model with general conductivity

In this paper we study a simplified transient energy-transport model in semiconductors with a general conductivity and the Dirichlet boundary conditions on an interval. By using a new iterative scheme, we prove the global existence and uniqueness of strong solutions provided that the variation of the temperature is small. Also, the existence and stability of stationary solutions are proved if the temperature is large.     

On kink and anti-kink wave solutions of Schrodinger equation with distributed delay

This paper deals with existence problem of traveling wave solutions of a class of nonlinear Schr\"{o}dinger equation having distributed delay with a strong generic kernal. By using the geometric singular perturbation theory and the Melnikov function method, we establish results of the existence of kink and anti-kink wave solutions of the nonlinear Schr\"{o}dinger equation with time delay when the average delay is sufficiently small.     

Pseudoparabolic equations with additive noise and applications

In this work we present some results on the Cauchy problem for a general class of linear pseudoparabolic equations with additive noise. We consider questions of existence and uniqueness of mild and strong solutions and well posedness for this problem. We also prove the existence and uniqueness of mild and strong solutions for a related perturbed Cauchy problem and we investigate the continuity of the solution with respect to a small parameter. The abstract results are illustrated using examples from electromagnetics and heat conduction. Copyright © 2008 John Wiley & Sons, Ltd.     

On the Stationary Navier–Stokes Equations in the Half-Plane

We consider the stationary incompressible Navier–Stokes equation in the half-plane with inhomogeneous boundary condition. We prove the existence of strong solutions for boundary data close to any Jeffery–Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery–Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove the existence of weak solutions, as well as weak–strong uniqueness for small data and provide numerical simulations.     

Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control

In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the H¹-energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in H²-sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data.     

On some nonlinear evolution equations with strong dissipation,III

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with strong dissipation studied in previous papers [5, 6], where we proved the existence of global classical solutions with small data applying small energy techniques. This time, we prove the existence of a set of initial values which guarantees the solution to be global. We know the set is not bounded in the escalated energy spaces (Sobolev spaces). For the purpose, we establish approximate equations with another dissipative term which give a devised penalty to the solutions and lead the solutions to be bounded for all t > 0. Therefore we give an improvement to existence theories of equations describing a local statement of balance of momentum for materials for which the stress is related to strain and strain rate. These have been studied by many authors (cf. Greenberg et al. 19], Greenberg [10], Davis [3], Clements [2], Andrews [1], Yamada [12], Webb [13], etc.).     

TRAVELING WAVES IN A BIOLOGICAL REACTION-DIFFUSION MODEL WITH STRONG GENERIC DELAY KERNEL AND NON-LOCAL EFFECT

In this paper,we consider the reaction diffusion equations with strong generic delay kernel and non-local effect,which models the microbial growth in a flow reactor.The existence of traveling waves is established for this model.More precisely,using the geometric singular perturbation theory,we show that traveling wave solutions exist provided that the delay is sufficiently small with the strong generic delay kernel.     

Stochastic 3D tamed Navier–Stokes equations: Existence,uniqueness and small time large deviation principles

In this paper, we first give a direct approach to the existence and uniqueness of strong solutions to stochastic 3D tamed Navier–Stokes equations in case of Dirichlet boundary conditions for the Stokes–Laplacian. Then we prove a small time large deviation principle for the solutions.     

Large time behavior of solutions to semilinear systems of wave equations

This paper is concerned with the initial value problem for semilinear systems of wave equations. First we show a global existence result for small amplitude solutions to the systems. Then we study asymptotic behavior of the global solution. We underline that ``modified' free profiles are obtained for all global solutions to the systems even in the case where the free profile might not exist. Moreover, we prove non–existence of any free profiles for the global solution in some cases where the effect of the nonlinearity is strong enough. The first author was partially supported by Grant-in-Aid for Science Research (14740114), JSPS.     

Traveling wave solutions for Schrödinger equation with distributed delay

The paper is devoted to study of traveling waves of nonlinear Schrödinger equation with distributed delay by applying geometric singular perturbation theory, differential manifold theory and the regular perturbation analysis for a Hamiltonian system. Under the assumptions that the distributed delay kernel is strong general delay kernel and the average delay is small, we first investigate the existence of solitary wave solutions by differential manifold theory. Then by utilizing the regular perturbation analysis for a Hamiltonian system, we explore the periodic traveling wave solutions.     

Lp-estimates for the Strong Solutions of Elliptic Equations of Nondivergent Type

We investigate the second derivatives L^p-estimates for the strong solutions of second order linear elliptic equations in nondivergencc form Lu = f in the case in which the leading coefficients of L are not continuous. The L^p-estimates for small p are obtained if L is uniformly elliptic. Furthermore, if the leading coefficients of L belong to W^{1,n}, then we get the second derivatives L^p-estimates for large p. The existence of the strong solutions of the homogeneous Dirichlet problem is also considered.     

Existence of strong solutions and global attractors for the coupled suspension bridge equations

In this paper, we show the existence of the strong solutions for the coupled suspension bridge equations. Furthermore, existence of the strong global attractors is investigated using a new semigroup scheme. Since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal.     

Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

In this paper, we first address the space‐time decay properties for higher‐order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions for small initial data in some scaling invariant function spaces. The smallness conditions are somehow weaker than those presented by Brandolese and Schonbek. We further investigate the asymptotic profiles and decay properties of these strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Local and global strong solutions to continuous coagulation-fragmentation equations with diffusion

We consider the diffusive continuous coagulation-fragmentation equations with and without scattering and show that they admit unique strong solutions for a large class of initial values. If the latter values are small with respect to a suitable norm, we provide sufficient conditions for global-in-time existence in the absence of fragmentation.     

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.     

Existence of Entropy Solutions to Two-Dimensional Steady Exothermically Reacting Euler Equations

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function ?(T) is assumed to have a positive lower bound. We first consider the Cauchy problem (the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave (weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.     

Strong solutions of periodic parabolic problems with discontinuous nonlinearities

We study the problem of finding time-periodic solutions of a parabolic equation with the homogeneous Dirichlet boundary condition and with a discontinuous nonlinearity. We assume that the nonlinearity is equal to the difference of two superpositionally measurable functions nondecreasing with respect to the state variable. For such a problem, we prove the principle of lower and upper solutions for the existence of strong solutions without additional constraints on the “jumping-up” discontinuities in the nonlinearity. We obtain existence theorems for strong solutions of this class of problems, including theorems on the existence of two nontrivial solutions.     

Unique solvability for a class of full non-Newtonian fluids of one dimension with vacuum

We prove the local existence and uniqueness of the strong solutions for a class of full non-Newtonian fluids in one space dimension with the hypotheses that the initial data are small in some sense and satisfy some compatibility conditions. The initial density need not be positive, which means that we allow the initial vacuum.     

Unique solvability for a class of full non-Newtonian fluids of one dimension with vacuum

We prove the local existence and uniqueness of the strong solutions for a class of full non-Newtonian fluids in one space dimension with the hypotheses that the initial data are small in some sense and satisfy some compatibility conditions. The initial density need not be positive, which means that we allow the initial vacuum.