The Navier–Stokes problem modified by an absorption term

It is well known that for the classical Navier–Stokes problem the best one can obtain is some decays in time of power type. With this in mind, we consider in this work, the classical Navier–Stokes problem modified by introducing, in the momentum equation, the absorption term |u|^σ?2 u, where σ > 1. For the obtained problem, we prove the existence of weak solutions for any dimension N ≥ 2 and its uniqueness for N = 2. Then we prove that, for zero body forces, the weak solutions extinct in a finite time if 1 < σ < 2 and exponentially decay in time if σ = 2. In the special case of a suitable force field which vanishes at some instant, we prove that the weak solutions extinct at the same instant provided 1 < σ < 2. We also prove that for non-zero body forces decaying at a power-time rate, the solutions decay at analogous power-time rates if σ > 2. Finally, we prove that for a general non-zero body force, the weak solutions exponentially decay in time for any σ > 1.     

On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations

This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive parameter λ and a real function f(u) which satisfies a superlinear growth condition just in a neighborhood of zero. By proving an a priori estimate (for a suitable class of solutions) we are able to avoid further restrictions on the behavior of f(u) at infinity in order to prove, for λ sufficiently large, the existence of one-sign and sign-changing solutions. Minimax methods are employed to establish this result.     

Velocity and temperature distribution in a system consisting of a fluid layer overlying a layer of porous medium

The velocity and temperature distribution in a system consisting of a fluid layer overlying a layer of porous medium is investigated in the presence of buoyancy and surface tension forces. The analysis indicates that buoyancy and surface tension forces are additive and aid the flow by counteracting the effect of permeability. The temperature distribution is sensitive to the variation of the aspect ratio, and conductivities of the media. Further, the inclusion of the viscous dissipation term markedly affects the temperature distribution.     

Theoretical study of a Bénard-Marangoni problem

In this paper we prove the existence of strong solutions for the stationary Bénard-Marangoni problem in a finite domain flat on the top, bifurcating from the basic heat conductive state. The Bénard-Marangoni problem is a physical phenomenon of thermal convection in which the effects of buoyancy and surface tension are taken into account. This problem is modelled with a system of partial differential equations of the type Navier-Stokes and heat equation. The boundary conditions include crossed boundary conditions involving tangential derivatives of the temperature and normal derivatives of the velocity field. To define tangential derivatives at the boundary, intended in the trace sense, it is necessary order two derivatives in the interior of the domain and thus the boundary term contains as high derivatives as the interior term. We overcome this difficulty by considering the weak formulation, and transforming the boundary integral into an equivalent integral defined in the whole domain. This allows us to reformulate the weak problem with a temperature having only order one weak derivatives. Concerning regularity results, we obtain strong solutions for the stationary Bénard-Marangoni problem.     

Transient nonlinear optically-thick radiative–convective double-diffusive boundary layers in a Darcian porous medium adjacent to an impulsively started surface: Network simulation solutions

A boundary-layer model is described for the two-dimensional nonlinear transient thermal convection heat and mass transfer in an optically-thick fluid in a Darcian porous medium adjacent to an impulsively started vertical surface, in the presence of significant thermal radiation and buoyancy forces in an (X¹,Y¹,t¹) coordinate system. An algebraic approximation is employed to simplify the integro-differential equation of radiative transfer for unidirectional flux normal to the plate into the boundary-layer regime, by incorporating this flux term in the energy conservation equation. The conservation equations are non-dimensionalized into an (X,Y,T) coordinate system and solved using the Network Simulation Method (NSM), a robust numerical technique which demonstrates high efficiency and accuracy. The transient variation of non-dimensional streamwise velocity component (u) and temperature (T) and concentration (C) functions is computed for various selected values of Stark number (radiation–conduction interaction parameter) and Darcy number. Transient velocity (u) and steady-state local skin friction (τ_X) are also studied for various thermal Grashof number (Gr), species Grashof number (Gm), Schmidt number (Sc) and Stark number (N) values. These computations for the infinite permeability case (Da → ∞) are compared with previous finite difference solutions [Prasad et al. Int J Therm Sci 2007;46(12):1251–8] and shown to be in excellent agreement. An increase in Darcy number is seen to accelerate the flow and boost velocity. A decrease in Stark number (corresponding to an increase in thermal radiation heat transfer contribution) is shown to increase the velocity values. Temperature function is observed to fall in value with a rise in Da and increase with decrease in N (corresponding to an increase in thermal radiation heat transfer contribution). Applications of the study include rocket combustion chambers, astrophysical flows, spacecraft thermal fluid dynamics in debris-laden environments (cosmic dust), heat transfer in forest fire spread, geochemical contamination and ceramic materials processing.     

Time periodic and almost time periodic solutions to rotating stratified fluids subject to large forces

Consider the 3D incompressible Boussinesq equations for rotating stably stratified fluids. It is shown that this set of equations possesses a unique time periodic or almost time periodic solutions for external forces satisfying these properties, which, however, do not necessarily need to be small. An explicit bound on the size of the external force, depending on the buoyancy frequency N, is given, which then allows for the unique existence of time periodic or almost periodic solutions. In particular, the size of the external forces can be taken large with respect to the buoyancy frequency. The approach depends crucially on the dispersive effect of the rotation and the stable stratification.     

Asymptotics of Damped Periodic Motions with Random Initial Speed

We consider motion on the circle, possibly with friction and external forces, the initial velocity being a large random variable. We prove that under various assumptions the probability law of the stopping position of the motion converges to a distribution depending only on the motion equation. Here the time of stopping is either a constant or the first time instant at which the velocity vanishes, and the initial velocity is of the form αU + β, where U is a fixed random variable and α and/or β tend to infinity.     

A semilinear elliptic PDE not in divergence form via variational methods

In this paper we consider a semilinear equation driven by an operator not in divergence form. Precisely, the principal part of the operator is in divergence form, but it has also a lower order term depending on Du. While the right-hand side of the equation satisfies superlinear and subcritical growth conditions at zero and at infinity. The problem has not a variational structure, but, despite that, we use variational techniques in order to prove an existence and regularity result for the equation.     

临界群与二阶差分方程解的多重性

研究了一类二阶非线性差分方程两点边值问题解的多重性.当该问题的非线性项在无穷远点具有特殊的渐近线性性质时,利用变分方法,结合临界群与Morse理论,同时考虑正、负能量泛函的临界点,不论该问题是否发生共振,均证明了它至少存在两个非零解.     

Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow

We consider a strictly hyperbolic system of balance laws in one space variable, that represents a simple model for a fluid flow in the presence of phase transitions. The state variables are specific volume, velocity and mass-density fraction λ of the vapor in the fluid. A reactive source term drives the dynamics of the phase mixtures; such a term depends on a relaxation parameter and involves an equilibrium pressure, allowing for metastable states.First we prove the global existence of weak solutions to the Cauchy problem, where the initial datum for λ is close either to 0 or 1 (the pure phases) and has small total variation, while the initial variations of pressure and velocity are not necessarily small. Then we consider the relaxation limit and prove that the weak solutions of the full system converge to those of the reduced system.     

Existence and decay of solutions in full space to Navier-Stokes equations with delays

We consider the Navier-Stokes equations with delays in Rⁿ,2≤n≤4. We prove existence of weak solutions when the external forces contain some hereditary characteristics and uniqueness when n=2. Moreover, if the external forces satisfy a time decay condition we show that the solution decays at an algebraic rate.     

Autoregulated Impulse Point Heating of a Finite Medium

We consider a finite heat conducting medium whose boundary is maintained at zero temperature and, moreover, to which the same amount of heat is supplied at a certain point at the instant when the temperature at this point decreases to a given level. Up to an arbitrary shift in time, we prove the existence and uniqueness of a periodic regime with a unique heat pulse during each period. We present an efficient algorithm for constructing this regime if the medium is either an n-dimensional ball heated at the center or an interval heated at an arbitrary point.     

Blow-up theorem for semilinear wave equations with non-zero initial position

One of the features of solutions of semilinear wave equations can be found in blow-up results for non-compactly supported data. In spite of finite propagation speed of the linear wave, we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak. This was first observed by Asakura (1986) [2] finding out a critical decay to ensure the global existence of the solution. But the blow-up result is available only for zero initial position having positive speed.In this paper the blow-up theorem for non-zero initial position by Uesaka (2009) [22] is extended to higher-dimensional case. And the assumption on the nonlinear term is relaxed to include an example, |u|^p−1u. Moreover the critical decay of the initial position is clarified by example.     

Uniform boundary stabilization of the finite difference space discretization of the 1−d wave equation

The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of the energy of the classical semi-discrete system in which the 1?d Laplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing size h. Actually, the decay rate tends to zero as h goes to zero. Then we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. Our method of proof relies essentially on discrete multiplier techniques.     

Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity

Abstract

In this paper we consider a semilinear heat equation (in a bounded domain Ω of ? ^N ) with a nonlinearity that has a superlinear growth at infinity. We prove the existence of a control, with support in an open set ω ? Ω, that insensitizes the L ² ? norm of the observation of the solution in another open subset 𝒪 ? Ω when ω ∩ 𝒪 ≠ ?, under suitable assumptions on the nonlinear term f(y) and the right hand side term ξ of the equation. The proof, involving global Carleman estimates and regularizing properties of the heat equation, relies on the sharp study of a similar linearized problem and an appropriate fixed-point argument. For certain superlinear nonlinearities, we also prove an insensitivity result of a negative nature. The crucial point in this paper is the technique of construction of L ^r -controls (r large enough) starting from insensitizing controls in L ².     

Remarks on the two-dimensional Benjamin equation

We generalize some recent results proved for the KP equation to the generalized Benjamin equation. First, we establish that the Cauchy problem cannot be solved by an iteration method. As a consequence, the flow map fails to be smooth. The second goal is to prove that the zero-mass constraint is satisfied at any non-zero time even it is not satisfied at the initial time.     

Existence and uniqueness for a mathematical model in superfluidity

In this paper we propose a model to study superfluidity by considering as state variables the order parameter, describing the concentration of the superfluid phase, the velocity of the superfluid and the absolute temperature. We assume that the order parameter satisfies a Ginzburg–Landau equation and that the velocity is decomposed as the sum of a normal and a superfluid component. The heat equation provides the evolution equation for the temperature. We prove that this model is consistent with the principles of thermodynamics. Well‐posedness of the resulting initial and boundary value problem is shown. Copyright © 2008 John Wiley & Sons, Ltd.     

DECAY OF ENERGY FOR A DISSIPATIVE ANISOTROPIC ELASTIC SYSTEM

In this article, we study the large-time behavior of energy for a N-dimensional dissipative anisotropic elastic system. By means of multiplicative techniques, energy method, and Zuazua’s estimate technique, we prove the decay property of energy for anisotropic elastic system.     

Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system

We prove a stability result of constant equilibria for the three dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter ε while the incompressible part of the initial velocity is assumed to be small compared to ε. We then get a unique global smooth solution. We also prove a uniform in ε time decay rate for these solutions. Our approach allows to combine the parabolic energy estimates that are efficient for the viscous equation at ε fixed and the dispersive techniques (dispersive estimates and normal forms) that are useful for the inviscid irrotational system.     

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.