Existence results for viscous polytropic fluids with vacuum

We consider the full Navier-Stokes equations for viscous polytropic fluids with nonnegative thermal conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover our results hold for both bounded and unbounded domains.     

Global well-posedness for the 2-D nonhomogeneous incompressible MHD equations with large initial data

In this paper, we consider the 2-D nonhomogeneous incompressible magnetohydrodynamic equations with variable viscosity and variable conductivity. We obtain the global existence of solutions for this system with initial data in the scaling invariant Besov spaces and without size restriction for the initial velocity and magnetic field.     

Large Data Existence Result for Unsteady Flows of Inhomogeneous Shear-Thickening Heat-Conducting Incompressible Fluids

We consider unsteady flows of inhomogeneous, incompressible, shear-thickening and heat-conducting fluids where the viscosity depends on the density, the temperature and the shear rate, and the heat conductivity depends on the temperature and the density. For any values of initial total mass and initial total energy we establish the long-time existence of weak solution to internal flows inside an arbitrary bounded domain with Lipschitz boundary.     

Global existence of strong solutions to the Cauchy problem for a 1D radiative gas

We consider a one-dimensional radiation hydrodynamics model in the case of the equilibrium diffusion approximation which is described by the compressible Navier-Stokes system with the additional terms in the pressure and internal energy respectively, which embody the effect of radiation. Under the physical growth conditions on the heat conductivity, we establish the existence and uniqueness of strong solutions to the Cauchy problem with large initial data, where the initial density and velocity may have differing constant states at infinity. Moreover, we show that if there is no vacuum in the initial density, then, the vacuum and concentration of the density will never occur in any finite time.     

Thermodynamic restrictions on the relaxation functions of the theory of heat conduction with finite wave speeds

In this paper we establish the restrictions imposed by the Second law of Thermodynamics on the relaxation functions which arise in the theory of heat conduction with finite wave speeds. We show that (i) the initial values of the energy relaxation function and the heat flux relaxation function are non-negative, (ii) the initial slope of the heat flux relaxation function is non-positive, and (iii) the equilibrium conductivity is non-negative. These results have important implications with regard to the behavior of waves and the uniqueness of solutions.     

A boundary value problem for the spherically symmetric motion of a pressureless gas with a temperature‐dependent viscosity

We consider an initial‐boundary value problem for the equations of spherically symmetric motion of a pressureless gas with temperature‐dependent viscosity µ(θ) and conductivity κ(θ). We prove that this problem admits a unique weak solution, assuming Belov's functional relation between µ(θ) and κ(θ) and we give the behaviour of the solution for large times. Copyright © 2009 John Wiley & Sons, Ltd.     

Low Mach number limit for the non-isentropic Navier-Stokes equations

We study the low Mach number limit of the local in time solutions to the compressible Navier-Stokes equations with zero heat conductivity coefficient as the Mach number tends to zero. A uniform existence result for the one-dimensional initial-boundary value problem is proved provided that the initial data are “well-prepared” in the sense that the temporal derivatives up to order two are bounded initially.     

Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity

Studied in this paper is the Cauchy problem of the two-dimensional magnetohydrodynamics system with inhomogeneous density and electrical conductivity. It is shown that the 2-D incompressible inhomogeneous magnetohydrodynamics system with a constant viscosity is globally well-posed for a generic family of the variations of the initial data and an inhomogeneous electrical conductivity. Moreover, it is established that the system is globally well-posed in the critical spaces if the electrical conductivity is homogeneous.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE THREE-DIMENSIONAL FULL COMPRESSIBLE QUANTUM EQUATIONS

We consider the quantum Navier-Stokes equations for the viscous, compressible, heat conducting fluids on the three-dimensional torus T~3. The model is based on a system which is derived by Jungel, Matthes and Milisic [15]. We made some adjustment about the relation of the viscosities of quantum terms.The viscosities and the heat conductivity coefficient are allowed to depend on the density, and may vanish on the vacuum. By several levels of approximation we prove the global-in-time existence of weak solutions for the large initial data.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE

We consider the quantum Navier-Stokes equations for the viscous, compressible, heat conducting fluids on the three-dimensional torus T³. The model is based on a system which is derived by Jungel, Matthes and Milisic [15]. We made some adjustment about the relation of the viscosities of quantum terms. The viscosities and the heat conductivity coefficient are allowed to depend on the density, and may vanish on the vacuum. By several levels of approximation we prove the global-in-time existence of weak solutions for the large initial data.     

Global well-posedness and exponential stability of solutions for the viscous radiative and reactive gas

This paper is concerned with the global well-posedness and exponential stability of solutions to a one-dimensional model for the viscous radiative and reactive gas with higher-order kinetics. We prove that under rather general assumptions on the heat conductivity κ, for any large smooth initial data, the problem admits a unique global classical solution. Moreover, the solution will exponentially decay to the unique steady state as time goes to infinity.     

Optimization of properties of multicomponent isotropic composites

We suggest a method for the calculation of the extremal conductivity of composites under some natural assumptions concerning their microstructure. The method is based on the principle of consecutive assembling of binary mixtures by addition of infinitely small amounts of one of the initial compounds to the already-assembled isotropic composite. This process is assumed to produce an optimal isotropic binary mixture at each step, which is performed by the Hashin-Shtrikman procedure. We are seeking a suitable sequence of compounds to be added to the mixture in order to minimize its resultant conductivity. A solution is given to the corresponding optimization problems for both finite number and infinite number of initial compounds taken in prescribed concentrations. We also describe the microstructure of the optimal composites. The results can be used for the optimal design of elastic and heat-conducting constructions.Dedicated to G. LeitmannThe authors have accepted with pleasure the invitation to participate in the JOTA special issue dedicated to Professor G. Leitmann. The papers by Professor Leitmann are well known in the USSR. They have greatly influenced the formation of the authors' scientific viewpoints. The first author has learned much from Professor Leitmann in the course of his work in translating two books by Professor Leitmann published in the USSR; the second author has also benefited from these translations. Methods of optimization, whose development has been influenced so much by Professor Leitmann's contributions, have penetrated into a number of fields. Here, we apply these methods to a problem in the mechanics of composite materials.     

A real time algorithm for the location search of discontinuous conductivities with one measurement

We consider an inverse problem for finding the anomaly of discontinuous electrical conductivity by one current‐voltage observation. We develop a real time algorithm for determining the location of the anomaly. This new idea is based on the observation of the pattern of a simple weighted combination of the input current and the output voltage. Combined with the size estimation result, this algorithm gives a good initial guess for Newton‐type schemes. We give the rigorous proof for the location search algorithm. Both the mathematical analysis and its numerical implementation indicate our location search algorithm is very fast, stable and efficient. © 2001 John Wiley & Sons, Inc.     

On the Global Well-Posedness of 3-D Boussinesq System with Variable Viscosity

In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coefficient is sufficiently close to some positive constant in L∞and the initial velocity is small enough in B_(3,1)~0(R~3). With some thermal conductivity in the temperature equation and with linear buoyancy force θe3 on the velocity equation in the Boussinesq system, the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L~1(R~3)and B_(3,1)~0(R~3) respectively.     

Blowup solutions in a problem for the nonlinear heat equation on a small interval

The article considers a one-dimensional quasi-linear heat equation with a volume heat source and a nonlinear thermal conductivity. The analysis is conducted for parameter values where selfsimilar solutions of the equations evolve in an LS-regime with blowup. Heat localization is observed in this case, and the combustion process in the developed stage is in the form of simple or complex structures of contracting half-width. We study the evolution dynamics of various initial distributions and their achievement of the self-similar regime, and also the dependence of the size of the localization region on the shape of the initial compactly supported distribution. The possibility of cyclic evolution of solutions against the background of overall growth with blowup is demonstrated. We particularly focus on the case when the size of the spatial region is much less than the characteristic size of the localization region, and heat flow is obstructed by the physical boundaries. In this case all initial perturbations achieve the self-similar regime, but the corresponding scenario has certain specific features. We present an example of formation of a complex spatial structure that evolves with blowup on a small interval. Translated from Prikladnaya Matematika i Informatika, No. 29, 2008, pp. 88–112.     

Shape Methods for the Transmission Problem with a Single Measurement

In the current work, we consider the inverse conductivity problem of recovering inclusion with one measurement. First, we use conformal mapping techniques for determining the location of the anomaly and estimating its size. We then get a good initial guess for quasi-Newton type method. The inverse problem is treated from the shape optimization point of view. We give a rigorous proof for the existence of the derivative of the state function and of shape functionals. We consider both least squares fitting and Kohn and Vogelius functionals. For the numerical implementation, we use a parameterization of shapes coupled with a boundary element method. Several numerical examples indicate the superiority of the Kohn and Vogelius functional over least squares fitting.     

Global existence of strong solutions to compressible Navier–Stokes system with degenerate heat conductivity in unbounded domains

In one-dimensional unbounded domains, we prove the global existence of strong solutions to the compressible Navier–Stokes system for a viscous and heat conducting ideal polytropic gas, when the viscosity is a constant and the heat conductivity is proportional to a positive power of the temperature. Note that the conditions imposed on the initial data are the same as those of the constant heat conductivity case (Kazhikhov, A. V. Siberian Math. J. 23 [1982], 44-49) and can be arbitrarily large. Therefore, our result generalizes Kazhikhov's result for the constant heat conductivity case to the degenerate and nonlinear one.     

Transverse electrical conductivity of a quantum collisional plasma in the Mermin approach

We derive formulas for the transverse electrical conductivity and the permittivity in a quantum collisional plasma using the kinetic equation for the density matrix in the relaxation approximation in the momentum space. We show that the derived formula becomes the classical formula when the Planck constant tends to zero and that when the electron collision rate tends to zero (i.e., the plasma becomes collisionless), the derived formulas become the previously obtained Lindhard formulas. We also show that when the wave number tends to zero, the quantum conductivity becomes classical. We compare the obtained conductivity with the conductivity obtained by Lindhard and with the classical conductivity     

Global wellposedness of magnetohydrodynamics system with temperature-dependent viscosity

The initial boundary value problem of the one-dimensional magneto-hydrodynamics system, when the viscosity, thermal conductivity, and magnetic diffusion coefficients are general smooth functions of temperature, is considered in this article. A unique global classical solution is shown to exist uniquely and converge to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1.     

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the typey″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.