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.     

Quantum Lévy Laplacian and associated heat equation

As a non-commutative extension of the Lévy Laplacian for entire functions on a nuclear space, we define the quantum Lévy Laplacian acting on white noise operators. We solve a heat type equation associated with the quantum Lévy Laplacian and study its relation to the classical Lévy heat equation. The solution to the quantum Lévy heat equation is obtained also from a normal-ordered white noise differential equation involving the quadratic quantum white noise.     

Quantum double suspension and spectral triples

In this paper we are concerned with the construction of a general principle that will allow us to produce regular spectral triples with finite and simple dimension spectrum. We introduce the notion of weak heat kernel asymptotic expansion (WHKAE) property of a spectral triple and show that the weak heat kernel asymptotic expansion allows one to conclude that the spectral triple is regular with finite simple dimension spectrum. The usual heat kernel expansion implies this property. The notion of quantum double suspension of a C^?-algebra was introduced by Hong and Szymanski. Here we introduce the quantum double suspension of a spectral triple and show that the WHKAE is stable under quantum double suspension. Therefore quantum double suspending compact Riemannian spin manifolds iteratively we get many examples of regular spectral triples with finite simple dimension spectrum. This covers all the odd-dimensional quantum spheres. Our methods also apply to the case of noncommutative torus.     

On the Navier-Stokes Equations with Energy-Dependent Nonlocal Viscosities

We discuss the mathematical modeling of incompressible viscous flows for which the viscosity depends on the total dissipation energy. In the two-dimensional periodic case, we begin with the case of temperature-dependent viscosities with very large thermal conductivity in the heat convective equation, in which we obtain the Navier-Stokes system coupled with an ordinary differential equation involving the dissipation energy as the asymptotic limit. Letting further the latent heat to vanish, we derive the Navier-Stokes equations with a nonlocal viscosity depending on the total dissipation of energy. Bibliography: 7 titles.Dedicated to V. A. Solonnikov on the occasion of his 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 71–91.     

On stationary flows of asymmetric fluids with heat convection

Existence, uniqueness and regularity of solutions of equations describing stationary flows of viscous incompressible isotropic fluids with an asymmetric stress tensor have been considered recently.⁵ In this paper we extend the results of Reference 5 to include heat convection in the hydrodynamic model. We show that the boundary value problem (1.1)–(1.6) below has solutions in appropriate Sobolev spaces, provided the viscosities v and c_a + c_d are sufficiently large. The proof is based on a fixed point argument. Moreover, we show that the solutions are unique if the heat conductivity κ is large enough.     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

Nonlinear analysis of capillary instability with heat and mass transfer

The nonlinear capillary instability of the cylindrical interface between the vapor and liquid phases of a fluid is studied when there is heat and mass transfer across the interface, using viscous potential flow theory. The fluids are considered to be viscous and incompressible with different kinematic viscosities. Both asymmetric and axisymmetric disturbances are considered. The analysis is based on the method of multiple scale perturbation and the nonlinear stability is governed by first-order nonlinear partial differential equation. The stability conditions are obtained and discussed theoretically as well as numerically. Regions of stability and instability have been shown graphically indicating the effect of various parameters. It has been observed that the heat and mass transfer has stabilizing effect on the stability of the system in the nonlinear analysis for both axisymmetric as well as asymmetric disturbances.     

OPTIMAL DECAY RATE OF THE COMPRESSIBLE QUANTUM NAVIER-STOKES EQUATIONS

For quantum fluids governed by the compressible quantum Navier-Stokes equations in $\Bbb R^3$ with viscosity and heat conduction, we prove the optimal $L^p-L^q$ decay rates for the classical solutions near constant states. The proof is based on the detailed linearized decay estimates by Fourier analysis of the operators, which is drastically different from the case when quantum effects are absent.     

Spectral Dimension of Liouville Quantum Gravity

This paper is concerned with computing the spectral dimension of (critical) 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove that the spectral dimension is 1 via an exact expression for the boundary Liouville Brownian motion and heat kernel. Then we treat the 2d-case via a decomposition of time integral transforms of the Liouville heat kernel into Gaussian multiplicative chaos of Brownian bridges. We show that the spectral dimension is 2 in this case, as derived by physicists (see Ambjørn et al. in JHEP 9802:010, 1998) 15 years ago.     

SPH方法中的Riemann解与人工粘性

本文描述了光滑粒子动力学方法的人工粘性和Riemann解方法,分析了Godunov方法与传统人工粘性方法的耗散项．给定一种人工粘性,总可以找到一种相应的Riemann解法器,使得它们的耗散项在形式上几乎相同．本文利用各种近似Riemann解构造了相应的新的人工粘性．这些新的粘性,无需人工调节粘性系数．本文完成了多个数值试验,比较了使用传统人工粘性方法与Riemann解方法的不同．采用新的人工粘性,辅助热通量粘性,可以获得令人满意的计算结果．     

Evaluation of Casimir Energies Through Spectral Functions

We consider applications of elliptic differential operators and their associated spectral functions in quantum field theory problems. The role of zeta functions and traces of heat kernels in the regularization of Casimir energies is emphasized, and the renormalization procedure is discussed with simple examples.     

Brownian motion, quantum corrections and a generalization of the Hermite polynomials

The nonequilibrium evolution of a Brownian particle, in the presence of a “heat bath” at thermal equilibrium (without imposing any friction mechanism from the outset), is considered. Using a suitable family of orthogonal polynomials, moments of the nonequilibrium probability distribution for the Brownian particle are introduced, which fulfill a recurrence relation. We review the case of classical Brownian motion, in which the orthogonal polynomials are the Hermite ones and the recurrence relation is a three-term one. After having performed a long-time approximation in the recurrence relation, the approximate nonequilibrium theory yields irreversible evolution of the Brownian particle towards thermal equilibrium with the “heat bath”. For quantum Brownian motion, which is the main subject of the present work, we restrict ourselves to include the first quantum correction: this leads us to introduce a new family of orthogonal polynomials which generalize the Hermite ones. Some general properties of the new family are established. The recurrence relation for the new moments of the nonequilibrium distribution, including the first quantum correction, turns out to be also a three-term one, which justifies the new family of polynomials. A long-time approximation on the new three-term recurrence relation describes irreversible evolution towards equilibrium for the new moment of lowest order. The standard Smoluchowski equations for the lowest order moments are recovered consistently, both classically and quantum-mechanically.     

Quantum trajectories of an oscillator interacting with an electromagnetic field, a classical force, and a heat bath

We consider a solvable problem describing the dynamics of a quantum oscillator interacting with an electromagnetic field, a classical force, and a heat bath. We propose a general method for solving Markovian master equations, the method of quantum trajectories. We construct the stochastic evolution operator involving the stochastic analogue of the Baker-Hausdorff formula and calculate the system density matrix for an arbitrary initial state. As a physical application, we evaluate the influence of the environment at a finite temperature on the accuracy of measuring a weak classical force by the interference method. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 444–459, March, 2009.     

Two-time correlation functions in an exactly solvable spin-boson model

We consider a spin-boson model describing the dephasing process in an open quantum system and obtain exact expressions for the two-time spin correlation function and the decoherence function applicable for any values of the coupling constants. We show that the initial statistical correlations between the dynamical system and the heat bath considerably affect the time dependence of the decoherence function.     

On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate

We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. The existence of weak solutions for steady flows of such fluids subject to homogeneous Dirichlet boundary conditions is established in Franta, Málek, Rajagopal [M. Franta, J. Málek, K.R. Rajagopal, On steady flows of fluids with pressure- and shear- dependent viscosities, Proc. Roy. Soc. A Math. Phys. Eng. Sci. 461 (2055) (2005) 651–670]. In this paper we treat non-homogeneous Dirichlet boundary conditions, assuming either that the normal part of velocity on the boundary is equal to zero or that the boundary data are small. We also relax the requirement concerning how to fix the pressure. Such a model has relevance to some important engineering applications.     

Null controllability for the heat equation with singular inverse-square potentials

We prove the null controllability of the heat equation perturbed by a singular inverse-square potential arising in quantum mechanics and combustion theory. This is done within the range of subcritical coefficients of the singular potential, provided the control acts on an annular set around the singularity. Our proof uses a splitting argument on the domain, decomposition in spherical harmonics, new Carleman inequalities and refined Hardy inequalities.     

Decay of Quantum Correlations on a Lattice by Heat Kernel Methods

We prove some estimates of the correlation of two local observables in quantum lattice models at high temperature. For that, we describe the heat kernel of the Hamiltonian for finite subsets of the lattice, which may tend to the whole lattice. (In other words, we study the heat kernel of a Schr?dinger in large dimension.) Submitted: February 22, 2007. Accepted: May 3, 2007.     

Quantum Laplacian and Applications

In this paper, we give an explicit solution of some linear quantum white noise differential equations by applying the convolution calculus on a suitable distribution space. In particular, we give an integral representation for the solution of the quantum heat equation.     

Exergetic efficiency optimization for an irreversible quantum Brayton refrigerator with spin systems

The purpose of this paper is to study the optimal performance for an irreversible quantum Brayton refrigerator with spin systems, which consists of two isomagnetic field branches connected by two irreversible adiabatic branches. The time evolution of the total magnetic moment M is determined by solving the generalized quantum master equation of an open system in the Heisenberg picture. The time of two irreversible adiabatic processes is considered based on finite-rate evolution in this paper. The optimization region (or criteria) for an irreversible quantum Brayton refrigerator with spin systems is obtained. The relationship between the exergetic efficiency ε_E and dimensionless cooling load R for the irreversible quantum Brayton refrigerator with heat leakage and other irreversibility losses are derived.