Phonon hydrodynamics and its applications in nanoscale heat transport

We consider the distinguished limits of the phase field equations and prove that the corresponding free boundary problem is attained in each case. These include the classical Stefan model, the surface tension model (with or without kinetics), the surface tension model with zero specific heat, the two phase Hele–Shaw, or quasi-static, model. The Hele–Shaw model is also a limit of the Cahn–Hilliard equation, which is itself a limit of the phase field equations. Also included in the distinguished limits is the motion by mean curvature model that is a limit of the Allen–Cahn equation, which can in turn be attained from the phase field equations.

Convergence of the phase field model to its sharp interface limits

The problem of the derivation of hydrodynamics from the Boltz- mann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few in- stances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman-Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad's moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equa- tion. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short-wave limit and the viscosity modifica- tion at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert's 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton's iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the Boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts. Finally, a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the Boltzmann equation in the future.

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Hilbert's 6th problem: Exact and approximate hydrodynamic manifolds for kinetic equations

The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman--Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad's moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short-wave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert's 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton's iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the Boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts. Finally, a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the Boltzmann equation in the future.

/pdf/hilbert-s-6th-problem-exact-and-approximate-hydrodynamic-27szdeufwd.pdf

Hilbert's 6th Problem: Exact and Approximate Hydrodynamic Manifolds for Kinetic Equations

Global classical solution of Muskat free boundary problem

We consider a new concept of weak solutions to the phase-field equations with a small parameter e characterizing the length of interaction. For the standard situation of a single free interface, this concept (in contrast with the common one) leads to the well-known Stefan–Gibbs–Thomson problem as e→0. For the case of a large number M(e) (M(e)→∞ as e→0) of free interfaces, which corresponds to the ‘wave-train’ interpretation of a ‘mushy region’, this concept allows us to obtain the limit problem as e→0.

Hugoniot-type conditions and weak solutions to the phase-field system

This monograph consists of two volumes and is devoted to second-order partial differential equations (mainly, equations with nonnegative characteristic form). A number of problems of qualitative theory (for example, local smoothness and hypoellipticity) are presented.

Equations with nonnegative characteristic form. I

The purpose of this paper is to investigate problems of the Navier-Stokes approximation to kinetic equations in terms of the so-called Chapman-Enskog projection. One considers properties of the Chapman-Enskog projection for the Cauchy problem for moment approximations of the kinetic equation and primarily the Chapman-Enskog projection for the Boltzmann-Peierls kinetic equation. The existence of the Chapman-Enskog projection for the Cauchy problem is proved for the phase space of conservative variables (phenomena of nonlinear diffusion) and for the phase space of physical variables (the second sound projection).

Navier-Stokes approximation and problems of the Chapman-Enskog projection for kinetic equations

UDC 517.9 We consider the Cauchy problem for the one-dimensional Carleman equation with bounded energy and periodic initial data and obtain the local equilibrium conditions. We prove exponential stabilization to the equilibrium state. Bibliography :1 0titles. Illustrations :3 figures. Dedicated to N. N. Uraltseva

Local Equilibrium of the Carleman Equation

We consider one-dimensional Carleman and Godunov–Sultangazin equations and obtain local equilibrium conditions for solutions of the Cauchy problem with finite energy and periodic initial data. Moreover, we prove the exponential stabilization to the equilibrium state.

E. V. Radkevich

Papers

Hugoniot-type conditions and weak solutions to the phase-field system

Equations with nonnegative characteristic form. I

Navier-Stokes approximation and problems of the Chapman-Enskog projection for kinetic equations

Local Equilibrium of the Carleman Equation

On the Nature of Local Equilibrium in the Carleman and Godunov–Sultangazin Equations