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Register a new userOn semi-classical limit of spatially homogeneous quantum Boltzmann equation: weak convergence
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It is expected in physics that the homogeneous quantum Boltzmann equation with Fermi-Dirac or Bose-Einstein statistics and with Maxwell-Boltzmann operator (neglecting effect of the statistics) for the weak coupled gases will converge to the homogeneous Fokker-Planck-Landau equation as the Planck constant $hbar$ tends to zero. In this paper and the upcoming work cite{HLP2}, we will provide a mathematical justification on this semi-classical limit. Key ingredients into the proofs are the new framework to catch the {it weak projection gradient}, which is motivated by Villani cite{V1} to identify the $H$-solution for Fokker-Planck-Landau equation, and the symmetric structure inside the cubic terms of the collision operators.
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This paper is dedicated to the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier-Stokes equation. One of the difficulty of the problem is to deal with the degenerate viscosity, together with the lack of integrability on the velocity. Our method is based on the construction of weak solutions that are renormalized in the velocity variable. The existence, and stability of these solutions do not need the Mellet-Vasseur inequality.
Departing from the weak solution, we prove the uniqueness, smoothing estimates and the global dynamics for the non cutoff spatially homogeneous Boltzmann equation with moderate soft potentials. Our results show that the behavior of the solution(including the production of regularity and the longtime behavior) can be {it characterized quantitatively} by the initial data at the large velocities, i.e.(i). initially polynomial decay at the large velocities in $L^1$ space will induce the finite smoothing estimates in weighted Sobolev spaces and the polynomial convergence rate (including the lower and upper bounds) to the equilibrium; (ii). initially the exponential decay at the large velocities in $L^1$ space will induce $C^infty$ regularization effect and the stretched exponential convergence rate. The new ingredients of the proof lie in the development of the localized techniques in phase and frequency spaces and the propagation of the exponential momentum.
Let M = R n or possibly a Riemannian, non compact manifold. We consider semi-excited resonances for a h-differential operator H(x, hD x ; h) on L 2 (M) induced by a non-degenerate periodic orbit $gamma$ 0 of semi-hyperbolic type, which is contained in the non critical energy surface {H 0 = 0}. By semi-hyperbolic, we mean that the linearized Poincar{e} map dP 0 associated with $gamma$ 0 has at least one eigenvalue of modulus greater (or less) than 1, and one eigenvalue of modulus equal to 1, and by non-degenerate that 1 is not an eigenvalue, which implies a family $gamma$(E) with the same properties. It is known that an infinite number of periodic orbits generally cluster near $gamma$ 0 , with periods approximately multiples of its primitive period. We construct the monodromy and Grushin operator, adapting some arguments by [NoSjZw], [SjZw], and compare with those obtained in [LouRo], which ignore the additional orbits near $gamma$ 0 , but still give the right quantization rule for the family $gamma$(E).
We consider a space-homogeneous gas of {it inelastic hard spheres}, with a {it diffusive term} representing a random background forcing (in the framework of so-called {em constant normal restitution coefficients} $alpha in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $alpha in [alpha_*,1)$ for some constructive $alpha_* in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $alpha in [alpha_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.
Following earlier work, we view two dimensional non-linear sigma model with target space $cM$ as a single particle relativistic quantum mechanics in the corresponding free loop space $cLM$. In a natural semi-classical limit ($hbar=alpha to 0$) of this model the wavefunction localizes on the submanifold of vanishing loops which is isomorphic to $cM$. One would expect that the relevant semi-classical expansion should be related to the tubular expansion of the theory around the submanifold and an effective dynamics on the submanifold is obtainable using Born-Oppenheimer approximation. In this work we develop a framework to carry out such an analysis at the leading order in $alpha$-expansion. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar of $cM$. The steps leading to this result are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) where we discuss its tubular expansion and how that is related to a semi-classical expansion of the Hamiltonian. Then we study an explicit construction of the relevant tubular neighborhood in $cLM$ using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of $cM$ which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model we arrive at the final result for LSQM.
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