Nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation

In was shown in Ruan et al. (2008) [3] that rarefaction waves for the generalized KdV-Burgers-Kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small. The main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation. In our results, we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity, the initial perturbation can be chosen arbitrarily in , while for a general smooth nonlinear flux function, we need to ask for the L²-norm of the initial perturbation to be small but the L²-norm of the first derivative of the initial perturbation can be large and, consequently, the -norm of the initial perturbation can also be large.     

On the Heegaard Floer homology of branched double-covers

Let be a link. We study the Heegaard Floer homology of the branched double-cover Σ(L) of S³, branched along L. When L is an alternating link, of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E² term is a suitable variant of Khovanov's homology for the link L, converging to the Heegaard Floer homology of Σ(L).     

Phase space analysis on some black hole manifolds

The Schwarzschild and Reissner-Nordstrøm solutions to Einstein's equations describe space-times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space-time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L⁶ norm in space decays like . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an ? loss of angular derivatives.     

Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum

The asymptotic behavior of solutions of the damped compressible Euler equations is conjectured to obey to the famous porous media equations (PMES). The previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic. In present paper, we prove that the L^∞ entropy weak solution with vacuum, obtained by the compensated compactness theory, converges strongly in space to the unique similarity solution of the related PME, as time goes to infinity.     

Lipschitz and piecewise-C regularity for scalar minimizers of affine simple integrals

Lipschitz, piecewise-C¹ and piecewise affine regularity is proved for AC minimizers of the “affine” integral , under general hypotheses on , , and with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ?(·) are lsc and may be both locally unbounded (e.g., not in L_loc¹), provided their quotient ?/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.     

Second-order semilinear elliptic inequalities in exterior domains

We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients in exterior domains in . For W(x)?|x|^−σ at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.     

The asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map acting on a von Neumann algebra—or more generally a dual operator system—we show that there is a unique reversible system (i.e., a complete order automorphism α of a dual operator system N) that captures all of the asymptotic behavior of L, called the asymptotic lift of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n×n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W^∗-dynamical system (N,Z), and we identify (N,Z) as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed algebra Nα. In general, we show the action of the asymptotic lift is trivial iff L is slowly oscillating in the sense that

     

Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L^′ be R-modules. We investigate the properties of the functors and . For instance, we show the following:

(a)

if L and L^′ are artinian, then is artinian, and is noetherian over the completion ;

(b)

if L is artinian and L^′ is Matlis reflexive, then , , and are Matlis reflexive.

Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.     

Reduced Gutzwiller formula with symmetry: Case of a Lie group

We consider a classical Hamiltonian H on R2d, invariant by a Lie group of symmetry G, whose Weyl quantization is a selfadjoint operator on L²(Rd). If χ is an irreducible character of G, we investigate the spectrum of its restriction to the symmetry subspace of L²(Rd) coming from the decomposition of Peter-Weyl. We give semi-classical Weyl asymptotics for the eigenvalues counting function of in an interval of R, and interpret it geometrically in terms of dynamics in the reduced space R2d/G. Besides, oscillations of the spectral density of are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of R2d.     

L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.     

Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L^(1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L^(1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L^(1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L^(1,∞) if and only if the sequence of singular numbers {s_n(x)}_n?1 (in the descending order and counting the multiplicities) satisfies . In this case, our characterization amounts to saying that a positive element x∈L^(1,∞) is measurable if and only if exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space , where the space is the closure of all finite rank operators in L^(1,∞) in the norm ∥.∥_(1,∞).     

Polynomial sequences of bounded operators

The basic results of spectral theory are obtained using the sequence of powers of a bounded linear operator T,T²,…,Tⁿ,…. In this paper, we replace the powers Tⁿ by certain polynomials p_n(T), and make use of special properties of the polynomial sequence to derive some new results concerning operators. For example, using an arbitrary polynomial sequence , we obtain “binomial” spectral radii and semidistances, which reduce, in the case of the sequence of powers, to the usual spectral radius and semidistance.     

Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains

We study the existence and the asymptotic behavior of positive solutions for the parabolic equation on D×(0,∞), where is a some unbounded domain in and V belongs to a new parabolic class J^∞ of singular potentials generalizing the well-known parabolic Kato class at infinity P^∞ introduced recently by Zhang. We also show that the choice of this class is essentially optimal.     

Three positive solutions of semilinear elliptic equations in the half space with a hole

In this paper, assume that h is nonnegative and ‖hL²_‖>0, we prove that if ‖hL²_‖ is sufficiently small, then there are at least three positive solutions of Eq. (1) in , where D is a C1,1 bounded domain in .