共查询到20条相似文献,搜索用时 712 毫秒
1.
Gonzalo Contreras 《Calculus of Variations and Partial Differential Equations》2001,13(4):427-458
For convex superlinear lagrangians on a compact manifold M we characterize the Peierls barrier and the weak KAM solutions of the Hamilton-Jacobi equation, as defined by A. Fathi [9],
in terms of their values at each static class and the action potential defined by R. Ma né [14]. When the manifold M is non-compact, we construct weak KAM solutions similarly to Busemann functions in riemannian geometry. We construct a compactification
of by extending the Aubry set using these Busemann weak KAM solutions and characterize the set of weak KAM solutions using this
extended Aubry set.
Received: 13 November 2000 / Accepted: 4 December 2000 / Published online: 25 June 2001 相似文献
2.
In the setting of the Weyl quantization on the flat torus \(\mathbb{T}^n \) , we exhibit a class of wave functions with uniquely associated Wigner probability measure, invariant under the Hamiltonian dynamics and with support contained in weak KAM tori in phase space. These sets are the graphs of Lipschitz-continuous weak KAM solutions of negative type of the stationary Hamilton-Jacobi equation. Such Wigner measures are, in fact, given by the Legendre transform of Mather’s minimal probability measures. 相似文献
3.
Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in L1‐spaces 下载免费PDF全文
We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L1‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L1‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L1‐spaces. For soft potentials, in L1‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap. 相似文献
4.
Rodrigo Bissacot Eduardo Garibaldi 《Bulletin of the Brazilian Mathematical Society》2010,41(3):321-338
Let σ: Σ → Σ be the left shift acting on Σ, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of σ-invariant Borel probabilities that maximize the integral of a given locally Hölder continuous potential A: Σ → ?. Under certain conditions, we are able to show not only that A-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions). 相似文献
5.
Cyril Imbert 《Journal of Differential Equations》2005,211(1):218-246
In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C2 in space and C1 in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part. 相似文献
6.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1997,324(3):265-270
We prove the existence of weak solutions of the spatially homogeneous Boltzmann equation without angular cut-off assumption for inverse sth power molecules with s ≥ 7/3, and general initial data with bounded mass, kinetic energy and entropy. Next, we show the convergence of these solutions to solutions of the Landau-Fokker-Planck equation when the collision kernel concentrates around the value π/2 相似文献
7.
The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions. 相似文献
8.
We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞. 相似文献
9.
Manuel Del Pino Jean Dolbeault Ivan Gentil 《Journal of Mathematical Analysis and Applications》2004,293(2):375-388
The equation ut=Δp(u1/(p−1)) for p>1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal Lp-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focus on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the Lp-Euclidean logarithmic Sobolev inequality. A large deviation result based on a Hamilton-Jacobi equation and also related to the Lp-Euclidean logarithmic Sobolev inequality is then stated. 相似文献
10.
Xuguang Lu 《Journal of Differential Equations》2008,245(7):1705-1761
For general initial data we prove the global existence and weak stability of weak solutions of the Boltzmann equation for Fermi-Dirac particles in a periodic box for very soft potentials (−5<γ?−3) with a weak angular cutoff. In particular the Coulomb interaction (γ=−3) with the weak angular cutoff is included. The conservation of energy and moment estimates are also proven under a further angular cutoff. The proof is based on the entropy inequality, velocity averaging compactness of weak solutions, and various continuity properties of general Boltzmann collision integral operators. 相似文献
11.
In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem. 相似文献
12.
Dragoş Iftimie 《Comptes Rendus Mathematique》2002,334(1):83-86
We consider the limit α→0 for the equation of the second grade fluids. We prove that weak convergence of the solutions to a weak solution of the Navier–Stokes equation holds under the assumption that the initial data weakly converges in L2. To cite this article: D. Iftimie, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 83–86 相似文献
13.
Kazuo Kobayasi 《Journal of Differential Equations》2003,189(2):383-395
We prove the equivalence of weak solutions and entropy solutions of an elliptic-parabolic-hyperbolic degenerate equation with homogeneous Dirichlet conditions and initial conditions. As a result of the equivalence, we obtain the L1-contraction principle and uniqueness of weak solutions of elliptic-parabolic degenerate equations. 相似文献
14.
In this paper, we prove that a quasi-periodic linear differential equation in sl(2,?) with two frequencies (α,1) is almost reducible provided that the coefficients are analytic and close to a constant. In the case that α is Diophantine we get the non-perturbative reducibility. We also obtain the reducibility and the rotations reducibility for an arbitrary irrational α under some assumption on the rotation number and give some applications for Schrödinger operators. Our proof is a generalized KAM type iteration adapted to all irrational α. 相似文献
15.
The space L2(0,1) has a natural Riemannian structure on the basis of which we introduce an L2(0,1)-infinite-dimensional torus T. For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Weak KAM theorem. As an application, we obtain existence of absolute action-minimizing solutions of prescribed rotation number for the one-dimensional nonlinear Vlasov system with periodic potential. 相似文献
16.
Jingyu Li 《Journal of Mathematical Analysis and Applications》2008,348(1):244-254
We study the regularity of the solutions for the wave equations with potentials that are time-dependent and singular. The size of the potentials is exactly a function of the spatial dimension rather than being small enough in the known results. Based on a weighted L2 estimate for the solutions, we prove the local regularity and the Strichartz estimates. The solvability of the equation is also studied. 相似文献
17.
Ping Zhang 《Applications of Mathematics》2006,51(4):427-466
In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and
some related problems. We first introduce the main tools, the L
p
Young measure theory and related compactness results, in the first section. Then we use the L
p
Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear
wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove
the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed.
In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic
equation, which is also the so-called vortex density equation arising from sup-conductivity. 相似文献
18.
For the equation −Δu=||xα|−2|up−1, 1<|x|<3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent 2∗=2N/(N−2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions. 相似文献
19.
This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity. This result extends some well-known results on Duffing equations to impulsive Duffing equations. 相似文献
20.
Properties of solutions of the Tricomi problem for the Lavrent’ev-Bitsadze equation at corner points
A. V. Rogovoy 《Differential Equations》2013,49(12):1650-1654
We consider the Tricomi problem for the Lavrent’ev-Bitsadze equation for the case in which the elliptic part of the boundary is part of a circle. For the homogeneous equation, we introduce a new class of solutions that are not continuous at the corner points of the domain and construct nontrivial solutions in this class in closed form. For the inhomogeneous equation, we introduce the notion of an n-regular solution and prove a criterion for the existence of such a solution. 相似文献