Almost everywhere well-posedness of continuity equations with measure initial data

Ambrosio, Luigi; Figalli, Alessio

doi:10.1016/j.crma.2010.01.018

Cited by 10 publications

(13 citation statements)

References 8 publications

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“…In [7] we were able to achieve the existence of limit points even when Coulomb singularities and crossings are present, namely assuming only that U b satisfies (5), (6), and (a) when Coulomb singularities but no crossings are present, namely assuming that U b ∈ C 1 . If one wishes to improve (a) and (b), trying to prove a full convergence result as ε ↓ 0 under weaker regularity assumptions on b (say ∇U ∈ W 1,p or ∇U ∈ BV out of Coulomb singularities), one faces the difficulty that the continuity equation (7) is well posed only in good functional spaces like L ∞ + [0, T ]; L 1 ∩ L ∞ (R d ) (see [13], [1], [12]).…”

Section: Introductionmentioning

confidence: 99%

Semiclassical limit of quantum dynamics with rough potentials and well‐posedness of transport equations with measure initial data

Ambrosio

Figalli

Friesecke

et al. 2011

Comm Pure Appl Math

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In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential U , which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness, and stability results for the flow in the space of measures induced by the continuity equation.

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Section: Introductionmentioning

confidence: 99%

Semiclassical limit of quantum dynamics with rough potentials and well‐posedness of transport equations with measure initial data

Ambrosio

Figalli

Friesecke

et al. 2011

Comm Pure Appl Math

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“…The dependance of the mass imbalance on Planck's constant is not easily visible from Figure 11. To clarify the situation we run several numerical experiments for a variety of values of and θ : θ = (2,3,4,6,8,9,12,14,17,18,21,23)/24, = 5 · 10 −1 , 10 −1 , 5 · 10 −2 , 10 −2 , 5 · 10 −3 .…”

Section: Collision Of Two Wave Packetsmentioning

confidence: 99%

“…As has been highlighted in [17], the regularity of the potential is not as important as the overall behavior of the underlying classical flow, φ t : (x, k) → (X(t), K(t)), where It is well known that the flow φ t defined by (2) is well-defined for all (x, k) ∈ R 2d as long as V ∈ C 1,1 (R d ), and that the problem (2) has weak solutions (possibly many) for all (x, k) ∈ R 2d as long as V ∈ C 1 (R d ). This basic observation puts nicely in context why any V / ∈ C 1,1 (R d ) is called non-smooth.…”

Section: Introductionmentioning

confidence: 99%

Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents

Athanassoulis

Katsaounis

Kyza

2016

Communications in Mathematical Sciences

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Abstract. Semiclassical asymptotics for linear Schrödinger equations with non-smooth potentials give rise to illposed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P. L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as −|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for −|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM.

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“…[15,13,14], and more recently in strong topology in [6,7]). More precisely, it is well-known that the limit dynamics of the Schrödinger equation is related to the Liouville equation 4) and, roughly speaking, the above results state that: (A) If U is of class C 2 and there exists a sequence ε k → 0 such that W ε k ρ 0,ε k converges in the sense of distribution to some (nonnegative) measure µ 0 , then W ε k ρ ε k t → (Φ t ) # µ 0 (the convergence is again in the sense of distribution), where Φ t is the (unique) flow map associated to the Hamiltonian system ẋ = p, p = −∇U (x) (1.5) so that µ t := (Φ t ) # µ 0 is the unique solution to (1.4) (here and in the sequel, # denotes the push-forward, so that µ t (A) = µ 0 (Φ −1 t (A)) for all A ⊂ R 2n Borel). (B) If U is of class C 1 and there exists a sequence ε k → 0 such that the curve t → W ε k ρ ε k t converges in the sense of distribution to some curve of (nonnegative) measure t → µ t , then µ t solves (1.4).…”

Section: )mentioning

confidence: 99%

Semiclassical limit for mixed states with singular and rough potentials

Figalli¹,

Ligabò²,

Paul³

2012

Indiana Univ. Math. J.

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We consider the semiclassical limit for the Heisenberg-von Neumann equation with a potential which consists of the sum of a repulsive Coulomb potential, plus a Lipschitz potential whose gradient belongs to BV ; this assumption on the potential guarantees the well posedness of the Liouville equation in the space of bounded integrable solutions. We find sufficient conditions on the initial data to ensure that the quantum dynamics converges to the classical one. More precisely, we consider the Husimi functions of the solution of the Heisenberg-von Neumann equation, and under suitable assumptions on the initial data we prove that they converge, as ε → 0, to the unique bounded solution of the Liouville equation (locally uniformly in time).

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Almost everywhere well-posedness of continuity equations with measure initial data

Cited by 10 publications

References 8 publications

Semiclassical limit of quantum dynamics with rough potentials and well‐posedness of transport equations with measure initial data

Semiclassical limit of quantum dynamics with rough potentials and well‐posedness of transport equations with measure initial data

Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents

Semiclassical limit for mixed states with singular and rough potentials

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