Time-dependent Family Research Articles

Exactly solvable time-dependent oscillator family

This paper proposes a method for constructing exactly solvable time-dependent oscillator families based on the invariant operator method and the duality method. An exactly solvable time-dependent oscillator family contains an infinite number of exactly solvable time-dependent oscillators. As a byproduct, we also construct exactly solvable Ermakov-Pinney equation families.

Microscopic state of BHs and an exact one body method for binary dynamics in general relativity

In gravitational collapses, the horizon and singularity’s realisation in the finite future of the proper time used co-moving observer happens in the future of infinitely far away future of the normal time used outside probe. To the latter the horizon and singularity defined in the singularity theorem are physical realities only in the sense of uncertainty principle and ensemble interpretation. We provide two exact time dependent solution families to the Einstein equation and show that they form a pair of complementary description for the microscopic state of black holes by showing that the Bekenstein–Hawking entropy formula follows properly from their canonical wave function’s degeneracy. We also develop an eXact One Body method for general relativity two-body dynamics whose conservative part calls no post newtonian approximation as input and applies to the full three stages of black hole binary merger events. By this method, we analytically calculate the gravitational wave forms following from such merger processes. In the case black holes carry exact and apriori horizon and singularity our wave forms agree with those following from conventional effective one body method but exhibit more consistent late time behaviour. In the case black holes carry only asymptotic horizon and extended inner structure thus experiencing banana shape deformation as the merger occurs, our wave forms exhibit all features especially the late time quasi-normal mode type oscillation seen in real observations.

Comparison theorems for evolution inclusions with maximal monotone operators. $L^2$-theory

An evolution inclusion with time-dependent family of maximal monotone operators in considered in a separable Hilbert space. If the elements with minimum norm of the family of maximal monotone operators satisfy certain growth conditions, then the domains of definition of this family are closed convex sets. Hence the sweeping process is well defined, whose values are the normal cones of the domains of definition of maximal monotone operators. It is shown that if the sweeping process has a solution for each single-valued perturbation from the space of integrable functions, then the evolution inclusion with the maximal monotone operators and single-valued perturbations from the space of integrable functions is also solvable. Quite general conditions in terms of the properties of the family of maximal monotone operators that ensure the existence of solutions for the sweeping process are presented. All results obtained and the approach presented are new. They are used to prove an existence theorem for evolution inclusions with multivalued perturbations, whose values are closed nonconvex sets. Bibliography: 19 titles.

Periodic delay orbits and the polyfold implicit function theorem

We consider differential delay equations of the form \partial_tx(t) = X_{t}(x(t - \tau)) in \mathbb{R}^n , where (X_t)_{t\in S^1} is a time-dependent family of smooth vector fields on \mathbb{R}^n and \tau is a delay parameter. If there is a (suitably non-degenerate) periodic solution x_0 of this equation for \tau=0 , that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by \tau . However, it seems difficult to prove this using the classical implicit function theorem, since the equation above, considered as an operator, is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer–Wysocki–Zehnder (2009, 2021) to overcome this problem in a natural setup.

Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma

We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form dut−Ltutdt=N(ut)dt+∑i=1dFi(ut)dXti where (Lt)t∈[0,T] is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while X≡(X,X) is a two-step (non-necessarily geometric) rough path of Hölder regularity γ>1/3. Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity depending on that of the rough path X). As a technical tool, we introduce a version of the multiplicative sewing lemma, which allows to construct the so called product integrals in infinite dimensions. We later use it to construct a semigroup analogue for the non-autonomous linear PDEs as well as show how to deduce the semigroup version of the usual sewing lemma from it.

Self-similar solutions to kinetic-type evolution equations: beyond the boundary case

We study the asymptotic behavior as $t \to \infty $ of a time-dependent family $(\mu _{t})_{t \geq 0}$ of probability measures on $\mathbb {R}$ solving the kinetic-type evolution equation $\partial _{t} \mu _{t} + \mu _{t} = Q(\mu _{t})$ where $Q$ is a smoothing transformation on $\mathbb {R}$. This problem has been investigated earlier, e.g. by Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928–1961, 2012] and Bogus, Buraczewski and Marynych [Stochastic Process. Appl. 130(2):677–693, 2020]. Combining the refined analysis of the latter paper, which provides a probabilistic description of the solution $\mu _{t}$ as the law of a suitable random sum related to a continuous-time branching random walk at time $t$, with recent advances in the analysis of the extremal positions in the branching random walk we are able to solve the case that has been left open until now. In the course of our work, we significantly weaken the assumptions in the literature that guarantee the existence (and uniqueness) of a solution to the evolution equation $\partial _{t} \mu _{t} + \mu _{t} = Q(\mu _{t})$.

An optimal transport approach for solving dynamic inverse problems in spaces of measures

In this paper we propose and study a novel optimal transport based regularization of linear dynamic inverse problems. The considered inverse problems aim at recovering a measure valued curve and are dynamic in the sense that (i) the measured data takes values in a time dependent family of Hilbert spaces, and (ii) the forward operators are time dependent and map, for each time, Radon measures into the corresponding data space. The variational regularization we propose is based on dynamic (un-)balanced optimal transport which means that the measure valued curves to recover (i) satisfy the continuity equation,i.e., the Radon measure at timetis advected by a velocity fieldvand varies with a growth rateg, and (ii) are penalized with the kinetic energy induced byvand a growth energy induced byg. We establish a functional-analytic framework for these regularized inverse problems, prove that minimizers exist and are unique in some cases, and study regularization properties. This framework is applied to dynamic image reconstruction in undersampled magnetic resonance imaging (MRI), modelling relevant examples of time varying acquisition strategies, as well as patient motion and presence of contrast agents.

Averaging Principles for Nonautonomous Two-Time-Scale Stochastic Reaction-Diffusion Equations with Jump

In this paper, we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven by Poisson random measures. The coefficients of the equation are assumed to be functions of time, and some of them are periodic or almost periodic. Therefore, the Poisson term needs to be processed, and a new averaged equation needs to be given. For this reason, the existence of time-dependent evolution family of measures associated with the fast equation is studied and proved that it is almost periodic. Next, according to the characteristics of almost periodic functions, the averaged coefficient is defined by the evolution family of measures, and the averaged equation is given. Finally, the validity of the averaging principle is verified by using the Khasminskii method.

A solvable model of the breakdown of the adiabatic approximation

Let L ≥ 0 and 0 < ɛ ≪ 1. Consider the following time-dependent family of 1D Schrödinger equations with scaled harmonic oscillator potentials iε∂tuε=−12∂x2uε+V(t,x)uε, uɛ(−L − 1, x) = π−1/4 exp(−x2/2), where V(t, x) = (t + L)2x2/2, t < − L, V(t, x) = 0, − L ≤ t ≤ L, and V(t, x) = (t − L)2x2/2, t > L. The initial value problem is explicitly solvable in terms of Bessel functions. Using the explicit solutions, we show that the adiabatic theorem breaks down as ɛ → 0. For the case L = 0, complete results are obtained. The survival probability of the ground state π−1/4 exp(−x2/2) at microscopic time t = 1/ɛ is 1/2+O(ε). For L > 0, the framework for further computations and preliminary results are given.

A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection–diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland’s dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

Hill’s equation, tire tracks and rolling cones

Louis Poinsot has shown in 1854 that the motion of a rigid body, with one of its points fixed, can be described as the rolling without slipping of one cone, the ‘body cone’, along another, the ‘space cone’, with their common vertex at the fixed point. This description has been further refined by the second author in 1996, relating the geodesic curvatures of the spherical curves formed by intersecting the cones with the unit sphere in Euclidean , thus enabling a reconstruction of the motion of the body from knowledge of the space cone together with the (time dependent) magnitude of the angular velocity vector. In this article we show that a similar description exists for a time dependent family of unimodular matrices in terms of rolling cones in 3-dimensional Minkowski space and the associated ‘pseudo spherical’ curves, in either the hyperbolic plane H2 or its Lorentzian analog H1,1. In particular, this yields an apparently new geometric interpretation of Schrödinger’s (or Hill’s) equation in terms of rolling without slipping of curves in the hyperbolic plane.

Self-similar solutions of kinetic-type equations: The boundary case

For a time dependent family of probability measures (ρt)t⩾0 we consider a kinetic-type evolution equation ∂ϕt∕∂t+ϕt=Q̂ϕt where Q̂ is a smoothing transform and ϕt is the Fourier–Stieltjes transform of ρt. Assuming that the initial measure ρ0 belongs to the domain of attraction of a stable law, we describe asymptotic properties of ρt, as t→∞. We consider the boundary regime when the standard normalization leads to a degenerate limit and find an appropriate scaling ensuring a non-degenerate self-similar limit. Our approach is based on a probabilistic representation of probability measures (ρt)t⩾0 that refines the corresponding construction proposed in Bassetti and Ladelli, (2012).

Super-Ricci flows for metric measure spaces

We introduce the notions of ‘super-Ricci flows’ and ‘Ricci flows’ for time-dependent families of metric measure spaces (X,dt,mt)t∈I. The former property is proven to be stable under suitable space-time versions of mGH-convergence. Uniformly bounded families of super-Ricci flows are compact. In the spirit of the synthetic lower Ricci bounds of Lott–Sturm–Villani for static metric measure spaces, the defining property for super-Ricci flows is the ‘dynamic convexity’ of the Boltzmann entropy Ent(.|mt) regarded as a functions on the time-dependent geodesic space (P(X),Wt)t∈I. For Ricci flows, in addition a nearly dynamic concavity of the Boltzmann entropy is requested.Alternatively, super-Ricci flows will be studied in the framework of the Γ-calculus of Bakry–Émery–Ledoux and equivalence to gradient estimates will be derived.For both notions of super-Ricci flows, also enforced versions involving an ‘upper dimension bound’ N will be presented.

Averaging Principle for Nonautonomous Slow-Fast Systems of Stochastic Reaction-Diffusion Equations: The Almost Periodic Case

We study the validity of an averaging principle for a slow-fast system of stochastic reaction-diffusion equations. We assume here that the coefficients of the fast equation depend on time, so that the classical formulation of the averaging principle in terms of the invariant measure of the fast equation is no longer available. As an alternative, we introduce the time-dependent evolution family of measures associated with the fast equation. Under the assumption that the coefficients in the fast equation are almost periodic, the evolution family of measures is almost periodic. This allows us to identify the appropriate averaged equation and prove the validity of the averaging limit.

Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels

Backward compact dynamics is deduced for a non-autonomous Benjamin-Bona-Mahony (BBM) equation on an unbounded 3D-channel. A backward compact attractor is defined by a time-dependent family of backward compact, invariant and pullback attracting sets. The theoretical existence result for such an attractor is derived from the backward flattening property, and this property is proved to be equivalent to the backward asymptotic compactness in a uniformly convex Banach space. Finally, it is shown that the BBM equation has a backward compact attractor in a Sobolev space under some suitable assumptions, such as, backward translation boundedness and backward small-tail. Both spectrum decomposition and cut-off technique are used to give all required backward uniform estimates.

Generalized Pauli channels and a class of non-Markovian quantum evolution

We analyze the quantum evolution represented by a time-dependent family of generalized Pauli channels. This evolution is provided by the random decoherence channels with respect to the maximal number of mutually unbiased bases. We derive the necessary and sufficient conditions for the vanishing back-flow of information.

Approximation of pseudo-differential flows

Given a classical symbol $M$ of order zero, and associated semiclassical operators ${\rm op}_\varepsilon(M),$ we prove that the flow of ${\rm op}_\varepsilon(M)$ is well approximated, in time $O(|\ln \varepsilon|),$ by a pseudo-differential operator, the symbol of which is the flow $\exp(t M)$ of the symbol $M.$ A similar result holds for non-autonomous equations, associated with time-dependent families of symbols $M(t).$ This result was already used, by the author and co-authors, to give a stability criterion for high-frequency WKB approximations, and to prove a strong Lax-Mizohata theorem. We give here two further applications: sharp semigroup bounds, implying nonlinear instability under the assumption of spectral instability at the symbolic level, and a new proof of sharp G\r{a}rding inequalities.

Ricci flow on contact manifolds

This paper is devoted to Ricci flow on contact manifolds. We define the contact curvature flow and establish a short time existence. Meanwhile, we study a contact Ricci soliton and prove that every solution of the unnormalized contact curvature flow is a selfsimilar solution corresponding to a contact Ricci soliton which is a steady soliton. Finally we show that a time dependent family of contact Einstein, Sasakian, K-contact, or η-Einstein 1-forms ηt is a solution of the normalized contact curvature flow if it is a conformal variation of an initial 1-form η0.

Global Well-Posedness and Pullback Attractors for an Incompressible Non-Newtonian Fluid with Infinite Delays

This paper studies a two-dimensional incompressible non-Newtonian fluid with infinite delays. The authors first prove the global well-posedness of the solutions and then establish the existence of the pullback $$\mathcal {D}$$ -attractor for a universe which is composed of time-dependent families under some suitable assumptions.

Reaction–diffusion equations with non-autonomous force in [formula omitted] and delays under measurability conditions on the driving delay term

In this paper we analyze the existence of solutions for a reaction–diffusion problem with hereditary effects and a time-dependent force term with values in H−1. The main novelty is that the delay term may be driven by a function under very minimal assumptions, namely, just measurability. This is due to the fact that we only deal with a phase-space of functions continuous in time, allowing this general setting, which might be more useful when less regularity is known in the hereditary mechanism. After that, we obtain uniform estimates and asymptotic compactness properties (via an energy method) that allow us to ensure the existence of pullback attractors for the associated process to the problem. Actually, we obtain two different families of minimal pullback attractors, namely, those of fixed bounded sets but also for a class of time-dependent families (universe) given by a tempered condition. Finally, from comparison results, we establish relations among them, and under suitable additional assumptions we conclude that these families of attractors are in fact the same object.