An official website of the United States government
We construct a broad class of bounded potentials of the one-dimensional Schroedinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hoelder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.
more »
« less
This content will become publicly available on January 1, 2026
Existence and higher regularity of statistically steady states for the stochastic Coleman-Gurtin equation
We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way.
more »
« less
- Award ID(s):
- 2206491
- PAR ID:
- 10637932
- Publisher / Repository:
- AIMS
- Date Published:
- Journal Name:
- Evolution Equations and Control Theory
- Volume:
- 14
- Issue:
- 6
- ISSN:
- 2163-2480
- Page Range / eLocation ID:
- 1367 to 1411
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This work concerns the local convergence theory of Newton and quasi-Newton methods for convex-composite optimization: where one minimizes an objective that can be written as the composition of a convex function with one that is continuiously differentiable. We focus on the case in which the convex function is a potentially infinite-valued piecewise linear-quadratic function. Such problems include nonlinear programming, mini-max optimization, and estimation of nonlinear dynamics with non-Gaussian noise as well as many modern approaches to large-scale data analysis and machine learning. Our approach embeds the optimality conditions for convex-composite optimization problems into a generalized equation. We establish conditions for strong metric subregularity and strong metric regularity of the corresponding set-valued mappings. This allows us to extend classical convergence of Newton and quasi-Newton methods to the broader class of nonfinite valued piecewise linear-quadratic convex-composite optimization problems. In particular, we establish local quadratic convergence of the Newton method under conditions that parallel those in nonlinear programming.more » « less
-
null (Ed.)We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium enhancement in the random diffusion of criminal agents may exert visible relaxation effects. It is shown that sufficient regularity of the non-negative source terms in the system and a sufficiently strong nonlinear enhancement ensure that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is globally bounded under mild additional conditions on the source terms. These results are supplemented by numerical evidence which illustrates smoothing effects in solutions with sharply structured initial data in the presence of such porous medium-type diffusion and support the existence of singular structures in the linear diffusion case, which is the type of diffusion proposed in Ref. 29.more » « less
-
ABSTRACT Vitrimers with self‐healing, recycling, and remolding capabilities are changing the paradigm for thermoset polymer design. In the past several years, vitrimers that exhibit shape memory effects and are curable by ultraviolet (UV) light have made significant progress in the realm of 4D printing. Herein, we report a molecular dynamics (MD) modeling framework to model UV curable shape memory vitrimers. We used our framework and compared our modeling results with one UV curable shape memory vitrimer found in the literature, bisphenol A glycerolate dimethacrylate. The comparison showed reasonable agreement between the modeling and experimental results in terms of thermomechanical and shape memory properties, along with self‐healing efficiency. It was found that during recycling, it was important for the network to percolate through a majority of the system to get reasonably high recovery stress and recycling efficiency. Once this was achieved, a topological descriptor that was found to represent the compactness of the network was identified as having a very high correlation with recovery stress and recycling efficiency for networks that percolated 70% or more of the monomers in a system.more » « less
-
In this paper we introduce a constructive approach to study well-posedness of solutions to stochastic uid-structure interaction with stochastic noise. We focus on a benchmark problem in stochastic uidstructure interaction, and prove the existence of a unique weak solution in the probabilistically strong sense. The benchmark problem consists of the 2D time-dependent Stokes equations describing the ow of an incompressible, viscous uid interacting with a linearly elastic membrane modeled by the 1D linear wave equation. The membrane is stochastically forced by the time-dependent white noise. The uid and the structure are linearly coupled. The constructive existence proof is based on a time-discretization via an operator splitting approach. This introduces a sequence of approximate solutions, which are random variables. We show the existence of a subsequence of approximate solutions which converges, almost surely, to a weak solution in the probabilistically strong sense. The proof is based on uniform energy estimates in terms of the expectation of the energy norms, which are the backbone for a weak compactness argument giving rise to a weakly convergent subsequence of probability measures associated with the approximate solutions. Probabilistic techniques based on the Skorohod representation theorem and the Gyongy-Krylov lemma are then employed to obtain almost sure convergence of a subsequence of the random approximate solutions to a weak solution in the probabilistically strong sense. The result shows that the deterministic benchmark FSI model is robust to stochastic noise, even in the presence of rough white noise in time. To the best of our knowledge, this is the rst well-posedness result for stochastic uid-structure interaction.more » « less
