Higher Order Regularity Research Articles

A variational approach to frame-indifferent quasistatic viscoelasticity of rate type.

The three-dimensional dynamical model for nonlinear viscoelasticity of strain-rate type is investigated in a quasistatic setting under the assumption of higher-order regularity of the deformation, which in the literature is referred to as the case of non-simple materials. The existence of weak solutions is proven using a time-discretization technique while respecting the condition of dynamical frame indifference. Some observations on frame indifference for strain-rate-type stresses are made, and corrections are proposed for some related work in the literature. Finally, a counterexample is given to show that the assumed higher-order regularity is necessary in order to obtain the required compactness.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Efficient equilibrium-based stress recovery for isogeometric laminated Euler–Bernoulli curved beams

Laminated curved composite parts, used, e.g., in the spar and ribs in aircraft and wind turbine blades, are typically subjected to high interlaminar stresses. This work focuses on a two-step procedure to study laminated Euler–Bernoulli curved beams discretized via Isogeometric Analysis (IGA). First, we solve a (planar) Euler–Bernoulli curved beam formulation in primal form to obtain the tangential and transverse displacements. This formulation features high-order PDEs, which we can straightforwardly approximate using either an IGA-Galerkin or an IGA-collocation approach. Starting from the obtained displacement solution, which accounts for bending-stretching coupling, we can directly compute the normal stress only, while we do not have information concerning the transverse shear stress state, typically responsible for delamination. However, by imposing equilibrium in strong form in a curvilinear framework which eases the post-processing, eliminating the need for coordinate changes, we can easily recover interlaminar transverse shear stresses at locations of interest. Such a posteriori step requires calculating the high-order displacement derivatives in the equilibrium equations and, therefore, demands once again higher-order regularity that can be easily fulfilled by exploiting the high-continuity properties of IGA. Extensive numerical tests prove the effectiveness of the proposed approach, which is also aided by the IGA’s superior geometric approximation.

On regularity conditions in higher-order state-constrained control problems

This work is aimed at the study of optimal control problems with state constraints of higher order. Such problems are due to multi-level dynamical control systems of triangle-type which are often encountered in various engineering applications, notably in Robotics. In the framework of the proposed investigation, a notion of higher-order regularity with respect to the state constraints (termed as k-regularity) is formulated. Under such a condition, a strengthened variant of Pontryagin’s maximum principle is derived in which the non-triviality condition is pointwise and thereby allows for the informative maximum condition. An example of a 3-order control problem is considered, which demonstrates the essentiality of k-regularity for pointwise non-triviality and also shows how the presented necessary optimality conditions can be applied.

Rates of convergence for regression with the graph poly-Laplacian

In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset {xi}i=1n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x_i\\}_{i=1}^n$$\\end{document} and a set of noisy labels {yi}i=1n⊂R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{y_i\\}_{i=1}^n\\subset \\mathbb {R}$$\\end{document} we let un:{xi}i=1n→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_n{:}\\{x_i\\}_{i=1}^n\\rightarrow \\mathbb {R}$$\\end{document} be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When yi=g(xi)+ξi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y_i = g(x_i)+\\xi _i$$\\end{document}, for iid noise ξi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi _i$$\\end{document}, and using the geometric random graph, we identify (with high probability) the rate of convergence of un\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_n$$\\end{document} to g in the large data limit n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document}. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.

Embedded boundary conditions for shear-deformable plate bending

An efficient procedure for embedding kinematic boundary conditions of shear-deformable plate bending is based on stabilized variational formulations, obtained by Nitsche’s approach for enforcing surface constraints. Effective use of the plate thickness as scaling in the rotational constraint terms leads to a uniform problem statement with a single stabilization parameter. An alternative form of the plate kinematics precludes the vulnerability of computation based on the standard Reissner–Mindlin formulation to shear locking, yet raises other challenges in the form of higher-order regularity and unconventional boundary conditions. The embedded boundary Nitsche approach provides a framework that easily accommodates these features. The accuracy of the approach for both plate formulations is verified by representative computations with bicubic B-splines, exhibiting optimal rates of convergence, and robust performance with respect to values of the parameters.

Higher-order error estimates for physics-informed neural networks approximating the primitive equations

Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates are a priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the H^s norm during the training.

Higher-Order Boundary Layers and Regularity for Stokes Systems over Rough Boundaries

Higher-Order Boundary Layers and Regularity for Stokes Systems over Rough Boundaries

Multipatch isogeometric analysis for geometrically exact shell elements using B-bar method and Bézier extraction

In the present study, a multipatch isogeometric analysis method for geometrically exact shell problems is proposed. Generalized curvilinear coordinates are used for the first-order shear-deformable geometrically exact shell elements. This general tensor-based shell formulation can be directly linked to a computer-aided design system such as nonuniform rational basis spline (NURBS), which is represented by two parameters. Nitsche’s method for patch-to-patch connections in conforming or nonconforming meshes is adopted. The compatibility conditions between NURBS patches in Nitsche’s coupling are weakly imposed using the continuity of displacements, rotations, and stress resultants. The final algebraic equation for the solution fields has a simple symmetric form with previously derived strain–displacement relation matrices and material modulus for membrane, bending, and transverse shear terms. To alleviate the locking phenomenon, the B-bar method, which utilizes the strain projected onto a lower-order field, is used in the isogeometric shell element. Reduced-order B-splines for strain fields are constructed based on tying points, which eased the locking phenomenon compared with the classical B-bar method. To further alleviate additional shell locking by a higher-order regularity, the Bézier extraction method is adopted to reduce the element regularity. The proposed combination of Nitsche’s coupling and B-bar projection demonstrates the superior accuracy and robustness in the multipatch isogeometric shell examples with a high convergence rate.

Higher-Order Regularity of the Free Boundary in the Inverse First-Passage Problem

Consider the inverse first-passage problem: Given a diffusion process $\{\frak{X}_{t}\}_{t\geqslant 0}$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a survival probability function $p$ on $[0,\infty)$, find a boundary, $x=b(t)$, such that $p$ is the survival probability that $\frak{X}$ does not fall below $b$, i.e., for each $t\geqslant 0$, $p(t)= \mathbb{P}(\{\omega\in\Omega\;|\; {\frak{X}}_s(\omega) \geqslant b(s),\ \forall\, s\in(0,t)\})$. In earlier work, we analyzed viscosity solutions of a related variational inequality, and showed that they provided the only upper semi-continuous (usc) solutions of the inverse problem. We furthermore proved weak regularity (continuity) of the boundary $b$ under additional assumptions on $p$. The purpose of this paper is to study higher-order regularity properties of the solution of the inverse first-passage problem. In particular, we show that when $p$ is smooth and has negative slope, the viscosity solution, and therefore also the unique usc solution of the inverse problem, is smooth. Consequently, the viscosity solution furnishes a unique classical solution to the free boundary problem associated with the inverse first-passage problem.

Nonlinear parabolic stochastic evolution equations in critical spaces part II

This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss–Simonett–Wilke (J Differ Equ 264(3):2028–2074, 2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical L^2-settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier–Stokes equations, reaction– diffusion equations and the Allen–Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results.

Global well-posedness for axisymmetric MHD equations with vertical dissipation and vertical magnetic diffusion

In this paper, the unique global strong axisymmetric solutions for the incompressible MHD equations with only vertical dissipation and vertical magnetic diffusion are established, where no smallness assumptions on the initial data are required. One of the key ingredients of the analysis is to obtain higher-order regularity estimates for velocity and magnetic fields, where the axisymmetry of the solutions and some new L 2 energy estimates play an important role.

Fractional Sobolev regularity for fully nonlinear elliptic equations

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number depending only on ellipticity constants and dimension, such that if u is a viscosity solution of then, with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that for a universal constant We believe our techniques are flexible and can be adapted to various models and contexts.

Isogeometric analysis for geometrically exact shell elements using Bézier extraction of NURBS with assumed natural strain method

In the present study, isogeometric analysis for geometrically exact shell considering the first-order shear deformation is studied. Since kinematics of the shell is formulated in the form of a general tensor in generalized curvilinear coordinates, isogeometric analysis of arbitrarily shaped shells is possible with a direct geometry link to NURBS. For the first-order shear deformable shell element, the assumed natural strain (ANS) method is normally used to alleviate membrane and transverse shear locking phenomenon. Due to a higher-order regularity between integration knot elements of NURBS basis function has, a locking phenomenon still appears even though the field consistent approach such as ANS is utilized. To solve this problem, we adopted Bézier extraction method which maps a piecewise C0 Bernstein polynomials basis onto NURBS basis to reduce element regularity for isogeometric shell analysis. For an arbitrarily shaped shell geometry generation using NURBS of the shell, the point data sets from the exact geometry shape can be used by the global interpolation. The test results of numerical examples using the developed method show the accuracy and robustness with a higher convergence rate of the proposed approach.

On the vanishing dissipation limit for the incompressible MHD equations on bounded domains

In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magnetohydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial boundary value problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher-order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD equation as the dissipations tend to zero.

Higher-order estimates of long-time solutions to the Kähler-Ricci flow

In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. By developing sharp new parabolic Schauder estimates on cylinders, we proved that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges in Cloc∞-topology away from singular fibers to a negative Kähler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fibers are biholomorphic to each other.

Cortical entrainment to hierarchical contextual rhythms recomposes dynamic attending in visual perception.

Temporal regularity is ubiquitous and essential to guiding attention and coordinating behavior within a dynamic environment. Previous researchers have modeled attention as an internal rhythm that may entrain to first-order regularity from rhythmic events to prioritize information selection at specific time points. Using the attentional blink paradigm, here we show that higher-order regularity based on rhythmic organization of contextual features (pitch, color, or motion) may serve as a temporal frame to recompose the dynamic profile of visual temporal attention. Critically, such attentional reframing effect is well predicted by cortical entrainment to the higher-order contextual structure at the delta band as well as its coupling with the stimulus-driven alpha power. These results suggest that the human brain involuntarily exploits multiscale regularities in rhythmic contexts to recompose dynamic attending in visual perception, and highlight neural entrainment as a central mechanism for optimizing our conscious experience of the world in the time dimension.

Higher-order regularity in local and nonlocal quantum gravity

In the present work we investigate the Newtonian limit of higher-derivative gravity theories with more than four derivatives in the action, including the non-analytic logarithmic terms resulting from one-loop quantum corrections. The first part of the paper deals with the occurrence of curvature singularities of the metric in the classical models. It is shown that in the case of local theories, even though the curvature scalars of the metric are regular, invariants involving derivatives of curvatures can still diverge. Indeed, we prove that if the action contains 2n+6 derivatives of the metric in both the scalar and the spin-2 sectors, then all the curvature-derivative invariants with at most 2n covariant derivatives of the curvatures are regular, while there exist scalars with 2n+2 derivatives that are singular. The regularity of all these invariants can be achieved in some classes of nonlocal gravity theories. In the second part of the paper, we show that the leading logarithmic quantum corrections do not change the regularity of the Newtonian limit. Finally, we also consider the infrared limit of these solutions and verify the universality of the leading quantum correction to the potential in all the theories investigated in the paper.

Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics

The (third order in time) JMGT equation [Jordan (J Acoust Soc Am 124(4):2491–2491, 2008) and Cattaneo (C Sulla conduzione del calore Atti Sem Mat Fis Univ Modena 3:83–101, 1948)] is a nonlinear (quasi-linear) partial differential equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second-order in time equation referred to as Westervelt equation. Replacing Fourier’s law by Maxwell–Cattaneo’s law gives rise to the third-order in time derivative scaled by a small parameter \(\tau >0\), the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper, we provide an asymptotic analysis of the third-order model when \(\tau \rightarrow 0 \). It is shown that the corresponding solutions converge in a strong topology of the phase space to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher-order regularity. This addresses an open question raised in [20], where a related JMGT equation has been studied and weak star convergence of the solutions when \(\tau \rightarrow 0\) has been established. Thus, our main contribution is showing strong convergence on infinite time horizon, along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the “smallness” of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.

Higher-Order Linearization and Regularity in Nonlinear Homogenization

We prove large-scale C^infty regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert’s 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian overline{L}, (ii) the commutation of higher-order linearization and homogenization, and (iii) large-scale C^{0,1}-type regularity for higher-order linearization errors. We consequently obtain a quantitative estimate on the scaling of linearization errors, a Liouville-type theorem describing the polynomially-growing solutions of the system of higher-order linearized equations, and an explicit (heterogenous analogue of the) Taylor series for an arbitrary solution of the nonlinear equations—with the remainder term optimally controlled. These results give a complete generalization to the nonlinear setting of the large-scale regularity theory in homogenization for linear elliptic equations.

Attractors with higher-order regularity of stochastic reaction–diffusion equations on time-varying domains

Random attractors and their higher-order regularity properties are studied for stochastic reaction–diffusion equations on time-varying domains. Some new a priori estimates for the difference of solutions near the initial time and the continuous dependence in initial data in [Formula: see text] are proved. Then attraction of the random attractors in the higher integrability space [Formula: see text] for any [Formula: see text] and the regular space [Formula: see text] is established.