Generalizing Mutational Equations for Uniqueness of Some Nonlocal 1<Superscript>st</Superscript>-Order Geometric Evolutions

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ? ^N evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.     

On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions

Regularity of the solution for the wave equation with constant propagation speed is conserved with respect to time, but such a property is not true in general if the propagation speed is variable with respect to time. The main purpose of this paper is to describe the order of regularity loss of the solution due to the variable coefficient by the following four properties of the coefficient: “smoothness”, “oscillations”, “degeneration” and “stabilization”. Actually, we prove the Gevrey and C^∞ well‐posedness for the wave equations with degenerate coefficients taking into account the interactions of these four properties. Moreover, we prove optimality of these results by constructing some examples (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hyperfunctions and (analytic) hypoellipticity

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known “sum of squares” operators, which satisfy Hörmander’s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).     

Sur les équations différentielles ordinaires et les équations de transport

We study in this Note ordinary differential equations for divergence-free vector-fields with a limited regularity. We first observe that it is equivalent to solve the associated transport equations (i.e. Liouville equations). Then, we show existence, uniqueness, and stability results for generic vector-fields in L¹ or for “piecewise” W^1.1 vector-fields.     

Stochastic differential equations

When a system is acted upon by exterior disturbances, its time-development can often be described by a system of ordinary differential equations, provided that the disturbances are smooth functions. But for sound reasons physicists and engineers usually want the theory to apply when the noises belong to a larger class, including for example “white noise.” If the integrals in the system derived for smooth noises are reinterpreted as Itô integrals, the equations make sense; but in nonlinear cases they often fail to describe the time-development of the system. In this paper (extending previous work of the author) a calculus is set up for stochastic systems that extends to a theory of differential equations. When the equations are known that describe the development of the system when noises are smooth, an extension to the larger class of noises is proposed that in many cases gives results consistent with the smooth-noise case and also has “robust” solutions, that change by small amounts when the noises undergo small changes. This is called the “canonical” extension.Nevertheless, there are certain systems in which the canonical equations are inappropriate. A criterion is suggested that may allow us to distinguish when the canonical equations are the right choice and when they are not.     

On the quasi-stationary Maxwell equations with monotone characteristics in a multiply connected domain

The quasi-stationary Maxwell equations for a magnetic core with a non-linear characteristic and homeomorphic to a three dimensional torus are reduced to a unique evolution equation involving a monotone operator. This reduction is based on the properties of some functional spaces suited to the study of the operators “curl” and “div,” with different kinds of boundary and period conditions. Some existence, uniqueness and regularity results are proved.     

A theory of regularity structures

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the \(\Phi ^4_3\) Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of \(3\) -dimensional ferromagnets near their critical temperature.     

Abstract differential equations with almost-periodic solutions

In this paper we present some quite simple results concerning almost-periodic solutions of abstract differential equations. We start with a general proposition about linear equations, then we establish some new facts about bounded or relatively compact solutions which become almost-periodic; finally, we study “separation from 0 properties” of non-trivial almost-periodic solutions for equations with bounded or unbounded operator coefficients.     

Necessary conditions without differentiability assumptions in optimal control

We derive an existence theorem coupled with necessary conditions for the relaxed problem of the optimal control of ordinary differential equations in which the cost functional, the restrictions, and the right hand sides are Lipschitz-continuous (but not necessarily differentiable) in their dependence on the state variables. Our approach is based on the use of special mollifiers to approximate Lipschitz-continuous functions with C¹ functions and the subsequent study of the behavior of necessary conditions for the approximating problems as these mollifiers tend to the δ-distribution. Our generalization of the relaxed Pontryagin maximum principle and of the transversality conditions has a canonical form obtained by replacing, in the “old” expressions, the partial derivatives with finite difference quotients at neighboring arguments, and then applying limiting processes and convexification. Somewhat stronger necessary conditions are obtained by representing the Lipschitz-continuous functions as compositions, some of whose factors may be differentiable. The application of the canonical necessary conditions is illustrated by examples.     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.     

The asymptotics of solutions of singularly perturbed functional differential equations: Distributed and concentrated delays are different

This paper illustrates the differences between systems with distributed delays and systems having only concentrated delays in what concerns the asymptotic rates of solutions of singularly perturbed linear retarded functional differential equations. An example of a system with distributed delays shows that the introduction of a “slow” variable coupled with the “fast” variable may decrease the asymptotic rates of solutions observed when the perturbation parameter is close to zero. Such a situation cannot happen for ordinary differential equations, or even for differential-difference equations.     

The Global Random Attractor for a Class of Stochastic Porous Media Equations

We prove new L ²-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of “ζ-monotonicity” for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.     

Shape regularity conditions for polygonal/polyhedral meshes,exemplified in a discontinuous Galerkin discretization

This article concerns shape regularity conditions on arbitrarily shaped polygonal/polyhedral meshes. In (J. Wang and X. Ye, A weak Galerkin mixed finite element method for second‐order elliptic problems, Math Comp 83 (2014), 2101–2126), a set of shape regularity conditions has been proposed, which allows one to prove important inequalities such as the trace inequality, the inverse inequality, and the approximation property of the L² projection on general polygonal/polyhedral meshes. In this article, we propose a simplified set of conditions which provides similar mesh properties. Our set of conditions has two advantages. First, it allows the existence of “small” edges/faces, as long as the shape of the polygon/polyhedron is regular. Second, coupled with an extra condition, we are now able to deal with nonquasiuniform meshes. As an example, we show that the discontinuous Galerkin method for Laplacian equations on arbitrarily shaped polygonal/polyhedral meshes, satisfying the proposed set of shape regularity conditions, achieves optimal rate of convergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 308–325, 2015     

Differential analysis of polarity: Polar Hamilton-Jacobi,conservation laws,and Monge Ampère equations

We develop a differential theory for the polarity transform parallel to that of the Legendre transform, which is applicable when the functions studied are “geometric convex”, namely, convex, non-negative, and vanish at the origin. This analysis establishes basic tools for dealing with this duality transform, such as the polar subdifferential map, and variational formulas. Another crucial step is identifying a new, non-trivial, sub-class of C ² functions preserved under this transform. This analysis leads to a new method for solving many new first order equations reminiscent of Hamilton–Jacobi and conservation law equations, as well as some second order equations of Monge–Ampère type. This article develops the theory of strong solutions for these equations which, due to the nonlinear nature of the polarity transform, is considerably more delicate than its counterparts involving the Legendre transform. As one application, we introduce a polar form of the homogeneous Monge–Ampère equation that gives a dynamical meaning to a new method of interpolating between convex functions and bodies. A number of other applications, e.g., to optimal transport and affine differential geometry are considered in sequels.     

Existence results of second order impulsive functional differential equations with infinite delay and fractional damping

In this paper we prove the existence, uniqueness, regularity and continuous dependence of mild solutions for second order impulsive functional differential equations with infinite delay and fractional damping in Banach spaces. We generalize the existence theorem of integer order differential equations to the fractional order case. The results obtained here improve and generalize some known results.     

Generalizing Mutational Equations for Uniqueness of Some Nonlocal 1st-Order Geometric Evolutions

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ ^N evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.      

On Regularity Property of Retarded Ornstein–Uhlenbeck Processes in Hilbert Spaces

In this work, some regularity properties of mild solutions for a class of stochastic linear functional differential equations driven by infinite-dimensional Wiener processes are considered. In terms of retarded fundamental solutions, we introduce a class of stochastic convolutions which naturally arise in the solutions and investigate their Yosida approximants. By means of the retarded fundamental solutions, we find conditions under which each mild solution permits a continuous modification. With the aid of Yosida approximation, we study two kinds of regularity properties, temporal and spatial ones, for the retarded solution processes. By employing a factorization method, we establish a retarded version of the Burkholder–Davis–Gundy inequality for stochastic convolutions.     

Behaviour at time t = 0 of the solutions of semi-linear evolution equations

We investigate the regularity at time t = 0 of the solutions of linear and semi-linear evolutions equations (including the Stokes and Navier-Stokes equations), Necessary and sufficient conditions on the data for an arbitrary order of regularity are given (the classical “compatibility conditions”). In the case of the Stokes and Navier-Stokes equations the compatibility conditions which we find are global conditions on the data. The presentation given here seems to improve and generalize the known results even in the simplest case of linear evolution equations.     

Some properties of a uniformly linearly independent sequence of subspaces

The concept of a uniformly linearly independent sequence, due to R.M. Elkin, is a useful notion. Convergence theory of iterative processes for solving nonlinear equations or optimization problems in $\text{R}$ⁿ is an example of a discipline which has benefited from the use of this notion. The purpose of this paper is to present some properties of a uniformly linearly independent sequence of subspaces of $\text{R}$ⁿ. The properties derived were motivated by convergence results of Elkin for “block univariate relaxation” methods.     

Existence and stability for some partial neutral functional differential equations with infinite delay

In this paper, we study a class of partial neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. A principle of linearized stability is also established in the autonomous case. To illustrate our abstract results, we conclude this work by an example.