Uniform Regularity Research Articles

On uniform regularity and strong regularity

ABSTRACTWe investigate uniform versions of (metric) regularity and strong (metric) regularity on compact subsets of Banach spaces, in particular, along continuous paths. These two properties turn out to play a key role in analyzing path-following schemes for tracking a solution trajectory of a parametric generalized equation or, more generally, of a differential generalized equation (DGE). The latter model allows us to describe in a unified way several problems in control and optimization such as differential variational inequalities and control systems with state constraints. We study two inexact path-following methods for DGEs having the order of the grid error and , respectively. We provide numerical experiments, comparing the schemes derived, for simple problems arising in physics. Finally, we study metric regularity of mappings associated with a particular case of the DGE arising in control theory. We establish the relationship between the pointwise version of this property and its counterpart in function spaces.

Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime

We consider a kinetic-fluid model with random initial inputs which describes disperse two-phase flows. In the light particle regime, using energy estimates, we prove the uniform regularity in the random space of the model for random initial data near the global equilibrium in some suitable Sobolev spaces, with the randomness in the initial particle distribution and fluid velocity. By hypocoercivity arguments, we prove that the energy decays exponentially in time, which means that the long time behavior of the solution is insensitive to such randomness in the initial data. Then we consider the generalized polynomial chaos stochastic Galerkin method (gPC-sG) for the same model. For initial data near the global equilibrium and smooth enough in the physical and random spaces, we prove that the gPC-sG method has spectral accuracy, uniformly in time and the Knudsen number, and the error decays exponentially in time.

Uniform regularity for the incompressible Navier-Stokes system with variable density and Navier boundary conditions

We investigate the uniform regularity for the nonhomogeneous incompressible Navier-Stokes system with Navier boundary conditions and the inviscid limit to the Euler system. It is shown that there exists a unique strong solution of the Navier-Stokes equations in an interval of time that is uniform with respect to the viscosity parameter. The uniform estimate in conormal Sobolev spaces is established. Based on the uniform estimate, we show the convergence of the viscous solutions to the inviscid ones in L ∞ ( [ 0 , T ] × Ω ) L^\infty ([0,T]\times \Omega ) . This improves the result obtained by Ferreira et al. [SIAM J. Math. Anal. Vol. 45, No. 4, (2013), pp. 2576-2595], where L ∞ ( [ 0 , T ] , L 2 ( Ω ) ) L^\infty ([0,T],L^2(\Omega )) convergence was proved.

Existence and continuity of solution trajectories of generalized equations with application in electronics

We consider a special form of parametric generalized equations arising from electronic circuits with AC sources and study the effect of perturbing the input signal on solution trajectories. Using methods of variational analysis and strong metric regularity property of an auxiliary map, we are able to prove the regularity properties of the solution trajectories inherited by the input signal. Furthermore, we establish the existence of continuous solution trajectories for the perturbed problem. This can be achieved via a result of uniform strong metric regularity for the auxiliary map.

Uniform regularity for free-boundary Navier–Stokes equations with surface tension

We study the zero-viscosity limit of free boundary Navier–Stokes equations with surface tension in unbounded domain thus extending the work of Masmoudi and Rousset [Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations, Arch. Ration. Mech. Anal. (2016), doi:10.1007/s00205-016-1036-5] to take surface tension into account. Due to the presence of boundary layers, we are unable to pass to the zero-viscosity limit in the usual Sobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at the boundary should blow-up. To deal with this problem, we solve the free boundary problem in the so-called Sobolev co-normal spaces (after fixing the boundary via a coordinate transformation). We prove estimates which are uniform in the viscosity. And after inviscid limit process, we get the local existence of free-boundary Euler equation with surface tension. Our main idea is to use Dirichlet–Neumann operator and time-derivatives.

Uniform regularity for a 3D time-dependent Ginzburg–Landau model in superconductivity

In this paper, we prove the uniform-in-σ regularity for 3D time-dependent Ginzburg–Landau model in superconductivity in the case of Coulomb gauge. Here σ is the normal conductivity of the superconducting material.

Metrically Regular Differential Generalized Equations

In this paper we consider a control system coupled with a generalized equation, which we call a differential generalized equation (DGE). This model covers a large territory in control and optimization, such as differential variational inequalities, control systems with constraints, as well as necessary optimality conditions in optimal control. We study metric regularity and strong metric regularity of mappings associated with DGE by focusing in particular on the interplay between the pointwise versions of these properties and their infinite-dimensional counterparts. Metric regularity of a control system subject to inequality state-control constraints is characterized. A sufficient condition for local controllability of a nonlinear system is obtained via metric regularity. Sufficient conditions for strong metric regularity in function spaces are presented in terms of uniform pointwise strong metric regularity. A characterization of the Lipschitz continuity of the control part of the solution mapping as a function of time is established. Finally, a path-following procedure for a discretized DGE is proposed for which an error estimate is derived.

Soil enzymes as biodiagnostics indicator of heavy metal pollution of urbanozem

The article presents a comparative analysis of the impact of the introduction of different doses of copper and cadmium on the activity of redox enzymes of urbanozem, collected from different territories of Ufa. The studies established the inverse relationship of the activity of catalase and polyphenol oxidase, and the direct one of the activity of peroxidase that depends on the doses of heavy metals, that allows to recommend their use as bioindicator of pollution of urbanozem with these metals. The reaction of the studied enzymes on the introduction of heavy metals is an indicator of their toxicity to living things at the molecular level. Comparative analysis of the impact of cadmium and copper in different doses on the activity of soil enzymes did not reveal a uniform regularity. Each of the metals showed their toxicity in different ways depending on the duration of their impact.

Uniform regularity for the free surface compressible Navier–Stokes equations with or without surface tension

In this paper, we investigate the uniform regularity of solutions to the three-dimensional isentropic compressible Navier–Stokes system with free surfaces and study the corresponding asymptotic limits of such solutions to that of the compressible Euler system for vanishing viscosity and surface tension. It is shown that there exists a unique strong solution to the free boundary problem for the compressible Navier–Stokes system in a finite time interval which is independent of the viscosity and the surface tension. The solution is uniformly bounded both in [Formula: see text] and a conormal Sobolev space. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. Based on such uniform estimates, the asymptotic limits, to the free boundary problem for the ideal compressible Euler system with or without surface tension as both the viscosity and the surface tension tend to zero, are established by a strong convergence argument.

Criteria of pointwise and uniform directional Lipschitz regularities on tensor products of Schauder functions

In Ben Slimane and Ben Braiek (2012) and Ben Slimane (2012), we found characterizations of the pointwise and uniform directional regularities of a multi-parameter function in terms of decay rates of either anisotropic Triebel wavelet coefficients or continuous Calderón anisotropic wavelet transform. The purpose of this paper is twofold. We first use a result of Kamont (1996) to provide an easier criterium of uniform directional Lipschitz regularity by decay conditions on the coefficients of the function in a tensor product Schauder basis. As a consequence, we deduce the characterization of the local critical directional Lipschitz regularity. We apply our results for both multidimensional parameter fractional Wiener field in Rd and Sierpinski cascade function. We then obtain criteria of pointwise directional Lipschitz regularity by decay conditions on either two progressive difference in the given direction or the coefficients of the function in a tensor product Schauder basis.

Entropy and a convergence theorem for Gauss curvature flow in high dimension

We prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in C^\infty -topology to a smooth strictly convex soliton as t goes to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of an entropy functional for convex bodies.

Regularity for the evolution of p-harmonic maps

This paper presents our study of regularity for p-harmonic map heat flows. We devise a monotonicity-type formula of scaled energy and establish a criterion for a uniform regularity estimate for regular p-harmonic map heat flows. As application we show the small data global in the time existence of regular p-harmonic map heat flow.

Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem

For the singularly perturbed system $$\begin{aligned} \varDelta u_{i,\beta }=\beta u_{i,\beta }\sum _{j\ne i}u_{j,\beta }^2, \quad 1\le i\le N, \end{aligned}$$ we prove that flat segregated interfaces are uniformly Lipschitz as \(\beta \rightarrow +\infty \). As a byproduct of the proof we also obtain the optimal lower bound near flat interfaces, $$\begin{aligned} \sum _iu_{i,\beta }\ge c\beta ^{-1/4}. \end{aligned}$$ .

Uniform regularity and vanishing viscosity limit for the chemotaxis‐Navier‐Stokes system in a 3D bounded domain

We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis‐Navier‐Stokes system with Navier boundary condition for velocity field and Neumann boundary condition for cell density and chemical concentration in a 3D bounded domain. It is shown that there exists a unique strong solution of the incompressible chemotaxis‐Navier‐Stokes system in a finite time interval, which is independent of the viscosity coefficient. Moreover, this solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible chemotaxis‐Euler system.

Uniform regularity and convergence of phase-fields for Willmore’s energy

We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure $$\mu $$ . For this purpose, we introduce a suitable notion of essentially uniform convergence. This mode of convergence is a natural generalisation of uniform convergence that precisely describes the convergence of phase fields in three dimensions. More in detail, we show that, in three space dimensions, points close to which the phase fields stay bounded away from a pure phase lie either in the support of the limiting mass measure $$\mu $$ or contribute a positive amount to the limiting Willmore energy. Thus there can only be finitely many such points. As an application, we investigate the Hausdorff limit of level sets of sequences of phase fields with bounded energy. We also obtain results on boundedness and $$L^p$$ -convergence of phase fields and convergence from outside the interval between the wells of a double-well potential. For minimisers of suitable energy functionals, we deduce uniform convergence of the phase fields from essentially uniform convergence.

Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity

This paper concerns with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization. We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with $L^2$ boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity and uses a large-scale Rellich estimate obtained in \cite{Shen-2016}.

Local regularity and compactness for the p-harmonic map heat flows

Abstract We study a geometric analysis and local regularity for the evolution of p {{p}} -harmonic maps, called p {{p}} -harmonic map heat flows. Our main result is to establish a criterion for a uniform local regularity estimate for regular p {{p}} -harmonic map heat flows, devising some new monotonicity-type formulas of a local scaled energy. The regularity criterion obtained is almost optimal, comparing with that of the corresponding stationary case. As application we show a compactness of regular p {{p}} -harmonic map heat flows with energy bound.

Uniform regularity and vanishing viscosity limit for the compressible nematic liquid crystal flows

Uniform regularity and vanishing viscosity limit for the compressible nematic liquid crystal flows

The quasineutral limit of the Vlasov–Poisson equation in Wasserstein metric

In this work, we study the quasineutral limit of the one-dimensional Vlasov-Poisson equation for ions with massless thermalized electrons. We prove new weak-strong stability estimates in the Wasserstein metric that allow us to extend and improve previously known convergence results. In particular, we show that given a possibly unstable analytic initial profile, the formal limit holds for sequences of measure initial data converging sufficiently fast in the Wasserstein metric to this profile. This is achieved without assuming uniform analytic regularity.

Uniform Regularity for Linear Kinetic Equations with Random Input Based on Hypocoercivity

In this paper we study the effect of randomness in kinetic equations that preserve mass. Our focus is in proving the analyticity of the solution with respect to the randomness, which naturally leads to the convergence of numerical methods. The analysis is carried out in a general setting, with the regularity result not depending on the specific form of the collision term, the probability distribution of the random variables, or the regime the system is in and thereby is termed “uniform. Applications include the linear Boltzmann equation, the Bhatnagar--Gross--Krook (BGK) model, and the Carlemann model, among many others, and the results hold true in kinetic, parabolic, and high field regimes. The proof relies on the explicit expression of the high order derivatives of the solution in the random space, and the convergence in time is mainly based on hypocoercivity, which, despite the popularity in PDE analysis of kinetic theory, has rarely been used for numerical algorithms.