Maximal Regularity Research Articles

Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations

We study a type of third-order linear differential equations with variable and unbounded coefficients, which are defined in an infinite interval. We also consider a non-linear generalization with coefficients that depends on an unknown function. We establish sufficient conditions for the correctness of this linear equation and the maximal regularity estimate for their solution. Using these results, we prove the solvability of a nonlinear differential equation and estimate the norms of its terms.

Fully discrete best-approximation-type estimates inL∞ (I;L2(Ω)d) for finite element discretizations of the transient Stokes equations

AbstractIn this article, we obtain an optimal best-approximation-type result for fully discrete approximations of the transient Stokes problem. For the time discretization, we use the discontinuous Galerkin method and for the spatial discretization we use standard finite elements for the Stokes problem satisfying the discrete inf-sup condition. The analysis uses the technique of discrete maximal parabolic regularity. The results require only natural assumptions on the data and do not assume any additional smoothness of the solutions.

Maximal L1-Regularity of Generators for Bounded Analytic Semigroups in Banach Spaces

In this paper, we prove that the generator of any bounded analytic semigroup in (θ, 1)-type real interpolation of its domain and underlying Banach space has maximal L1-regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal L1-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous B sq,1 -type Besov spaces on domains of ℝn, n ≥ 2.

Maximal Lp-Lq regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation

This paper is concerned with the generalized resolvent estimate and the maximal Lp-Lq regularity of the linearized Oberbeck-Boussinesq approximation for unsteady motion of a drop in another fluid without surface tension, which is indispensable for establishing the well-posedness of the Oberbeck-Boussinesq approximation for the two incompressible liquids separated by a closed interface. We prove the existence of R-bounded solution operators for the model problems and the maximal Lp-Lq regularity for the system. The key step is to prove the maximal Lp-Lq regularity theorem for the linearized heat equation with the help of the R-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem.

Maximal Regularity for Compressible Two-Fluid System

We investigate a compressible two-fluid Navier-Stokes type system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. We are interested in regular solutions in a $L_p-L_q$ maximal regularity setting. We show that such solutions exists locally in time and, under additional smallness assumptions on the initial data, also globally. Our proof rely on appropriate transformation of the original problem, application of Lagrangian coordinates and maximal regularity estimates for associated linear problem.

Anisotropy and stratification effects in the dynamics of fast rotating compressible fluids

The primary goal of this paper is to develop robust methods to handle two ubiquitous features appearing in the modeling of geophysical flows: (i) the anisotropy of the viscous stress tensor and (ii) stratification effects.We focus on the barotropic Navier–Stokes equations with Coriolis and gravitational forces. Two results are the main contributions of the paper. Firstly, we establish a local well-posedness result for finite-energy solutions, via a maximal regularity approach. This method allows us to circumvent the use of the effective viscous flux, which plays a key role in the weak solutions theories of Lions–Feireisl and Hoff, but seems to be restricted to isotropic viscous stress tensors. Moreover, our approach is sturdy enough to take into account nonconstant reference density states; this is crucial when dealing with stratification effects. Secondly, we study the structure of the solutions to the previous model in the regime when the Rossby, Mach and Froude numbers are of the same order of magnitude. We prove an error estimate on the relative entropy between actual solutions and their approximation by a large-scale quasi-geostrophic flow supplemented with Ekman boundary layers. Our analysis holds for a large class of barotropic pressure laws.

Staffans–Weiss perturbations for maximal $$L^p$$-regularity in Banach spaces

In this paper, we show that the concept of maximal \(L^p\)-regularity is stable under a large class of unbounded perturbations, namely Staffans–Weiss perturbations. To that purpose, we first prove that the analyticity of semigroups is preserved under this class of perturbations, which is a necessary condition for the maximal regularity. In UMD spaces, \({\mathcal {R}}\)-boundedness is exploited to give conditions guaranteeing the maximal regularity. For Banach spaces, a condition is imposed to prove maximal regularity. Moreover, we apply the obtained results to perturbed boundary value problems.

A Global Regularity Result for the 2D Generalized Magneto-Micropolar Equations

In this paper, we proved the global (in time) regularity for smooth solution to the 2D generalized magneto-micropolar equations with zero viscosity. When there is no kinematic viscosity in the momentum equation, it is difficult to examine the bounds on the any derivatives of the velocity J ε u L 2 . In order to overcome the main obstacle, we find a new unknown quantity which is by combining the vorticity and the microrotation angular velocity; the structure of the system including the combined quantity obeys a Beale–Kato–Majda criterion. Moreover, the maximal regularity of parabolic equations together with the classic commutator estimates allows us to derive the H s estimates for solutions of the system.

Regularity properties of nonlocal fractional differential equations and applications

Abstract The regularity properties of nonlocal fractional elliptic and parabolic equations in vector-valued Besov spaces are studied. The uniform B p , q s B_{p,q}^{s} -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. In particular, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, the maximal regularity properties of the nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional parabolic equations are studied.

On Maximal Regularity Estimates for Discontinuous Galerkin Time-Discrete Methods

On Maximal Regularity Estimates for Discontinuous Galerkin Time-Discrete Methods

A refinement of Baillon’s theorem on maximal regularity

By Baillon’s theorem, it is known that maximal regularity with respect to the space of continuous functions is rare; it implies that either the semigroup generator involved is a bounded operator or the space considered contains $c_{0}$. We show that the l

Maximal regularity for time-stepping schemes arising from convolution quadrature of non-local in time equations

<p style='text-indent:20px;'>We study discrete time maximal regularity in Lebesgue spaces of sequences for time-stepping schemes arising from Lubich's convolution quadrature method. We show minimal properties on the quadrature weights that determines a wide class of implicit schemes. For an appropriate choice of the weights, we are able to identify the <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>-method as well as the backward differentiation formulas and the <inline-formula><tex-math id="M2">\begin{document}$ L1 $\end{document}</tex-math></inline-formula>-scheme. Fractional versions of these schemes, some of them completely new, are also shown, as well as their representation by means of the Grünwald–Letnikov fractional order derivative. Our results extend and improve some recent results on the subject and provide new insights on the basic nature of the weights that ensure maximal regularity.</p>

On the Maximal Lp-Lq Regularity Theorem for the Linearized Electro-Magnetic Field Equations with Interface Conditions

On the Maximal Lp-Lq Regularity Theorem for the Linearized Electro-Magnetic Field Equations with Interface Conditions

On the admissibility of observation for perturbed non-autonomous linear systems

In this paper, we are concerned with unbounded observation operators for perturbed non-autonomous evolution equations. First, we briefly survey the necessary background about the concept of maximal regularity. Second, we establish the admissibility invariance of observation operators under non-autonomous Miyadera-Voigt perturbations. Finally, we illustrate our abstract result with an example.

Video Quality Model of Compression, Resolution and Frame Rate Adaptation Based on Space-Time Regularities

Being able to accurately predict the visual quality of videos subjected to various combinations of dimension reduction protocols is of high interest to the streaming video industry, given rapid increases in frame resolutions and frame rates. In this direction, we have developed a video quality predictor that is sensitive to spatial, temporal, or space-time subsampling combined with compression. Our predictor is based on new models of space-time natural video statistics (NVS). Specifically, we model the statistics of divisively normalized difference between neighboring frames that are relatively displaced. In an extensive empirical study, we found that those paths of space-time displaced frame differences that provide maximal regularity against our NVS model generally align best with motion trajectories. Motivated by this, we built a new video quality prediction engine that extracts NVS features that represent how space-time directional regularities are disturbed by space-time distortions. Based on parametric models of these regularities, we compute features that are used to train a regressor that can accurately predict perceptual quality. As a stringent test of the new model, we apply it to the difficult problem of predicting the quality of videos subjected not only to compression, but also to downsampling in space and/or time. We show that the new quality model achieves state-of-the-art (SOTA) prediction performance on the new ETRI-LIVE Space-Time Subsampled Video Quality (STSVQ) and also on the AVT-VQDB-UHD-1 database.

The well-posedness of a parabolic non-autonomous semilinear equation

In this paper, we consider a class of semilinear non-autonomous equations, where the nonlinear term does not perfectly satisfy the Lipschitz condition. It means that the nonlinear function is defined as the composition of a global Lipschitz function and non-autonomous unbounded linear operators. This case is beyond the scope of standard Cauchy-Lipshitz theory and requires further treatment. First, we deal with the well-posedness of non-autonomous parabolic equations, from the point of view of maximal regularity. This allows, indirectly, to use a fixed point theorem to prove the result. Second, we will rely on the concept of Lebesgue extensions of non-autonomous observation operators to prove the admissibility of observation for semilinear non-autonomous systems.

Inverse problems on degenerate integro-differential equations

<p style='text-indent:20px;'>We study inverse problems on degenerate integro-differential equations without using maximal regularity property. This seems new at all in the related literature. Some applications to concrete partial differential equations are given.</p>

On a generalized diffusion problem: A complex network approach

<p style='text-indent:20px;'>In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution <i>via</i> operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors.</p>

The Moore-Penrose inverse of accretive operators with application to quadratic operator pencils

We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.

Maximal regularity and a singular limit problem for the Patlak\u2013Keller\u2013Segel system in the scaling critical space involving BMO

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter tau rightarrow infty . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces dot{B}^s_{q,sigma }({mathbb {R}}^n) and dot{F}^s_{q,sigma }({mathbb {R}}^n) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.