Maximal Regularity Estimates Research Articles

Weighted maximal regularity estimates of higher order elliptic systems on Lipschitz domains

For the homogeneous biharmonic equation Δ2u=0 in Ω with boundary data in WAk(∂Ω,ωσdS), the Banach space of the kth order Whitney Arrays on ∂Ω, we investigate values for σ in the power weightωσ=|z|σforσ>1−d such that the weighted L2 Dirichlet or regularity problem is solvable. Specifically, we show that there exists α>0 such that the Dirichlet and regularity problems are uniquely solvable for−min⁡(2+α,d−1)<σ<α, and−α<σ<2+α respectively. This gives a partially affirmative answer to a question posed by Shen (2005) [15].

Homogeneous Sobolev global-in-time maximal regularity and related trace estimates

Homogeneous Sobolev global-in-time maximal regularity and related trace estimates

The maximal regularity of the third‐order differential equation and its applications

In this work, we study the solvability and maximal regularity questions for the singular third‐order differential equation with unbounded coefficients and some applications. Unlike previously studied cases, the leading and intermediate coefficients of this equation can grow independently. We obtain sufficient coefficient conditions for correctness of this equation and compactness for inverse of the corresponding differential operator. We also prove the maximal regularity estimate for a generalized solution. Using these results, we obtain upper and lower estimates for the number of Kolmogorov ‐diameters of the set associated with the linear Korteweg–de Vries equation's solutions. We give an example and show that from the above inequalities follow two‐sided estimates of the Kolmogorov ‐diameters.

Solution formula for generalized two-phase Stokes equations and its applications to maximal regularity: Model problems

&lt;abstract&gt;&lt;p&gt;In this paper, we give a solution formula for the two-phase Stokes equations with and without surface tension and gravity over the whole space with a flat interface. The solution formula has already been considered by Shibata and Shimizu. However, we have reconstructed the formula so that we are able to easily prove resolvent and maximal regularity estimates. The previous work required the assumption of additional conditions on normal components. Here, although we consider normal components, the assumption is weaker than before. The method is based on an $ H^\infty $-calculus which has already been applied for the Stokes problems with various boundary conditions in the half-space.&lt;/p&gt;&lt;/abstract&gt;

ON ONE ELLIPTIC EQUATION WITH UNBOUNDED COEFFICIENTS

Abstract. The paper studies a second-order elliptic equation given on an infinite strip of width 2 and having variable intermediate and junior coefficients. We assume that the intermediate coefficients may not obey the potential. The problem is posed of finding a strong generalized solution of the equation that satisfies periodic conditions at the edges of the strip. The coefficients of the equation are smooth and not bounded functions. In terms of the coefficients themselves, the sufficient conditions for the existence and uniqueness of a solution to the problem posed are indicated, and a maximal regularity estimate for the solution is given. Their main difference from known results is the requirement of a different order of growth of intermediate coefficients for partial derivatives of an unknown function with respect to different variables. We also cover the case where the intermediate coefficients can grow at infinity faster than a linear function.

Singular limit problem for the Keller–Segel–Navier–Stokes system on torus

In this paper, we consider the singular limit problem of a new Keller–Segel–Navier–Stokes system with two nonlinear terms on the torus . By using the properties of the heat kernel, we derive the generalized maximal regularity estimates and solve this singular limit problem with combining the Banach fixed‐point theorem.

Global maximal regularity for equations with degenerate weights

In this paper we are concerned with global maximal regularity estimates for elliptic equations with degenerate weights. We consider both the linear case and the non-linear case. We show that higher integrability of the gradients can be obtained by imposing a local small oscillation condition on the weight and a local small Lipschitz condition on the boundary of the domain. Our results are new in the linear and non-linear case. We show by example that the relation between the exponent of higher integrability and the smallness parameters is sharp even in the linear or the unweighted case.

Causal sparse domination of Beurling maximal regularity operators

We prove boundedness of Calderön–Zygmund operators acting in Banach function spaces on domains, defined by the L1 Carleson functional and Lq (1 < q < ∞) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.

Global well-posedness for 2D fractional inhomogeneous Navier–Stokes equations with rough density

This paper is concerned with the global well-posedness issue of the two-dimensional (2D) incompressible inhomogeneous Navier–Stokes equations with fractional dissipation and rough density. By establishing the -maximal regularity estimate for the generalized Stokes system and using the Lagrangian approach, we prove the global existence and uniqueness of regular solutions for the 2D fractional inhomogeneous Navier–Stokes equations with large velocity field, provided that the initial density is sufficiently close to the constant 1 in and in the norm of some multiplier spaces. Moreover, we also consider the associated density patch problem, and show the global persistence of -regularity of the density patch boundary when the piecewise jump of density is small enough.

Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity

In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are Lp(Lq)-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8], the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model.

On the optimal Lq-regularity for viscous Hamilton–Jacobi equations with subquadratic growth in the gradient

This paper studies a maximal [Formula: see text]-regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a [Formula: see text] duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold [Formula: see text], [Formula: see text] being the dimension of the ambient space and [Formula: see text] the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range [Formula: see text].

Global existence of strong solutions to the kinetic Cucker-Smale model coupled with the two dimensional incompressible Navier-Stokes equations

&lt;p style='text-indent:20px;'&gt;In this paper, we investigate existence of global-in-time strong solutions to the Cauchy problem of the kinetic Cucker–Smale model coupled with the incompressible Navier–Stokes equations in the two dimensional space. By introducing a weighted Sobolev space and using the maximal regularity estimate on the linear non-stationary Stokes equations, we present a complete analysis on existence of global-in-time strong solutions to the coupled model, without any smallness assumptions on initial data.&lt;/p&gt;

Regularity results in 2D fluid–structure interaction

We study the interaction of an incompressible fluid in two dimensions with an elastic structure yielding the moving boundary of the physical domain. The displacement of the structure is described by a linear viscoelastic beam equation. Our main result is the existence of a unique global strong solution. Previously, only the ideal case of a flat reference geometry was considered such that the structure can only move in vertical direction. We allow for a general geometric set-up, where the structure can even occupy the complete boundary. Our main tool—being of independent interest—is a maximal regularity estimate for the steady Stokes system in domains with minimal boundary regularity. In particular, we can control the velocity field in W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} in terms of a forcing in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} provided the boundary belongs roughly to W3/2,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{3/2,2}.$$\\end{document} This is applied to the momentum equation in the moving domain (for a fixed time) with the material derivative as right-hand side. Since the moving boundary belongs a priori only to the class W2,2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2},$$\\end{document} known results do not apply here as they require a C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document}-boundary.

Global Unique Solutions for the Inhomogeneous Navier-Stokes equations with only Bounded Density, in Critical Regularity Spaces

We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where the initial density \(\rho _0\) is discontinuous and the initial velocity \(u_0\) has critical regularity. Assuming that \(\rho _0\) is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity \(u_0\) belongs to the critical homogeneous Besov space \(\dot{B}^{-1+2/p}_{p,1}({\mathbb {R}}^{2})\) \((1<p<2)\) and, in the three-dimensional case, if \(u_0\) is small in \(\dot{B}^{-1+3/p}_{p,1}({\mathbb {R}}^{3})\ (1<p<3)\). Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the case of large variations of density with, possibly, vacuum. Interestingly, our result implies that the Fujita-Kato type solutions constructed by Zhang (Adv Math 363:107007, 2020) are unique. Our work relies on interpolation results, time weighted estimates and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system.

Critical regularity issues for the compressible Navier–Stokes system in bounded domains

We are concerned with the barotropic compressible Navier–Stokes system in a bounded domain of mathbb {R}^d (with dge 2). In a critical regularity setting, we establish local well-posedness for large data with no vacuum and global well-posedness for small perturbations of a stable constant equilibrium state. Our results rely on new maximal regularity estimates—of independent interest—for the semigroup of the Lamé operator, and of the linearized compressible Navier–Stokes equations.

Various regularity estimates for the Keller-Segel-Navier-Stokes system in Besov spaces

We show the local well-posedness for the Keller-Segel-Navier-Stokes system with initial data in the scaling invariant Besov spaces, where the solution exists globally in time if the initial data is sufficiently small. We also reveal that the solution belongs to the Lorentz spaces in time direction, while the solution is smooth in space and time. Moreover, we obtain the maximal regularity estimates of solutions under the certain conditions. We further show that the solution has the additional regularities if the initial data has higher regularities. This result implies that global solutions decay as the limit t→∞ in the same norm of the space of the initial data. Our results on the Lorentz regularity estimates are based on the strategy by Kozono-Shimizu (2019) [26].

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.

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.

On Maximal Regularity Estimates for Discontinuous Galerkin Time-Discrete Methods

On Maximal Regularity Estimates for Discontinuous Galerkin Time-Discrete Methods

An $$L_p$$-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations

We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: $$\begin{aligned} du= \left( {\bar{a}}^{ij}(\omega ,t)u_{x^ix^j}+ f \right) dt + g^k dw^k_t, \quad t \in (0,T); \quad u(0,\cdot )=0, \end{aligned}$$ (1) where \(T \in (0,\infty )\), \(w^k\) \((k=1,2,\ldots )\) are independent Wiener processes, \(({\bar{a}}^{ij}(\omega ,t))\) is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$\begin{aligned} \kappa |\xi |^2 \le {\bar{a}}^{ij}(\omega ,t) \xi ^i \xi ^j \le K |\xi |^2 \quad \forall \;(\omega ,t,\xi ) \in \Omega \times (0,T) \times {\mathbf {R}}^d \end{aligned}$$for some \(\kappa , K \in (0,\infty )\), $$\begin{aligned} f \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP) \right) , \end{aligned}$$and $$\begin{aligned} g, g_x \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP; l_2) \right) \end{aligned}$$with \(2 \le r \le p < \infty \) and appropriate measurable conditions. Moreover, for the solution u, we obtain the following maximal regularity moment estimate $$\begin{aligned}&\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u_{xx}(t,x)|^r\right] \right) ^{p/r} dx dt \nonumber \\&\le N \bigg (\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |f(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \nonumber \\&\quad + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g_x(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \bigg ), \end{aligned}$$ (2) where N is a positive constant depending only on d, p, r, \(\kappa \), K, and T. As an application, for the solution u to (1), the rth moment \(m^r(t,x):=\mathbb {E}|u(t,x)|^r\) is in the parabolic Sobolev space \(W_{p/r}^{1,2}\left( (0,T) \times \mathbf {R}^d\right) \).