Maximal Lp-regularity Research Articles

Maximal Lp-regularity and H∞-calculus for block operator matrices and applications

Many coupled evolution equations can be described via 2×2-block operator matrices of the form A=[ABCD] in a product space X=X1×X2 with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator A can be seen as a relatively bounded perturbation of its diagonal part with D(A)=D(A)×D(D) though with possibly large relative bound. For such operators the properties of sectoriality, R-sectoriality and the boundedness of the H∞-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with A can be analyzed in maximal Ltp-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.

Periodic solutions for the Blackstock–Crighton–Westervelt equation

We investigate the Blackstock–Crighton–Westervelt equation which models nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We prove existence and regularity, in a Lp−Lq setting, of time-periodic solutions for a given sufficiently small time-periodic forcing data, and homogeneous Dirichlet boundary conditions over a cylindrical domain. We show maximal Lp-regularity for the abstract linearized model. We use techniques of operator-valued Fourier multiplier theorems combined with a generalized version of the implicit function theorem.

Well-posedness for magnetoviscoelastic fluids in 3D

We show that the system of equations describing a magnetoviscoelastic fluid in three dimensions can be cast as a quasilinear parabolic system. Using the theory of maximal Lp-regularity, we establish existence and uniqueness of local strong solutions and we show that each solution is smooth (in fact analytic) in space and time. Moreover, we give a complete characterization of the set of equilibria and show that solutions that start out close to a constant equilibrium exist globally and converge to a (possibly different) constant equilibrium. Finally, we show that every solution that is eventually bounded in the topology of the state space exists globally and converges to the set of equilibria.

Kinetic maximal Lp-regularity with temporal weights and application to quasilinear kinetic diffusion equations

We introduce the concept of kinetic maximal Lp-regularity with temporal weights and prove that this property is satisfied for the (fractional) Kolmogorov equation. We show that solutions are continuous with values in the trace space and prove, in particular, that the trace space can be characterized in terms of anisotropic Besov spaces. We further extend the property of kinetic maximal Lμp-regularity to the Kolmogorov equation with variable coefficients. Finally, we show how kinetic maximal Lμp-regularity can be used to obtain local existence of solutions to a class of quasilinear kinetic equations and illustrate our result with a quasilinear kinetic diffusion equation.

Error analysis of a fully discrete finite element method for variable density incompressible flows in two dimensions

An error estimate is presented for a fully discrete, linearized and stabilized finite element method for solving the coupled system of nonlinear hyperbolic and parabolic equations describing incompressible flow with variable density in a two-dimensional convex polygon. In particular, the error of the numerical solution is split into the temporal and spatial components, separately. The temporal error is estimated by applying discrete maximal Lp-regularity of time-dependent Stokes equations, and the spatial error is estimated by using energy techniques based on the uniform regularity of the solutions given by semi-discretization in time.

Strong time periodic solutions to Keller-Segel systems: An approach by the quasilinear Arendt-Bu theorem

It is shown that the classical as well as quasilinear Keller-Segel systems with non-degenerate diffusion possess for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution. The proof given relies on the existence of strong T-periodic solutions for the linearized system, its characterization in terms of maximal Lp-regularity of the underlying operator and a quasilinear version of the Arendt-Bu Theorem. The latter is of independent interest and yields the existence of strong T-periodic solutions to general quasilinear evolution equations under suitable conditions on the operators and the forcing terms.

Stability properties of stochastic maximal Lp-regularity

In this paper we consider Lp-regularity estimates for solutions to stochastic evolution equations, which is called stochastic maximal Lp-regularity. Our aim is to find a theory which is analogously to Dore's theory for deterministic evolution equations. He has shown that maximal Lp-regularity is independent of the length of the time interval, implies analyticity and exponential stability of the semigroup, is stable under perturbation and many more properties. We show that the stochastic versions of these results hold.

Maximal ℓp-regularity for discrete time Volterra equations with delay

ABSTRACT In this paper, we investigate the existence and uniqueness of solutions belonging to the vector-valued space by using Blunck's theorem on the equivalence between operator-valued -multipliers and the notion of R-boundedness for the discrete time Volterra equation with delay given by where A is a closed linear operator with domain defined on a Banach space X, and verifies suitable conditions such as 1-regularity. We characterize maximal -regularity of solutions of such problems in terms of the data and an spectral condition, and we provide optimal estimates. Moreover, we illustrate our results providing different models that label into our general scheme such as the discrete time wave and Kuznetsov equations.

Nonautonomous maximal Lp-regularity under fractional Sobolev regularity in time

We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common H\"older assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q(\Omega)$ we obtain maximal $L^p$-regularity for $p \ge 2$ and elliptic operators in divergence form with uniform $VMO$-modulus in space and $W^{\alpha,p}$-regularity for $\alpha > \frac{1}{2}$ in time.

Quasilinear Evolution Equations in LμP-Spaces with Lower Regular Initial Data

We study the Cauchy problem of the quasilinear evolution equations in Lμp-spaces. Based on the theories of maximal Lp-regularity of sectorial operators, interpolation spaces, and time-weighted Lp-spaces, we establish the local posedness for a class of abstract quasilinear evolution equations with lower regular initial data. To illustrate our results, we also deal with the second-order parabolic equations and the Navier-Stokes equations in Lp,q-spaces with temporal weights.

Critical spaces for quasilinear parabolic evolution equations and applications

We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal Lp-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier–Stokes problem, convection–diffusion equations, the Nernst–Planck–Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.

Non-autonomous maximal regularity for forms given by elliptic operators of bounded variation

We show maximal Lp-regularity for non-autonomous Cauchy problems provided the trace spaces are stable in some parameterized sense and the time dependence is of bounded variation. In particular on L2(Ω), for Lipschitz domains Ω and under mixed boundary conditions, we obtain maximal Lp-regularity for all p∈(1,2] for elliptic operators with coefficients aij:Ω→C satisfying aij(⋅,x)∈BV uniformly in x∈Ω.

On the quasi-geostrophic equations on compact closed surfaces in [formula omitted

The quasi-geostrophic equations on compact surfaces in R3 without boundary are studied by means of the theory of quasi-linear parabolic evolution equations based on maximal Lp-regularity. It is shown that the problem is globally strongly well-posed in Lq, the solutions regularize instantly and form a global semiflow in the proper state manifold, and, as time goes to infinity, a solution converges exponentially to a constant in the topology of the state space.

A method for obtaining time-periodic Lp estimates

We introduce a method for showing a prioriLp estimates for time-periodic, linear, partial differential equations set in a variety of domains such as the whole space, the half space and bounded domains. The method is generic and can be applied to a wide range of problems. We demonstrate it on the heat equation. The main idea is to replace the time axis with a torus in order to reformulate the problem on a locally compact abelian group and to employ Fourier analysis on this group. As a by-product, maximal Lp regularity for the corresponding initial-value problem follows without the notion of R-boundedness. Moreover, we introduce the concept of a time-periodic fundamental solution.

Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time

In this paper we study maximal Lp-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the Lp-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an Lp(Lq)-theory for such equations for p,qin (1, infty ). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.

Existence and maximal Lp-regularity of solutions for the porous medium equation on manifolds with conical singularities

ABSTRACTWe consider the porous medium equation on manifolds with conical singularities and show existence, uniqueness, and maximal Lp-regularity of a short-time solution. In particular, we obtain information on the short time asymptotics of the solution near the conical point. Our method is based on bounded imaginary powers results for cone differential operators on Mellin–Sobolev spaces and R-sectoriality perturbation techniques.

Maximal regularity of the spatially periodic stokes operator and application to nematic liquid crystal flows

We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal Lp-regularity of the periodic Laplace and Stokes operators and a local-intime existence theorem for quasilinear parabolic equations a la Clement-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group \(G: = \mathbb{R}^{n - 1} \times \mathbb{R}/L\mathbb{Z}\) to obtain an R-bound for the resolvent estimate. Then, Weis’ theorem connecting R-boundedness of the resolvent with maximal Lp regularity of a sectorial operator applies.

An Extrapolation Theorem in Non-Euclidean Geometries and its Application to Partial Differential Equations

We prove a generalization of an extrapolation theorem in the fashion of Garcia-Cuerva and Rubio de Francia towards R-boundedness on weighted Lebesgue spaces over locally compact abelian groups. This result can be applied to show maximal Lp regularity for differential operators that correspond to parabolic evolution equations subject to more general spatial geometries, for example the partially periodic Stokes operator. As a main tool, we generalize the classical Muckenhoupt theorem on maximal operators to locally compact abelian groups.

A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

In this paper we consider scalar parabolic equations in a general non-smooth setting emphasizing interface conditions and mixed boundary conditions. In particular, we study dynamics and diffusion on a Lipschitz interface and on the boundary, where the diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we consider diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal Lp-regularity and bounded H∞-calculus for the corresponding operator, providing well-posedness for a large class of initial conditions and external forces.

UMD Banach spaces and the maximal regularity for the square root of several operators

In this paper we prove that the maximal Lp-regularity property on the interval (0,T), T>0, for Cauchy problems associated with the square root of Hermite, Bessel or Laguerre type operators on L2(Ω,dμ;X), characterizes the UMD property for the Banach space X.