Maximal Regularity Research Articles

Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface

In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the $L_p$ in time $L_q$ in space maximal regularity class, ($2 < p < ∞$, $N < q < ∞$, and $2/p + N/q < 1$), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the $L_p$-$L_q$ framework for the linearized equations with the help of maximal $L_p$-$L_q$ regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

Mixed problems for degenerate abstract parabolic equations and applications

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

Numerical Analysis of Nonlinear Subdiffusion Equations

We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three technical tools: a fractional version of the discrete Gr\"onwall-type inequality, discrete maximal regularity, and regularity theory of nonlinear equations. We establish a general criterion for showing the fractional discrete Gr\"onwall inequality, and verify it for the L1 scheme and convolution quadrature generated by BDFs. Further, we provide a complete solution theory, e.g., existence, uniqueness and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term. Together with the known results of discrete maximal regularity, we derive pointwise $L^2(\Omega)$ norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order $O(h^2)$ (up to a logarithmic factor) and $O(\tau^\alpha)$, respectively, without any extra regularity assumption on the solution or compatibility condition on the problem data. The sharpness of the convergence rates is supported by the numerical experiments.

A triangular nonlinear reaction‐fractional diffusion system with a balance law

The aim of this paper is to establish a global existence result for a nonlinear reaction diffusion system with fractional Laplacians of different orders and a balance law. Our method of proof is based on a duality argument and a recent maximal regularity result due to Zhang.

Discrete maximal regularity and the finite element method for parabolic equations

Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for a parabolic equation. We derive various stability results in $L^p(0,T;L^q(\Omega))$ norm, $p,q\in (1,\infty)$ for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of G.~Dore and A.~Venni (On the closedness of the sum of two closed operators. \emph{Math.\ Z.}, 196(2):189--201, 1987). As an application, optimal order error estimates in that norm are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinearity and offer a new method for deriving optimal order error estimates. Some interesting auxiliary results including discrete Gagliardo-Nirenberg and Sobolev inequalities are also presented.

Maximal regularity for non-autonomous stochastic evolution equations in UMD Banach spaces

A non-autonomous stochastic linear evolution equation in UMD Banach spaces of type 2 is considered. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to a stochastic partial differential equation.

Navier–Stokes equations with external forces in Lorentz spaces and its application to the self-similar solutions

We show existence theorem of global mild solutions with small initial data and external forces in Lorentz spaces with scaling invariant norms. If the initial data have more regularity in another scaling invariant class, then our mild solution is actually the strong solution. The result on local existence of solutions for large data is also discussed. Our method is based on the maximal regularity theorem on the Stokes equations in Lorentz spaces. Then we apply our theorem to prove existence of self-similar solutions provided both initial data and external forces are homogeneous functions. Since we construct the global solution by means of the implicit function theorem, as a byproduct, its stability with respect to the given data is necessarily obtained.

Global solvability of the Navier–Stokes equations with a free surface in the maximal Lp-Lq regularity class

We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the N-dimensional Euclidean space for N≥2. The aim of this paper is to show the global solvability of the Navier–Stokes equations with a free surface, describing the above-mentioned motion, in the maximal Lp-Lq regularity class. Our approach is based on the maximal Lp-Lq regularity with exponential stability for the linearized equations, and also it is proved that solutions to the original nonlinear problem are exponentially stable.

Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains

We investigate the time-periodic Stokes equations with non-homogeneous divergence data in the whole space, the half space, bent half spaces and bounded domains. The solutions decompose into a well-studied stationary part and a purely periodic part, for which we establish $\mathrm{L}^{p}$ estimates. For the whole space and the half space case we use a reduction of the Stokes equations to $(n-1)$ heat equations. Perturbation and localisation methods yield the result on bent half spaces and bounded domains. A one-to-one correspondence between maximal regularity for the initial value problem and time periodic maximal regularity is proven, providing a short proof for the maximal regularity of the Stokes operator avoiding the notion of $\mathcal{R}$-boundedness. The results are applied to a quasilinear model governing the flow of nematic liquid crystals.

Nonlinear Calderón–Zygmund Theory in the Limiting Case

We prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Calderon and Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions. A prototype of the results obtained here tells for instance that if $$-{\rm div} \, (|Du|^{p-2}Du)=\mu \quad \mbox{in} \ \Omega\subset\mathbb{R}^n,$$ with $${\mu}$$ being a Borel measure with locally finite mass on the open subset $${\Omega\subset \mathbb{R}^n}$$ and $${p > 2-1/n}$$ , then $$|Du|^{p-2}Du \in W^{\sigma, 1}_{\rm{loc}}(\Omega)\quad \mbox{for \, every} \ \sigma \in (0,1).$$ The case $${\sigma=1}$$ is obviously forbidden already in the classical linear case of the Poisson equation $${-\triangle u=\mu}$$ .

Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: The convex subset case

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ. The associated Cameron–Martin space is denoted by H. Consider two sufficiently regular convex functions U:X→R and G:X→R. We let ν=e−Uμ and Ω=G−1(−∞,0]. In this paper we are interested in the W2,2 regularity of the weak solutions of elliptic equations of the type(0.1)λu−Lν,Ωu=f, where λ>0, f∈L2(Ω,ν) and Lν,Ω is the self-adjoint operator associated with the quadratic form(ψ,φ)↦∫Ω〈∇Hψ,∇Hφ〉Hdνψ,φ∈W1,2(Ω,ν). In addition we will show that if u is a weak solution of problem (0.1) then it satisfies a Neumann type condition at the boundary, namely for ρ-a.e. x∈G−1(0)〈Tr(∇Hu)(x),Tr(∇HG)(x)〉H=0, where ρ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure and Tr is the trace operator.

Boundedness in a quasilinear fully parabolic two-species chemotaxis system of higher dimension

This paper considers the following coupled chemotaxis system: {ut=∇⋅(ϕ(u)∇u)−χ1∇⋅(u∇w)+μ1u(1−u−a1v),vt=∇⋅(ψ(v)∇v)−χ2∇⋅(v∇w)+μ2v(1−a2u−v),wt=Δw−γw+αu+βv,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} u_{t}=\\nabla\\cdot(\\phi(u)\\nabla u)-\\chi_{1} \\nabla\\cdot(u\\nabla w)+\\mu_{1} u(1-u-a_{1} v), \\\\ v_{t}=\\nabla\\cdot(\\psi(v)\\nabla v)-\\chi_{2} \\nabla\\cdot(v\\nabla w)+\\mu_{2} v(1-a_{2}u-v), \\\\ w_{t}=\\Delta w-\\gamma w+\\alpha u+\\beta v, \\end{cases} $$\\end{document} with homogeneous Neumann boundary conditions in a bounded domain Omegasubsetmathbb{R}^{N} (Nge3) with smooth boundaries, where chi_{1}, chi_{2}, mu_{1}, mu_{2}, a_{1}, a_{2}, α, β and γ are positive. Based on the maximal Sobolev regularity, the existence of a unique global bounded classical solution of the problem is established under the assumption that both mu_{1} and mu_{2} are sufficiently large.

Runge–Kutta Time Discretization of Nonlinear Parabolic Equations Studied via Discrete Maximal Parabolic Regularity

For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge–Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the $$W^{1,\infty }$$ norm uniformly on bounded time intervals and, with an improved approximation order, in the parabolic energy norm. The proofs rely on discrete maximal parabolic regularity. This is used to obtain $$W^{1,\infty }$$ estimates, which are the key to the numerical analysis of these problems.

On a Navier–Stokes–Ohm problem from plasma physics

A model from electro-magneto-hydrodynamics describing a completely ionized gas, a plasma, is studied. Local well posedness of the problem in time-weighted $$L_p$$ spaces is obtained by means of maximal regularity of the linearized problem, and the induced local semiflow in the proper state space is constructed. If the underlying domain is bounded and simply connected, based on the principle of linearized stability, it is shown that the trivial solution of the problem is exponentially stable. Moreover, if the orbit of a solution is relatively compact in the state space, the solution exists globally, and it converges to the trivial equilibrium.

Discrete maximal regularity of time-stepping schemes for fractional evolution equations

In this work, we establish the maximal ell ^p-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order alpha in (0,2), alpha ne 1, in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank–Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis (Math Ann 319:735–758, 2001. doi:10.1007/PL00004457) and its discrete analogue due to Blunck (Stud Math 146:157–176, 2001. doi:10.4064/sm146-2-3). These results generalize the corresponding results for parabolic problems.

Conical maximal regularity for elliptic operators via Hardy spaces

Conical maximal regularity for elliptic operators via Hardy spaces

Global solution to the nematic liquid crystal flows with heat effect

The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain Ω⊂RN (N=2,3) are studied in this paper. Following Danchin's method in [7], we use a localization argument to recover the maximal regularity of Stokes equation with variable viscosity, by which we first prove the local existence of a unique strong solution, then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state. This paper also generalizes Hu–Wang's result in [21] to the non-isothermal case.

Global solutions to random 3D vorticity equations for small initial data

One proves the existence and uniqueness in (Lp(R3))3, 32<p<2, of a global mild solution to random vorticity equations associated to stochastic 3D Navier–Stokes equations with linear multiplicative Gaussian noise of convolution type, for sufficiently small initial vorticity. This resembles some earlier deterministic results of T. Kato [16] and are obtained by treating the equation in vorticity form and reducing the latter to a random nonlinear parabolic equation. The solution has maximal regularity in the spatial variables and is weakly continuous in (L3∩L3p4p−6)3 with respect to the time variable. Furthermore, we obtain the pathwise continuous dependence of solutions with respect to the initial data. In particular, one gets a locally unique solution of 3D stochastic Navier–Stokes equation in vorticity form up to some explosion stopping time τ adapted to the Brownian motion.

Domain variations and moving boundary problems

In the past few decades maximal regularity theory has successfully been applied to moving boundary problems. The basic idea is to reduce the system with varying domains to one in a fixed domain. This is done by a transformation (the so-called Hanzawa transformation) and yields a typically nonlocal and nonlinear coupled system of (evolution) equations. Well-posedness results can then often be established as soon as it is proved that the relevant linearization is the generator of an analytic semigroup or admits maximal regularity. To implement this program, it is necessary to somehow parametrize to space of boundaries/domains (typically the space of compact hypersurfaces $\Gamma$ in $\mathbb{R}^n$, in the Euclidean setting). This has traditionally been achieved by means of the already mentioned Hanzawa transformation. The approach, while successful, requires the introduction of a smooth manifold $\Gamma_\infty$ close to the manifold $\Gamma_0$ in which one cares to linearize. This prevents one to use coordinates in which $\Gamma_0$ lies at their "center". As a result formul{\ae} tend to contain terms that would otherwise not be present were one able to linearize in a neighborhood emanating from $\Gamma_0$ instead of from $\Gamma_\infty$. In this paper it is made use of flows (curves of diffeomorphisms) to obtain a general form of the relevant linearization in combination with an alternative coordinatization of the manifold of hypersurfaces, which circumvents the need for the introduction of a "phantom" reference manifold $\Gamma_\infty$ by, in its place, making use of a "phantom geometry" on $\Gamma_0$. The upshot is a clear insight into the structure of the linearization, simplified calculations, and simpler formul{\ae} for the resulting linear operators, which are useful in applications.

Maximal regularity in lp spaces for discrete time fractional shifted equations

In this paper, we are presenting a new method based on operator-valued Fourier multipliers to characterize the existence and uniqueness of ℓp-solutions for discrete time fractional models in the formΔαu(n,x)=Au(n,x)+∑j=1kβju(n−τj,x)+f(n,u(n,x)),n∈Z,x∈Ω⊂RN,βj∈Randτj∈Z,where A is a closed linear operator defined on a Banach space X and Δα denotes the Grünwald–Letnikov fractional derivative of order α>0. If X is a UMD space, we provide this characterization only in terms of the R-boundedness of the operator-valued symbol associated to the abstract model. To illustrate our results, we derive new qualitative properties of nonlinear difference equations with shiftings, including fractional versions of the logistic and Nagumo equations.