We consider autonomous and non-autonomous evolution equations on a time interval $[0,τ]$ in a Banach space $X$ with the non-standard time-boundary condition $u(0)=Φu(τ)$, where $Φ$ is a linear map on $X$. If $Φ=0$, this is an initial value problem, whereas $Φ=I$ corresponds to periodic boundary conditions, and $Φ=-I$ to antiperiodic boundary conditions. Our main point is to establish maximal $L^p$-regularity. In the non-autonomous case we consider two situations. The first concerns time-dependent operators with a fixed domain. In the second one we take $X=H$ a Hilbert space and consider evolution equations associated with non-autonomous forms. Of special interest is then maximal regularity in $H$ with a non-standard time-boundary condition.
We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common Hölder assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q(Ω)$ we obtain maximal $L^p$-regularity for $p \ge 2$ and elliptic operators in divergence form with uniform $VMO$-modulus in space and $W^{α,p}$-regularity for $α> \frac{1}{2}$ in time.
We show maximal $L^p$-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 $L^2$, we obtain for all $p \in (1,2]$ maximal $L^p$-regularity for non-autonomous elliptic operators with measurable coefficients.
We consider the problem of maximal regularity for non-autonomous Cauchy problems u ' (t) + A(t) u(t) = f (t), t $\in$ (0, $τ$ ] u(0) = u 0. The time dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space H. We are interested in J.L. Lions's problem concerning maximal regularity of such equations. We give a positive answer to this problem under minimal regularity assumptions on the forms. Our main assumption is that the forms are piecewise H 1 2 with respect to the variable t. This regularity assumption is optimal and our results are the most general…
This is a survey on recent progress concerning maximal regularity of non-autonomous equations governed by time-dependent forms on a Hilbert space. It also contains two new results showing the limits of the theory.
We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.
By Baillon's result, it is known that maximal regularity with respect to the space of continuous functions is rare; it implies that either the involved semigroup generator is a bounded operator or the considered space contains $c_{0}$. We show that the latter alternative can be excluded under a refined condition resembling maximal regularity with respect to $\mathrm{L}^{\infty}$.
An old problem due to J.-L. Lions going back to the 1960s asks whether the abstract Cauchy problem associated to non-autonomous forms has maximal regularity if the time dependence is merely assumed to be continuous or even measurable. We give a negative answer to this question and discuss the minimal regularity needed for positive results.
We consider the problem of maximal regularity for non-autonomous Cauchy problems $u'(t) + B(t)A(t)u(t) + P(t)u(t) = f(t), u(0) = u_0$ and $u'(t) + A(t)B(t)u(t) + P(t)u(t) = f (t), u(0) = u_0$. In both cases, the time dependent operators $A(t)$ are associated with a family of sesquilinear forms and the multiplicative left or right perturbations $B(t)$ as well as the additive perturbation $P(t)$ are families of bounded operators on the considered Hilbert space. We prove maximal $L_p$-regularity results and other regularity properties for the solutions of the previous problems under minimal regul…
In this paper we prove maximal $L^p$-regularity for a system of parabolic PDEs, where the elliptic operator $A$ has coefficients which depend on time in a measurable way and are continuous in the space variable. The proof is based on operator-theoretic methods and one of the main ingredients in the proof is the construction of an evolution family on weighted $L^q$-spaces.
We establish the existence and the uniqueness for the Boussinesq system in the whole 3D space in the critical space of continuous in time with values in the power 3 integrable in space functions for the velocity and square integrable in time with values in the power 3/2 integrable in space.
In this paper, we prove global-in-time $\dot{\mathrm{H}}^{α,q}$-maximal regularity for a class of injective, but not invertible, sectorial operators on a UMD Banach space X , provided $q\in(1,+\infty) , $α\in(-1+1/q,1/q)$. We also prove the corresponding trace estimate, so that the solution to the canonical abstract Cauchy problem is continuous with values in a not necessarily complete trace space.In order to put our result in perspective, we also provide a short review on L q-maximal regularity which includes some recent advances such as the revisited homogeneous operator and interpolation th…
Maximal regularity for time-fractional Schrödinger equations and application to nonlinear equations
Cadi Ayyad University（卡迪·阿亚德大学）；Faculty of Sciences Semlalia, Laboratory of Mathematics, Modeling and Automatic Systems（塞姆拉利亚理学院，数学、建模与自动系统实验室）；The Vanguard Center, University Mohammed VI Polytechnic（穆罕默德六世理工大学先锋中心）；Faculty of Sciences-Agadir, Lab-SIV, Ibn Zohr University（阿加迪尔理学院，SIV实验室，伊本·佐赫尔大学）；Department of Mathematics, Graduate School of Mathematical Sciences, The University of Tokyo（东京大学数理科学研究科数学系）；Department of Mathematics, Faculty of Science, Zonguldak Bulent Ecevit University（宗古尔达克布伦特·埃切维特大学理学院数学系）
End-point maximal $L^1$-regularity for parabolic initial-boundary value problems is considered. For the inhomogeneous Dirichlet and Neumann data, maximal $L^1$-regularity for initial-boundary value problems is established in time end-point case upon the homogeneous Besov space $\dot B_{p,1}^s({\mathbb R}^n_+)$ with $1< p< \infty$ and $-1+1/p<s\le 0$ as well as optimal trace estimates. The main estimates obtained here are sharp in the sense of trace estimates and it is not available by known theory on the class of UMD Banach spaces. We utilize a method of harmonic analysis, in particular, the a…
We consider the Cauchy problem for non-autonomous forms inducing elliptic operators in divergence form with Dirichlet, Neumann, or mixed boundary conditions on an open subset $Ω$ $\subseteq$ R n. We obtain maximal regularity in L 2 ($Ω$) if the coefficients are bounded, uniformly elliptic, and satisfy a scale invariant bound on their fractional time-derivative of order one-half. Previous results even for such forms required control on a time-derivative of order larger than one-half.
Linear reaction-diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal Lp regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain transversally.
We prove two maximal regularity results in spaces of continuous and Hölder continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^α$. This derivative is intended in the sense of Caputo and $α$ is taken in $(0, 2)$. In case $α= 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.
For $q>2, γ> 1$, we prove that maximal regularity of $L^q$ type holds for periodic solutions to $-Δu + |Du|^γ= f$ in $\mathbb{R}^d$, under the (sharp) assumption $q > d \frac{γ-1}γ$.
The issue of so-called maximal regularity is discussed within a Hilbert space framework for a class of evolutionary equations. Viewing evolutionary equations as a sums of two unbounded operators, showing maximal regularity amounts to establishing that the operator sum considered with its natural domain is already closed. For this we use structural constraints of the coefficients rather than semi-group strategies or sesqui-linear form methods, which would be difficult to come by for our general problem class. Our approach, although limited to the Hilbert space case, complements known strategies…
In this paper we consider maximal regularity for the vector-valued quasi-steady linear elliptic problems. The equations are the elliptic equation in the domain and the evolution equations on its boundary. We prove the maximal $L_p$-$L_q$ regularity for these problems and give examples that our results are applicable. The Lopatinskii--Shapiro and the asymptotic Lopatinskii--Shapiro conditions are important to get boundedness of solution operators.
Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of ${\mathcal G}^\infty$-regularity, which reduces to $C^\infty$-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of ${\mathcal G}^\infty$-regularity does not hold.
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.
Maximal $L^p$-$L^q$ regularity is proved for the strong, weak and very weak solutions of the inhomogeneous Stokes problem with Navier-type boundary conditions in a bounded domain $Ω$, not necessarily simply connected. This extends previous results of the authors (2017).
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 li…
In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q regularity with differentiability, i.e. the main theorem is isomorphism between the solution space and the data space using Besov and Triebel--Lizorkin spaces. The key is compatibility conditions for the initial data. We are able to get a unique smooth solution if the data satisfying compatibility conditions are smooth.
We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the $L_1$ Carleson functional and $L_q$ ($1<q<\infty$) 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.
In this paper, we show mathematical justification of the data assimilation of nudging type in $L^p$-$L^q$ maximal regularity settings. We prove that the approximate solution of the primitive equations by data assimilation converges to the true solution with exponential order on the Besov space $B^{2/q}_{q,p}(Ω)$ in the periodic layer domain $Ω= \mathbb{T}^2 \times (-h, 0)$.
We consider non-autonomous evolutionary problems of the form $u'(t)+A(t)u(t)=f(t)$, $u(0)=u_0,$ on $L^2([0,T];H)$, where $H$ is a Hilbert space. We do not assume that the domain of the operator $A(t)$ is constant in time $t$, but that $A(t)$ is associated with a sesquilinear form $a(t)$. Under sufficient time regularity of the forms $a(t)$ we prove well-posedness with maximal regularity in $L^2([0,T];H)$. Our regularity assumption is significantly weaker than those from previous results inasmuch as we only require a fractional Sobolev regularity with arbitrary small Sobolev index.
We review some results on abstract linear and nonlinear population models with age and spatial structure. The results are mainly based on the assumption of maximal $L_p$-regularity of the spatial dispersion term. In particular, this property allows us to characterize completely the generator of the underlying linear semigroup and to give a simple proof of asynchronous exponential growth of the semigroup. Moreover, maximal regularity is also a powerful tool in order to establish the existence of nontrivial positive equilibrium solutions to nonlinear equations by fixed point arguments or bifurca…
In this paper we investigate maximal $L^q$-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on a parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton-Jacobi equations, recently addressed in the stationary framework by the authors. Finally,…
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 $N\geq 2$ when the gravity is not taken into account. The aim of this paper is to show the global solvability of the Naiver-Stokes equations with a free surface, describing the above-mentioned motion, in the maximal $L_p\text{-}L_q$ regularity class. Our approach is based on the maximal $L_p\text{-}L_q$ regularity with exponential stability for the linearized equations, and solutions to the original nonlinear problem are also exponentia…
