Linear and quasilinear evolution equations in the context of weighted L p -spaces

. In 2004, the article Maximal regularity for evolution equations in weighted L p -spaces by J. Pr¨uss and G. Simonett [52] has been published in Archiv der Mathematik . We provide a survey of the main results of that article and outline some applications to semilinear and quasilinear parabolic evolution equations which illustrate their power. Mathematics


Introduction
In this survey article, we review regularity results for abstract linear parabolic evolution equations of the form u(t) + Au(t) = f (t), t > 0, u(0) = u 0 , (1.1) in the framework of time-weighted L p -spaces 1 < p < ∞, that emanated from the groundbreaking paper [52].Here X is a Banach space, A is a closed linear operator in X with dense domain D(A) and the data (f, u 0 ) are given.In [52], the concept of L p,µ -maximal regularity for the operator A has been introduced, see Definition 2.2 for details.One of the fundamental results of the article [52] states that this concept is independent of the parameter µ as long as µ > 1/p.Note that in case µ = 1 one ends up in the classical (unweighted) L p -maximal regularity class, see e.g.[17,55,60] which is just a selection.We will furthermore consider quasilinear parabolic evolution equations u(t) + A(u(t))u(t) = F (u(t)), t > 0, u(0 and apply the linear theory for (1.1) from [52], combined with the contraction mapping principle, which yields the existence and uniqueness of a local-intime solution of (1.2).Moreover, the advantage of working in time-weighted L p -spaces is underlined by some results on parabolic regularization and the global-in-time existence for solutions to (1.2).In Sections 3.3 & 3.4 of this survey article, we consider nonlinearities with a certain growth and so-called critical spaces.
In the last section we list some selected references which have been influenced by the article [52].
Finally, let us mention that for example in [4,6,14], weighted continuous function spaces of the form [t → t 1−µ f (t)] ∈ BU C((0, T ]; X), lim t→0+ t 1−µ f (t) X = 0 have been studied earlier.It is shown that this class allows for maximal regularity results if the Banach space X is replaced by a continuous interpolation space (X, D(A)) 0 θ,∞ of order θ ∈ (0, 1).

Linear evolution equations
In this section we review and discuss the setting and some of the main results of the article [52].
Proof.The proof of the first two assertions can be found in [52, Lemma 2.1] while for a proof of the last assertion, we refer to [53, Lemma 3.4.5].

Maximal regularity
Let X 0 , X 1 be Banach spaces such that X 1 ֒→ X 0 and X 1 is dense in X 0 .Suppose that A : X 1 → X 0 is a linear and closed operator.We consider the abstract evolution equation Following the lines of [52], we have the following Definition 2.2.Let 1 < p < ∞ and µ ∈ (1/p, 1].The operator A has the property of maximal L p,µ -regularity in X 0 if for each f ∈ L p,µ (R + ; X 0 ) there exists a unique solution 2).If this is the case, we write for short A ∈ MR p,µ (X 0 ) and The following important and fundamental theorem states that the concept of maximal L p,µ -regularity does not depend on µ ∈ (1/p, 1].[52]).For all 1 < p < ∞ and µ ∈ (1/p, 1] the following assertions are equivalent:
Remark 2.5.If A ∈ MR p (X 0 ), then −A is the generator of an exponentially stable analytic semigroup in X 0 , see e.g.[16,Theorem 2.2], [51,Proposition 1.2] or [53,Proposition 3.5.2].A characterization of A ∈ MR p (X 0 ) has been given by Weis in [60].It is based on the concept of R-boundedness in case X 0 is additionally of class UMD.
We proceed with a selection of examples for operators belonging to the class MR p (X 0 ).
• If X 0 is a Hilbert space and −A generates an exponentially stable analytic semigroup in X 0 , then A ∈ MR p (X 0 ) ([53, Theorem 3.5.7]).
• If X 0 is a real interpolation space, and −A generates an exponentially stable analytic semigroup in X 0 , then A ∈ MR p (X 0 ) ([53, Theorem 3.5.8]).

Trace spaces
We turn our attention to the abstract Cauchy problem (1.1) and ask under which assumptions on the initial value u 0 , there exists a unique solution u ∈ W 1 p,µ (R + ; X 0 ) ∩ L p,µ (R + ; X 1 ) of (1.1), provided A ∈ MR p (X 0 ) and f ∈ L p,µ (R + ; X 0 ).By the second assertion in Proposition 2.1, the trace operator tr : ).The following characterization of the trace space tr E 1,µ holds.Proposition 2.6.For all 1 < p < ∞ and all µ ∈ (1/p, 1] it holds that tr E 1,µ = (X 0 , X 1 ) µ−1/p,p (up to equivalent norms), where (X 0 , X 1 ) µ−1/p,p is the real interpolation space between X 0 and X 1 of exponent µ − 1/p.Moreover, the embedding Here BU C stands for the bounded and uniformly continuous functions.
Proof.[52,Proposition 3.1] With the help of this proposition, one can prove the following there exists a unique solution 1).Moreover, there exists a constant C > 0 being independent of (f, u 0 ) such that the estimate holds.
Proof.For the proof of the first assertion, see [52,Theorem 3.2].Concerning the estimate (2.3), note that by Proposition 2.6.Hence (2.3) follows from the first assertion and the open mapping theorem.
Remark 2.8.Theorem 2.7 asserts in particular, that the regularity of the initial value can be reduced by decreasing the exponent µ ∈ (1/p, 1] of the time-weight.This in turn implies that the number of compatibility conditions in the context of initial-boundary value problems for parabolic partial differential equations may be reduced to a minimum.

Local well-posedness and regularization
In this section, we consider quasilinear evolution equations of the form We are looking for solutions in the maximal regularity class is open and nonempty, C 1− stands for the locally Lipschitz continuous functions and B(X 1 , X 0 ) denotes the space of all bounded and linear operators from X 1 to X 0 .
Assuming A(u 0 ) ∈ MR p (X 0 ), by Theorem 2.7 and extension-restriction arguments, the mapping is well defined.Obviously, any fixed point of this mapping is a solution to (3.1) and vice versa.The contraction mapping principle then yields the following Then there exist T = T (u 0 ) > 0 and ε = ε(u 0 ) > 0, such that BXγ,µ (u 0 , ε) ⊂ V µ and such that problem (3.1) has a unique solution is valid.Here BXγ,µ (u 0 , ε) denotes the closed ball with with center u 0 and radius ε in the topology of X γ,µ .
Concerning the continuation of solutions in the weighted maximal regularity class, one can prove the following Corollary 3.3.Let the assumptions of Theorem 3.1 be satisfied and assume that The mapping Proof.[ Let us point out another advantage of working in the setting of weighted L p -spaces.To see the benefit, observe that for all τ, T ∈ (0, t + (u 0 )) with τ < T , the estimate for the solution u of (3.1) holds, hence ֒→ C((0, t + (u 0 )); X γ,1 ), (3.4) by Proposition 2.6.This shows that the solution u(t) of (1.2) with initial value u 0 ∈ X γ,µ = (X 0 , X 1 ) µ−1/p,p regularizes instantaneously for t ∈ (0, t + (u 0 )) provided µ < 1.

Nonlinearities with polynomial growth
The condition (3.2) is for quite general functions (A, F ) and does not account for nonlinearities having a certain growth.Consider for example the PDE (3.7) We compute (assuming that a is sufficiently regular) div(a(u)∇u) = a(u)∆u + a ′ (u)|∇u| 2 .
Therefore, we may also consider (3.7) in the form Let us compare (3.8) with the PDE For (3.8) and (3.9), we choose the setting X 0 = L q (Ω) and Then, as in Example 3.5, the trace space is computed to the result To solve (3.9) in this setting, we require X γ,µ ֒→ C(Ω), i.e. 2µ > 2/p + n/q.Note that this is possible if 2/p + n/q < 2, in order to ensure µ ∈ (1/p, 1].We turn back to (3.8).The terms a(u)∆u and f (u) are as in (3.9) while the remaining term a ′ (u)|∇u| 2 seems to induce additional conditions on the weight µ for solving (3.8).However, it is of fundamental importance to observe that the last term in (3.8) has a certain structure: it is of quadratic growth and of lower order compared to a(u)∆u.Such a class of nonlinearities with a certain growth behaviour has been considered in [40], [54] and [56].We write where F r and F s are the regular and singular part of F , respectively.In the sequel, we denote by X β = (X 0 , X 1 ) β , β ∈ (0, 1), the complex interpolation spaces.The precise assumptions on (A, F r , F s ) are as follows.
, where C denotes a constant which may depend on u i Xγ,µ .The case β j = µ − 1/p is only admissible if (H2) holds with X βj replaced by X γ,µ .

Critical spaces
Looking into the literature, there is no universally accepted definition of critical spaces.One possible definition may be based on the idea of a 'largest space of initial data such that a given PDE is well-posed.On the other hand, critical spaces are often introduced as 'scaling invariant spaces,' provided the underlying PDE enjoys a scaling.
We consider again the setting from Section 3.3.Note that the condition (H3) implies that the minimal value of µ is given by which we call the critical weight as long as µ c ∈ (1/p, 1].Theorem 3.8 shows in particular that we have local well-posedness of (3.10) for initial values in the spaces X γ,µ , provided (H1) holds for µ ∈ [µ c , 1].Therefore, it makes sense to name the space X γ,µc = (X 0 , X 1 ) µc−1/p,p the critical space for (3.10).The critical space X γ,µc enjoys the following properties.
• Generically, the critical space X γ,µc is the largest space such that the underlying evolution equation is well-posed for initial values in X γ,µc .A concrete counterexample is given in [54, Section 2.2].• If the underlying evolution equation admits a scaling, then the critical space X γ,µc is scaling invariant.This has been proven in [54, Section 2.3].A typical example is given by the Navier-Stokes equations in R n : which is invariant under the scaling (u λ (t, x), π λ (t, x)) := (λu(λ 2 t, λx), λ 2 π(λ 2 t, λx)).
Here u denotes the velocity field and π the pressure.• The critical spaces X γ,µc are invariant with respect to interpolationextrapolation scales (see e.g.[4] for the theory of those scales).This has been proven in [54,Section 2.4].Considering a PDE in a scale of function spaces gives great flexibility in choosing an appropriate setting for analyzing a given equation.
In particular, the above definition of X γ,µc encompasses the aforementioned properties of critical spaces from the literature.
Remark 3.10.Critical spaces for the Navier-Stokes equations with perfectslip as well as partial-slip boundary conditions haven been characterized in [57].Further examples in the context of critical spaces include • the Cahn-Hilliard equation • the Vorticity equation • Convection-Diffusion equations • Chemotaxis equations which can be found in [54,Sections 3 & 5].
In particular, the solution exists globally if u ∈ L p ((0, a); X µc ) for any finite number a with a ≤ t + (u 0 ).
We remind that X µc = (X 0 , X 1 ) µc denotes the complex interpolation space.Theorem 3.11 has in particular been applied in [57] and [58] to prove the global well-posedness of the Navier-Stokes equations in bounded domains of R 2 and on two-dimensional compact manifolds, respectively.

Further implications and applications of weighted L p -spaces
In this section, we give a selection of some further results and applications that were influenced by the article [52].
In [1], [45] and [48] anisotropic weighted function spaces have been studied.Several sharp embedding results as well as trace theorems have been proven therein.Those (anisotropic) function spaces are of significant importance in the theory of maximal regularity for parabolic boundary value problems as they appear e.g. as certain trace spaces.
Results on interpolation of weighted vector-valued function spaces with boundary conditions can be found in [5] and [42].Those interpolation spaces appear for instance in the computation of the trace spaces or critical spaces as soon as boundary conditions come into play.
For additional literature on function spaces we recommend [5] & [53] and the references listed therein.

Maximal regularity results
Deterministic results on optimal weighted L p -L q -regularity for parabolic boundary value problems with inhomogeneous boundary data can be found in [24], [44], [46] for the case p = q and in [36], [41] for the general case p = q.In [36], weights in the spatial variable are considered as well.We also refer to the articles [25] & [26] concerning maximal regularity results in time-weighted spaces for parabolic operators in R n having merely measurable coefficients and to the article [19] for parabolic boundary value problems in non-smooth domains.
In the probabilistic setting there has been recent progress on stochastic maximal L p -regularity in time-weighted spaces.In this context we want to mention the articles [2], [3] and [50].The first two articles can be seen as a stochastic version of [54] concerning critical spaces, while in [50], stochastic partial differential equations with VMO coefficients within the stochastic weighted L p -maximal regularity framework are considered.Finally, we want to mention the article [7] for stochastic maximal L p -regularity results in (weighted) tent spaces.For a comprehensive list of further references in the probabilistic setting, we refer to [2] and [50].
Note that by Theorem 2.3, classical (unweighted) L p -maximal regularity extrapolates to the L p,µ -spaces with power weights having a positive exponent.This has subsequently been extended to all Muckenhoupt weights in [29] and in [10], [11], [12] to more general weights.For a proof of those extrapolation results in terms of R-boundedness, we refer to [20].

Specific applications of time-weighted spaces
The aforementioned abstract or general results on weighted L p -maximal regularity have been applied to numerous concrete examples over the last years.Among the vast literature we want to mention the following references.
Within the dynamics of fluids, weighted L p -spaces and maximal regularity results have for instance been applied to • the Hibler sea ice model [9], • a system of PDEs for magnetoviscoelastic fluids [18], • incompressible and inhomogeneous fluids (with variable density) [21], • nematic liquid crystal flows via quasilinear evolution equations [34], • the phase-field Navier-Stokes equations [37], • the Navier-Stokes equations in unbounded domains with rough initial data [39], • rotating rigid bodies with a liquid-filled gap [43].
For the primitive equations of geophysical flows, which might be seen as a suitable approximation of the Navier-Stokes equations, we refer to [8] & [27] where critical spaces have been computed and analyticity of the solutions is proven.We further mention the articles [31] & [32] for surveys ranging from boundary layers and fluid structure interaction problems over free boundary value problems and liquid crystal flow to the primitive equations.
For a comprehensive overview of applications of weighted function spaces in the context of free boundary problems as e.g. for the two-phase Navier-Stokes equations or Stefan problems, see the monograph [53].Further applications of weighted function spaces include • regularity issues for the Cahn-Hilliard equation [22], • reaction-diffusion systems of Maxwell-Stefan type [30], • Keller-Segel systems in critical spaces [33], • bidomain operators [35].