Research Article | Open Access
Qinghua Zhang, "Quasilinear Evolution Equations in -Spaces with Lower Regular Initial Data", Journal of Function Spaces, vol. 2018, Article ID 2569080, 11 pages, 2018. https://doi.org/10.1155/2018/2569080
Quasilinear Evolution Equations in -Spaces with Lower Regular Initial Data
We study the Cauchy problem of the quasilinear evolution equations in -spaces. Based on the theories of maximal -regularity of sectorial operators, interpolation spaces, and time-weighted -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 -spaces with temporal weights.
In this paper, we will deal with the following quasilinear evolution equation:in a Banach space couple with densely. Take and , and denote by the real interpolation space between and (see [1, Ch. 3] or [2, Ch. 1]). Given or , let . And, for each , let . Suppose that is a continuous map, is a Caratheodory map, and . We will deal with this problem in the framework of maximal regularity of the sectorial operators in -spaces with temporal weights.
Suppose that is a sectorial operator defined in with the domain (refer to [3–5] for the definition of sectorial operator), we say has the maximal -regularity on , or in symbol, and we mean that, for each , the following Cauchy problemhas a unique solution in . And there is a constant , such that for with . It is easy to see that if , then for arbitrary . Especially, if , then the constant in (4) is independent of .
In general, a sectorial operator does not have the property of -maximal regularity, even though its negative generates an exponentially decaying analytic semigroup. However, if is a space, and admits bounded imaginary powers ( for some ), or for short (see [3, Ch. 3], [5, 6]), then has the -maximal regularity. Further investigation, involving the vector-valued multiplier theorem and -functional calculus of sectorial operators, showed that, in a space, -maximal regularity equals -sectoriality for a sectorial operator (see , [8, Ch. 3]). Recall that, given a bounded or exterior domain with compact type boundary, or a half space , for each , the Lebesgue space is of type, on which the elliptic differential operators with some boundary conditions are -sectorial (cf. [7, 9]). And the Stokes operator is of type with the power angle (see [10, 11]). Hence, we can make estimates for the parabolic evolution equations (see [12, 13]) and the Navier-Stokes equations (see [14, 15]).
As for the maximal regularity of a sectorial operator in -setting, or equivalently , it means that, for each , Pr. (2) has a unique solution with the estimateassociated with a -dependent constant . Prüss-Simonett in  showed that is equivalent to for all . After that, Köhne-Prüss-Wilke in  studied Pr. (1) of autonomous type (i.e., and ), including the local well posedness and long time behavior of the solutions, in -spaces.
Motivated by the theory of -regular solutions (cf. [18, 19]), this paper will deal with Pr. (1) of nonautonomous type in weighted -spaces. Unlike , here the regularity of the initial value is lower than that of the space where the nonlinearity acts; more precisely, lies in the space for some and maps into .
This paper is organized as follows: as preliminaries, in Section 2, we give some definitions and properties of the time-weighted Lebesgue and Sobolev spaces of abstract valued functions, and the maximal regular space and its closed subspace . In Section 3, we establish the local and global existence of and solutions in subcritical and critical cases, respectively, using estimate (4), together with the embedding properties of and the smoothing action of the operator . To illustrate the obtained results, in Section 4, we make estimates for the parabolic evolution equations and the Navier-Stokes equations, using the fractional order Sobolev space and Besov space as the state spaces.
Framework of our study can be incorporated into the theory of sectorial operators and analytic semigroups, together with time-weighted -spaces. Results obtained here have their meaning in the study of quasilinear evolution spaces driven by the sectorial operators. For the relative investigations on the quasilinear evolutions in or spaces with different focuses, please refer to [5, 16, 17, 20], with the references therein.
Let and ; definewith the norm , and with the norm , and its closed subspace with zero trace where denotes the trace of . We can also define the maximal regularity spaces Here are some interpolating and embedding properties of the and spaces, all of which can be found in [21, Ch. 1] or .(i) whenever , and , and(ii)For each ,(iii)For the indexes , , and ,where is a domain in and and , denote the Sobolev and Besov spaces with real indexes, respectively.
Suppose that is a sectorial operator with , and generates a uniformly bounded analytic semigroup . From [4, Ch. 5], we know that, for each and all , the norm is equivalent to
Introduce the continuous function space with time weight endowed with the normand its closed subspace
Proposition 1. with the imbedding constant independent of .
Proof. We will only prove ; for the other inclusion, please refer to . Taking any , we have that and , where and the constant is independent of (see [21, Ch. 1]). Notice that with the imbedding norm independent of , and we then have with , and the desired result comes.
Suppose that, for all ,where is independent of .
Proposition 2. For each , define . Then for all together with the estimatewhere and is the upper bounds of the semigroup on the infinite interval .
Proof. Firstly, for each , we can deduce that Secondly, for each and , we have that This relation, together with the dense imbedding and the upper bounds , leads to the desired inclusion .
Analogously, we can derive the following.
Proposition 3. and for all and , where .
Remark 4. From , we know that, for every sectorial operator lying in , the corresponding semigroup decays exponentially, and consequently the constant appearing in Propositions 1 and 2 is independent of , whence the subscript can be erased.
The following lemma tells us that the property of -maximal regularity can be preserved under small perturbations (refer to ).
Lemma 5. Given a sectorial operator with the constant as in (4), suppose that is closed in satisfying for some . Then with the bounds .
3. Main Results and Proofs
Firstly, we give some hypotheses on and used in this paper.
: is continuous. And, for each , there is a constant such that for all and all with .
for some .
: is an operator of Caratheodory type; that is,
for each , is strongly measurable,
for a.a. , is locally Lipschitz, and there is a number and a function for which holds for a.a. and all .,
It is easy to show that, under all the hypotheses listed above, the following functions associated with : , , , , and are all infinitely small as . Without any confusion arising, in the coming discussions, the first small quantity is denoted by , and other ones related to are denoted by uniformly.
We shall now discuss the local well posedness of Pr. (1) in two cases.
Case 1 (). For each , consider the auxiliary problemIntroduce the map : Then solution of the auxiliary problem (22) can be represented by and consequently solution of the original problem (1) is exactly the fixed point of .
For any , let where are small positive numbers. Taking any , using (16) and Proposition 2, we can derive thatand in the same way. Now we can give the a priori estimates for and in as follows: where and Hence
In the next step, we will show the local Lipschitz property of . For this purpose, take any and derive that ConsequentlyTakeand , select such that and , and letand then, from (29) and (31), we can deduce that, for all and , and . Thus by Banach’s contraction theorem, the nonlinear map has a unique fixed point, which is the unique local solution of Pr. (1) simultaneously.
Theorem 6. Suppose all the hypotheses upon and together with are satisfied, and then there exist and such that, for each , Pr. (1) has a unique solution in the space .
Now, we turn to deal with the continuation of the local solutions. Suppose that (ii) is strengthened as follows:
For all , .
Under this situation, by invoking Lemma 5, together with the continuity of , we can deduce that, for each precompact subset , there is a common constant , for which Ine. (4) is satisfied for all with . We can also assume that all the semigroups share the same bounds .
Taking any , and using Theorem 6, we can find a finite interval , on which Pr. (1) has a unique solution with the initial value . Denote by and . Evidently, verify hypotheses and , respectively. Notice that for arbitrary . , and then, following the same process as in , we can find a time for which there is a unique solution surviving on the maximal existing interval solving the following problem:
DefineThen for any and solves (1) on the interval . We can also conclude that could not be extended beyond anymore. Thus the function constructed above is exactly the solution of (1) arising out of with the maximal existing interval . If , then is called globally existing. If , then we obtain and does not exist in . In this case, we say that blows up at time .
Theorem 7. Suppose that , and there exists such that (i) is also satisfied with . Suppose also, on its maximal interval of existence , solution of Pr. (1) verifies Then , and survives globally.
Proof. We will prove it by contradiction. Suppose that , and then the set is bounded in and therefore precompact in since . Thus exists in . Define , and then can be extended continuously onto the whole interval . Assume that , and then Thus, by the continuity of and , we can deduce that uniformly for .
In addition, for arbitrary , there is a partition of : such that provided for all . Notice that for all ; we obtain uniformly for .
Now we can take an arbitrary sequence converging to from the left side, and consider the following problem:where and . Assume that (16) holds also with , and then, by repeating the reasoning process as in Theorem 6 times with the same , , , and , we obtain a small number , such that, on the common interval , Pr. (43) has a unique solution in . By the uniqueness, we can find that for , which means that can be extended beyond the singular point . This contradicts to the maximality of since for large enough. Thus the proof has been completed.
Case 2 (). In this case, we only deal with the Cauchy problem of the semilinear parabolic equation; namely,Here, is a sectorial operator lying in .