Journal of Function Spaces
Volume 2018 (2018), Article ID 2569080, 11 pages
https://doi.org/10.1155/2018/2569080

Research Article

Quasilinear Evolution Equations in -Spaces with Lower Regular Initial Data

Qinghua Zhang

Department of Mathematics, Nantong University, Nantong, China

Correspondence should be addressed to Qinghua Zhang; zhangqh1971@126.com

Received 7 February 2018; Accepted 12 March 2018; Published 2 May 2018

Academic Editor: Maria Alessandra Ragusa

Copyright © 2018 Qinghua Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 problem

has 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 [7], [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 estimate

associated with a -dependent constant . Prüss-Simonett in [16] showed that is equivalent to for all . After that, Köhne-Prüss-Wilke in [17] 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 [17], 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.

2. Preliminaries

Let and ; define

with 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 [22].

(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 norm

and its closed subspace

Proposition 1. with the imbedding constant independent of .

Proof. We will only prove ; for the other inclusion, please refer to [16]. 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 estimate

where 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 [5], 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 [5]).

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 .,

(iii) .

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 problem

Introduce 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 that

and 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

Consequently

Take

and , select such that and , and let

and 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 [17], we can find a time for which there is a unique solution surviving on the maximal existing interval solving the following problem:

Define

Then 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 .

Fix and , and suppose that inequalities (4) and (16) also hold for with replaced by . Take

as our work space, which is a Banach space endowed with the norm

and let . Consider the ball in :

For all , , and , by , we can deduce that

where . And

On the other hand, for each , we have

Therefore, for the corresponding function , we obtain

for all , which means that , and

Furthermore, for another function , we have

Take

and , select such that and , and let

and then, from the a priori estimates (52) and (53), we can derive that, for all and ,

Thus using Banach’s contraction theorem, we can conclude that the multiple map has a unique fixed point, which is the unique local solution of Pr. (44) exactly. That is, consider the following theorem.

Theorem 8. Assume that hypotheses are satisfied; then there exist and , such that, for each , Pr. (44) has a unique solution in the space .

Remark 9. Like Case 1, each solution of Pr. (44) can be extended onto a maximal interval such that does not exist in provided . Moreover if , and there exists such that

then this solution exists globally.

4. Examples

Example 10. Consider the second-order parabolic equation

Suppose that () is bounded domain with boundary, , with , and , and is a symmetric and positive definite matrix for every . Suppose also is a function of Caratheodory type; that is, is measurable for each , and is locally Lipschitz for a.e. . Moreover, there are numbers and a function , such that

for a.e. , all , and all .

Let and , where is the trace of on . Then the interpolate space can be described as

By Nirenberg-Gagliardo’s inequality, we have that provided

and whenever

Therefore, for each , there is correspondingly a constant such that, for any with , we have

where .

Given and , define

It is easy to show that , and, from [7, 12], we know that for all .

Now the numbers can be chosen to fit in our setting according to the variation of . In fact, if , then we take

if , then we take

if , then we take

where is arbitrary. In this setting, all Conditions (61), (62), , and are verified. Thus, using Theorems 6 and 7, we have the following.

Theorem 11. For the numbers chosen above, if the initial point (including the case ), there is a radius and a time depending only on the -norm of , such that, for each , there is a unique solution to Pr. (58) on the interval with and satisfying

And, for any two points , lying in the same ball , and , we have the following estimates:

for some .
Any such solution can be extended onto a maximal interval uniquely. Furthermore, if

or weakly , then , and the solution survives globally.

Remark 12. Since the regularity of the initial value is allowed to be lower than that of the space where the nonlinear map acts, here we need not require , and, consequently, condition which was used in [17] is replaced by the weaker one: in this paper.

Example 13. Let us consider the homogeneous Navier-Stokes equation

Here is a bounded domain with () boundary. The vector-valued function denotes the velocity of the flow, and the scalar function denotes the pressure, is the viscous coefficient, and is the body force. Denote

and let be the complement of in the space . Recall that, under the assumptions upon , we have that

where is the divergence of in the sense of distribution, and denotes the trace of such that for all , where denotes the unit outer normal vector at . Moreover we have the Helmholtz decomposition (see [23, Ch. III])

from which we obtain a continuous projection which is called the Helmholtz projection. Define

Recall that, for each , is a sectorial operator of type in , and consequently it has the -maximal regularity on the infinite interval (cf. [10]).

Let ; then (71) can be abstracted by

For each , denote ; then we obtain . Notice that (cf. [24, Ch. 7]), for every ,

where denotes the Lorenz space. And

From the above embedding relations, we can derive the a priori estimates for the nonlinear term as follows:

(1)If , , then

(2)If , , then

Here and . Therefore the nonlinear map verifies

provided

Pick ; then all the assumptions of for the operator are satisfied with provided . Thus, using Theorems 6 and 7 for the semilinear equations, we can derive the following.

Theorem 14. For each and , there exist depending on the norm of and , such that, for every and with , Pr. (76) has a unique solution on the interval with the initial condition verified. This solution satisfies

And for all and with , by denoting , we have that

for some .
If we denote by the maximal interval of existence of the solution to Pr. (76) extended uniquely by the local solution, then with

and , or and the solution exists globally.
Furthermore, for each , we have the following estimate:

for some .

Remark 15. If for some , then we can also derive that ; that is,

Let us consider a special case. Let , , and , and then we have

So if we take and , then we have and . Thus, for , we obtain . Compared to which was derived in [25], despite the fact that the upper bounds of the regularity of is lower than , but, as a compensation, the lower bounds of exponent are less than , which means that, here, solution exhibits a better time asymptotic property as . Notice that , and limit (87) turns to be . This is an -version of the limit (, ) which was derived in [19] by means of -regular solution. It is worth pointing out that, by using the maximal -regularity of the Stokes operator , the continuity assumptions of the forcing term has been removed here. Moreover, the asymptotic regular assumption upon as : which was used in [19] or which was used in [25] are replaced by a much more relaxed one: . In this sense, results obtained here can be regarded as useful supplements of those in [19, 25] for the case .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by Teaching Celebrities Cultivating Project of Nantong University (03080766) and Preresearch Project of Nantong University (17ZY01).

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