Some Classes of Function Spaces, Their Properties, and ApplicationsView this Special Issue
Research Article | Open Access
Shaolei Ru, Jiecheng Chen, "Frequency-Uniform Decomposition, Function Spaces , and Applications to Nonlinear Evolution Equations", Journal of Function Spaces, vol. 2013, Article ID 176596, 12 pages, 2013. https://doi.org/10.1155/2013/176596
Frequency-Uniform Decomposition, Function Spaces , and Applications to Nonlinear Evolution Equations
By combining frequency-uniform decomposition with (), we introduce a new class of function spaces (denoted by ). Moreover, we study the Cauchy problem for the generalized NLS equations and Ginzburg-Landau equations in .
1. Introduction and Notation
In this paper, we introduce a new space (denoted by ) by combining frequency-uniform decomposition with . In this new space, we discuss the semilinear estimates, Schwartz estimates, embedding properties between and Triebel-Lizorkin space , and so forth. Moreover, we study the well-posedness of NLS equations and Ginzburg-Landau equations in .
In recent decades, the well-posedness of Schrödinger equations and Ginzburg-Landau equations is developed quickly. In this paper, we mainly study the local Cauchy problem of NLS equations and Ginzburg-Landau equations in this new space . Many people have studied these problems in Bessel potential and Besov spaces, for example, see [1–9], and so forth. We know that Besov spaces and Triebel spaces are generated by combining Littlewood-Paley decomposition with and (one can refer to  for the properties of Besov and Triebel spaces.). Next, we give a brief introduction of these spaces (see [10, 11]).
To define the Littlewood-Paley decomposition, it is convenient to consider a radial function such that and to define the sequence by Since , we can easily see that , , and . Define where is the Fourier transform and is the inverse Fourier transform. is said to be Littlewood-Paley decomposition operator. Formally, we see that We now define the Besov space for and by and Triebel space for and , by
Besov and Triebel spaces had been formulated during 1960s–1980s, which have been widely applied in recent years. Besov spaces and Triebel spaces are introduced by combining Littlewood-Paly decomposition operator with the function spaces and . Similar to Besov spaces and Triebel spaces, one can use frequency-uniform decomposition and , to generate a new class of function spaces. Actually, the spaces introduced by frequency-uniform decomposition and are modulation spaces. Many people have studied the well-posedness of evolution equations in these spaces (e.g., see [12–16], etc.). In this paper, we will consider the spaces generated by the frequency-uniform decomposition and . Firstly, we will recall the definition of the frequency-uniform decomposition and modulation spaces. In the 1930s, Wiener  first introduced the frequency-uniform decomposition. So, sometimes we call it Wiener decomposition of that is roughly denoted by where denotes the characteristic function on the set . Since is a translation of , have the same localized structures in frequency space, which are said to be the frequency-uniform decomposition operators. Since can not make differential operations, one needs to redefine it by smooth truncation function (see [12, 14, 18, 19]).
Now we give an exact definition on modulation spaces. We first denote . Let be a smooth radial bump function adapted to with if and if . Let be a translation of : Since in the unit closed cube and is a covering of , one has that for all , . Then, we write It is easy to see that Hence, the set is nonvoid. Let be a function sequence. Denote which is said to be the frequency-uniform decomposition operator. For any , we set , . For any , , we denote is said to be a modulation space, which was first introduced by Feichtinger  in the case (Appendix Theorem E shows the properties on ). For any , , we denote was first introduced by Baoxiang et al. .
We now introduce a new class of function spaces (denoted by ) by combining the frequency-uniform decomposition with . For any , , , we denote Moreover, we denote
Theorem 1. Assume , , , and ; then for any initial data , , there exists such that the initial value problem has a unique solution Moreover, if , then where .
Theorem 2. Assume , , , and . Assume also , . Then, for any initial data , , , there exists such that the initial value problem (complex Ginzburg-Landau equation) has a unique solution Moreover, if , then where , .
This paper is organized as follows. In Section 2, we will state some properties on , which is useful to establish the following inclusions. Section 3 is devoted to considering the multiplication algebra of . Some dispersive smooth effects for the Schrödinger and Ginzburg-Landau semigroups will be given in Section 4. Theorems 1 and 2 will be proved in Section 5.
2. The Properties on
In order to study the Cauchy problem in , we first give some properties on as follows.
Proposition 3. Letting , , , one has(1)if , then (2)if , then (3)if , then
Proof. Because , , we can get the results of (1) directly. For (2), we know that
Then, letting , we get (2).
Finally, we prove (3). Let . The proof can be classified into the following two kinds of cases.
Case 1 . In this case, . We have Actually, noticing that , we have . So, the first part of the above inequality holds. On the other hand, by Minkowski’s inequality, we have This proves the second part.
Case 2 . By Minkowski’s inequality and , we have This finishes the proof of this proposition.
Proposition 4 (completeness). For any , , , one has the following:(1) is a quasi-Banach space. Moreover, if , , then is a Banach space;(2);(3)if , then is dense in .
Step 1. In this step, we prove that . Actually, by  and (Proposition 3), we immediately obtain the result.
Step 2. is a quasinormed space. We prove the completeness. Let be a Cauchy sequence in (with respect to a fixed quasinorm in ). Proposition 3 shows that is also a Cauchy sequence in . Because is a complete topological linear space, we can find a limit element . Then, converges to in if . On the other hand, is a Cauchy sequence in (, .). By Appendix Theorem B, it is also a Cauchy sequence in . This shows that the limiting element of in (which is the same as in ) coincides with . Now it follows by standard arguments that belongs to and that converges in to . Hence, is complete.
Step 3. We prove that is dense in if and . Let ; then we put Of course, . Consequently (by Appendix Theorem F), Lebesgue’s bounded convergence theorem proves that the right-hand side of the above inequality tends to zero if . Hence, approximates in . Next, we let with and . Let with . Then, approximates in with if . However this is also an approximation of in . This proves that is dense in .
Proposition 5 (equivalent norm). Assume , , , . Then, and generate equivalent norms on .
Proof. Firstly, we have the following translation equality: For the sake of convenience, we denote Then, we have By Appendix Theorem F, we have Then, by , we have By the above discussion, we get .
The inverse inequality can be proved similarly. Based on the above observations, one can also obtain that .
Theorem 6. Assume , , ; then one has
Proof. By the definition of , we have
In this proof, we used (, Young’s inequality) and Appendix Theorem G.
Theorem 7. Assume ; then one has
Proof. We first prove . Consider the following: The inverse inequality can be proved similarly (). Moreover, by the similar discussion as mentioned previously, we can also obtain that .
Theorem 8. Let , , . Then, one has
Proof. One can obtain the result directly by Appendix Theorem D and Proposition 3.
Theorem 9. Let , . One has
Proof. Let , . We can easily find that if , then . For , , by Appendix Theorem F, we have This proves the theorem.
Theorem 10. Let be Triebel space; then one has
Proof. By Appendix Theorem A, we know that
Then, by the embedding between and (Proposition 3), we have (for any given )
This proves the first result (, ).
On the other hand, by Appendix Theorem A, we have
Then, by the embedding between and (Proposition 3), we have
This implies the theorem.
Theorem 11. Let , . Then, one has
Proof. By the definition of , we have Then, by Appendix Theorem F, Theorem 7 and Hölder’s inequality, the term can be dominated by On the other hand, by the similar discussion as the above, we have Let From the above discussion, we get the result.
3. Multilinear Estimate in
It is well known that is a multiplication algebra if , see . The regularity indices, for which constitutes a multiplication algebra, are quite different from those of Besov space.
Theorem 13. Let , , , . If , then one has where is independent of .
Proof. Recall that we denote and . We have
By , we can obtain the decomposition of as follows:
Then, we have
It is easy to see that and
Then, we have
Case 1 . Based on the previous observations and Appendix Theorem F, we have where we used and the following Young’s inequality:
Case 2 . By the definition of and Appendix Theorem F, we have By the above discussion, we can obtain the claimed results.
Corollary 14. Let , , , . If , then one has where is independent of .
Proof. Let , where . Then, we have By induction, we can easily see that the claimed result holds.
4. Smooth Effects of the Schrödinger and Ginzburg-Landau Semigroups
Put Our aim is to derive the estimates of , in the spaces .
Theorem 15. Assume , , , ; then one has
Step 1. By Appendix Theorem C, for , we have
Thus, for , we have
Step 2. By Proposition 3, for , we have This proves the theorem.
Theorem 16. Assume , , , , ; then one has
Proof. By Theorem 15, we have This proves the theorem.
Theorem 17. Letting , one has which holds for all .
Proof. By  (Appendix Theorem H), we have Then, we have This proves the theorem.
Theorem 18. Assume , ; then one has
Proof. By Theorem 17, we have This proves the theorem.
Theorem 19. Assume , . Assume also , , , . Then, one has
Theorem 20. Let , , ; then which holds for all .
Step 1 (see ). Let and ; then we have . Actually, in view of the Leibnitz rule, we have
Let , and let . It is easy to see that
Let ; one easily sees that
Hence, we have
Combining the previous discussion, we have
Step 2. By Step 1, we have . Then by Appendix Theorem F, we can obtain that This proves the theorem.
Theorem 21. Assume , , , ; then one has
Proof. By Theorem 20, we have