Existence and Asymptotic Behaviour of Solutions of Differential and Integral Equations in Some Function SpacesView this Special Issue
Asymptotic Study of the 2D-DQGE Solutions
We study the regularity of the solutions of the surface quasi-geostrophic equation with subcritical exponent . We prove that if the initial data is small enough in the critical space , then the regularity of the solution is of exponential growth type with respect to time and its norm decays exponentially fast. It becomes then infinitely differentiable with respect to time and has value in all homogeneous Sobolev spaces for . Moreover, we give some general properties of the global solutions.
We consider the dissipative quasi-geostrophic equation with subcritical exponent : where , , is the unknown potential temperature, and is the divergence free velocity which is determined by the Riesz transformation of in the following sense: The critical homogeneous Sobolev space is and we have
In , we studied the existence of global solutions of if the initial data is small in the critical space and the subcritical exponent . In use of Theorem 4.2 in  with , we proved the following Theorem.
Theorem 1 (see ). For and , there exists a constant such that if the initial value problem has a unique solution in . Moreover,
We proved also the following result.
Theorem 2 (see ). Let .(i)If is a global solution of , then (ii)If is a global solution of , then
In this paper, we describe the long time behavior of these solutions with respect to the homogeneous Sobolev norm , for . We prove the following.
Theorem 3. There exists such that, for all , , and there exists a global solution such that, for all , for all , and
When the initial data is in and small enough in the homogeneous space , we prove that the Leray solution is also in all Sobolev spaces . Moreover, we describe the long time behavior of its homogeneous Sobolev norm , for . We state also the following.
Theorem 4. There exists such that, for all , , and there exists a global solution such that, for all ,
The paper is organized as follows. We start by recalling some preliminary background and stating useful preliminary results on Sobolev spaces. Sections 3 and 4 are devoted to the proof of the main results, Theorems 3 and 4. In Section 5, we give some general properties for any global solutions of the system .
2. Notations and Preliminary Results
2.1. Notations and Technical Lemmas
In this short section, we collect some notations and definitions that will be used later and we give some technical lemmas.(i)The Fourier transformation in is normalized as (ii)The inverse Fourier formula is (iii)For , denotes the usual nonhomogeneous Sobolev space on and its scalar product.(iv)For , denotes the usual homogeneous Sobolev space on and its scalar product.(v)The convolution product of a suitable pair of functions and on is given by (vi)For any Banach space , any real number , and any time , we will denote by the space of measurable functions such that .(vii)If and are two vector fields, we set (viii)For any subset of a set , denotes the characteristic function of .We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.
Lemma 5 (see ). Let , be two real numbers such that There exists a constant , such that, for all , If and , there exists a constant , such that, for all and ,
For the proof of the main results, we need the following lemma.
Lemma 6. Under the same conditions as in Theorem 3, for all and , where and and .
Remarks 7. (i) If , formula (16) gives (ii)If and , formula (16) gives (iii)If and , formula (16) gives
Proof of Lemma 6. By the Cauchy-Schwarz inequality, we get Using the weak derivatives properties, the elementary inequality , with and , and the product laws (Lemma 5), with , , and , we can dominate the nonlinear part of (20) as follows:
3. Proof of Theorem 3
To prove Theorem 3, we need the following result.
Proposition 8. There exists such that, for all , , and there exists a global solution such that
Proof. The proof is done in two steps.
First Step. For a nonnegative integer , Friedrich’s operator is defined by Consider the following approximate system on : Then, by the ordinary differential equations theory, the system has a unique maximal solution in the space , . Using the uniqueness and the fact that , we obtain and Taking the scalar product in , we obtain, for , It follows that, for all , , which implies that .
Now, taking scalar product in , we obtain
Using product law (15) with and , we obtain But Then,
Let For , by (30), we have then , and, for all , we have If we take the limit when goes to the infinity, we find a solution which satisfies which proves the first result of Theorem 3.
Second Step. Back to the approximate system, For , we define Then,
Using the classical inequality we let Taking the norm in , we obtain By Cauchy-Schwarz inequality, we have where Using product law (15) in the homogeneous Sobolev space with , , and , we obtain Then, To estimate the term , we use the Hölder inequality and we get The convex inequality , with and , gives Thus, where is in and .
Let and ; we set For , we have By Gronwall lemma, we get that, for all , For the given value of , we have that, for all , thus, It follows that and, for all , which proves formula (22), and the proof of Proposition 8 is finished.
Now we intend to study the behavior of the solution at infinity. We claim to prove that, for all , We can suppose that . We have where .
Using (22), we get and the proof of Theorem 3 is finished.
Remark 9. (i) Combining Theorems 2 and 3, we can obtain, for and ,
Indeed, from (34), , for all . For , we consider the following system: This system has a Leray unique solution that satisfies, for all , From the uniqueness of the solution, we have ; then For , we have Combining this inequality with the result of Theorem 2, we obtain the desired result.
(ii) If , we do not know if holds. But this result depends on the lower frequencies. Indeed, for and , we have By Theorem 3, we obtain
Then, for , , we have Then, to prove the result, It suffices to prove that
4. Proof of Theorem 4
First Step. Using the approximate system (24) and inequality (17), Then,
Second Step. From relation (22), we deduce that Then, the proof is achieved.
Remark 10. If and , we have
5. General Properties of Global Solutions
Theorem 11. Let be a global solution of such that Then, for some . Moreover, for all ,
Combining the energy estimate and the conclusion of Theorems 4 and 11, we get the following.
Theorem 12. Let be a global solution of such that
Then, for some .
Moreover, for all ,
Remarks 13. (a) Let and let be a global solution of such that
Using the Sobolev injection,
We conclude that, for all ,
(b) Let be a global solution of such that then, for all , Using the classical interpolation inequality and Theorem 12, we get
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.
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