Abstract
This paper focuses on the global existence and time-decay rates of the strong solution for the Boussinesq system with full viscosity in for . Under the initial assumption of with a small norm, and or and for some , global existence and uniqueness of the strong solution for the Boussinesq system is established. This solution is proven to obey the following estimates: for , for , and as for , and and as for , where if and if .
1. Introduction
This paper investigated the global existence, uniqueness, and time-decay rates of the strong solution for the Boussinesq system in as follows:
Here, the unknown vector-valued function represents the velocity of the flow of a fluid, while the unknown scalar functions and represent their temperature and inner pressure at the place and time . Moreover, and denote the initial temperature and initial velocity, respectively.
The Boussinesq system is a simplified model to simulate the motion of the ocean or the atmosphere. By the Boussinesq approximation, we can neglect the variation of the density of the fluid but the var biation of temperature, which causes a vortex buoyancy force in the system, where denotes the unit vertical vector. There are three physical constants associated with the fluid: is the proportion coefficient, is the viscous index of the fluid, and is the thermal confusion number. For the sake of simplicity, by means of rescaling of the unknowns, we always assume that .
Evidently, if , then system (1) reduces to the standard incompressible Navier–Stokes equations. This means that the classical theory for the Navier–Stokes equations may be extended to the Boussinesq system. Danchin and Paicu [1] and Cannon and DiBenedetto [2] showed that if the initial data of both lie in the energy space , then the Boussinesq system (1) permits a global weak solution lying in
This is an extension of the results for the Leray–Hopf weak solutions of the Navier–Stokes equations (cf. [3]). Brandolese and Schonbek [4] and Han [5] investigated the time-decay property of the weak solutions of (1) in space under the extra assumption .
Danchin and Paicu [6] readdressed the global existence of the weak solutions for (1) with a partial viscosity under the initial conditions: , , andfor some sufficient small constant , whereand is the real interpolation space between and (refer to [1], §7.24]), especially and for , called the Lorenz space. Sokrani [7] and Wang et al. [8] focused on the global posedness of the 3-D Boussinesq system with the application of axisymmetric data and spherical coordinates, respectively.
As for the strong solvability of the Boussinesq system, the authors of reference [1] proved that if the initial temperature is and for some with the conditionfor that is small enough, then system (1) with partial viscosity has a unique and globally existing strong solution.where denotes the homogeneous Besov space (cf. [[2], §2.3]). A similar result was obtained in [4] for (1) with full viscosity and . It was proved that under the initial assumptionfor some sufficiently small number of , system (1) has a unique strong solution of in the space
Considering that the associated function spaces are invariant under the following scaling transformation:respectively, the initial spaces , and , , and are called critical. In this sense, results obtained in [1, 4] can be viewed as the extensions of those in [9–11] for the Navier–Stokes equations, where the critical space or for the initial velocity was employed.
This paper also addresses the global solvability of the Boussinesq system (1) with full viscosity. Motivated by [1, 4, 6], we first establish the existence and uniqueness of results for the global integral solution of (1) under the hypothesis , andfor some that is small enough. Compared to the literature, though the initial space for is stronger than , the initial assumption is much cheaper.
By using the smoothing action of the heat semigroup on spaces for all , and the estimates of the operators , , and , this paper also describes the higher regularity of the weak solution. We will show that under the additional condition
The global integral solution of (1) has enough regularity to become a strong solution. Compared with [1, 4, 8], here, the extra assumption or is replaced with a simpler one. Noticing that there is no any other restriction on the lower bounds of , compared with (4), it gives a partial answer to [6] for the minimal conditions to guarantee the solvability of (1) in the case . In addition, time-decay properties of the higher order norms of the strong solution are also investigated in this paper.
To end this section, we give an outline of the paper. Preliminaries and main results are stated in Section 2. Section 3 is devoted to the proof of the existence and uniqueness of the weak and strong solution of the Boussinesq system with full viscosity, together with the estimates for the solution and its gradient. In Section 4, time-decaying properties of the temporal and higher-order spatial norms of the strong solution are presented.
2. Preliminaries and Main Results
For and , let and be the usual Lebesgue and Sobolev space of scalar- or vector-valued functions defined on with the norm denoted by and , respectively. Let be the collection of all scalar- or vector-valued functions whose components are smooth with compact supports, be its subset containing all the divergence free vector fields, and let be the completion of in the space for . We use to denote the Helmholtz projection from onto defined by . We also use to denote a universal positive constant which may change from line to line, but does not depend on the involved functions.
It is observed that every classical solution to (1) solves the following system of integral equations:
To deal with system (12), let us introduce some function spaces with temporal weights. Suppose that and is a Banach space, then we define the equation as
For the sake of convenience, in the following discussions, kernels of the operators , , , and are denoted by , , , and , respectively, where is the Gaussian kernel, and others are vector-valued functions associated with or . There are some estimates for these kernels for and (refer to [4]) referred to asandwhere and is the conjugate number of , i.e., .
As a direct consequence of (14), it follows that for , there is a constant such that
Remark 1. By the boundedness and differentiability of the heat semigroup on , we assert that for all and for all , we havewhere ([)) denotes the space of all continuous and bounded valued functions on [).
Furthermore, for , by the density of in , we can check that , and in addition to (16), it holds thatIn other words, .
Given that , we consider the fractional power of negative Laplacian with the index defined by (cf. [12])If , then there is a corresponding constant such thatfor and for a constant such thatfor all and (refer to [9, 11]), where naturally.
The following lemma plays an important role in checking the continuity of a function in space.
Lemma 2. Assume that such that , and and for . Given a scalar field , two vector fields and and a tensor field lying in verifies thatwhere for , and let
Then, the following conclusions hold true:(i)If, then , and,(ii)If, then
The derivation of the estimates for and is left in the proof of Theorem 4 with concrete exponents. Verification of the continuity of with respect to is performed in the proof of ([13], Lemma 2.5), and others can be made in a much similar way, so we omit the whole reasoning process here.
Lemma 3 (see [4]). Suppose that and are two Banach spaces, and are two bilinear operators, and is a linear operator, then corresponding to these operators there are three constants , such that
Then, for any and , ifthen, the following equation systemhas a unique solution which verifies that
We now present the main results of the paper.
Theorem 4. There is a small number and a constant such that for all with the restriction (10), system (12) has a unique global solution fulfilling the equation that follows:and
Here, endowed with the norm .
Remark 5. It is easy to check that every solution of system (12) verifies the following integral equations:Here, is arbitrary, and and are also arbitrary. Symbol denotes the tensor product of two vectors, while denotes the scalar product of the two tensors. In this sense, is called the very weak solution of the Bossinesq system (1) (refer to [14]).
Theorem 6. In addition to (10), we assume that condition (11) holds, then the global solution for (12) obtained in Theorem 4 satisfies the following properties:andfor a.e..
Furthermore, for and , we havewith the same estimates as in (30). For and , we also havewith the estimateswhere for some , if , and , if , and .
Remark 7. The function pair , which verifies (32), (33), and (34), where is replaced with some , is called strong solution of the Boussinesq system (1).
By using the decomposition method for the integral representation of the solution developed in [15, 13], we can also derive the asymptotic behavior and growth of the temporal derivative exponents and spatial Hessian of and , that is,
Theorem 8. Under the assumptions on the initial data (10) and (11), the strong solution also satisfiesandfor , where for and for , and is the associated inner pressure of the fluid.
3. Global Existence, Uniqueness, and Regularity of the Solution
We first give a proof of the existence and uniqueness of the integral solution for the Bossinesq system (1).
Proof of Theorem 4: we define two intersection spaceswith the norms
For , we define that
We also define that and . Under this setting, equation system (12) can be abstracted to (26).
We now investigate the boundedness of and one by one. First, via (15) and Young’s inequality, we have
Moreover, by taking , we can deduce that
As a direct consequence of the inclusion , we have
Thus, by invoking Lemma 2, we conclude that , and
Here, the constant is independent of and .
Similarly, for , we have
Sinceby invoking Lemma 2 again, we conclude that , andfor some constant which is independent of and .
Yet for and , we have
Due to (46), by invoking Lemma 2 the third time, we have , andfor some constant which is independent of and .
In addition, since and , by virtue of (16) and Remark 1, it follows that and , andwhere the constants depend only on .
Now, we apply Lemma 3 with , , and to assert that under the hypothesiscondition (25) is satisfied, and consequently, the integral equation team (12) admits a unique solution verifying (27). Thus, the global solvability and uniqueness of the integral solution to (12) are reached, where the small number and the constant appearing in (29) take the valuesrespectively. Finally, by means of interpolation we obtain the two estimates in (30).
Remark 9. Note that, and consequently , while as . Hence, methods employed here are not feasible any more to deal with the global existence of the weak solutions of (1) in the case .
Then, we turn to show the higher regularity of the integral solution to make it become a strong solution for (1). Proof of Theorem 6: for the sake of convenience, herein after, the two viscosity indices and are both normalised to 1. Without loss of generality, we assume that in the case of . We also assume that and . We take an exponent , and let and , then we have and . Note that the global solution for (12) is obtained as the limit of the approximate solutions , where solves the following equation system:and and . Consequently, we may derive all the desired estimates of by taking limits of the estimates of , while the latter can be made by the arguments of iteration.
First of all, by reviewing the proof of Theorem 4, we assert that (30) with and holds uniformly for , .
As for the reasoning basis, we can deduce thatThen, they verify the following estimates:andwhere inequality (20) with and the factfor are both employed, andThe estimate of is based on the inequality (14) for the kernel and the estimate in (30) for , that is,where constraint is needed. This result, together with the iterative method, leads to the following inequalities:Let , which is chosen in Theorem 4 is so small that , then by using (58), we obtainuniformly for .
The treatment of is slightly different. Due to the singularity of at , we divide the integral interval into two equal parts to derive the following equation:Also, by the arguments of iteration, we obtainfor all provided that .
We now turn to deal with . By employing (20) with and (29) and (60) for and (64), it can be deduced thatandWe place the abovementioned two estimates together to obtainThen, following the same derivation as in (66), we havefor the case where .
A similar procedure can be performed for , that iswhich jointly produceand consequentlyfor the same restriction on as mentioned above.
An estimate of can be made in two cases. In the case of , we haveIn the case of , we haveBoth cases lead to the same conclusionprovided is small enough.
Based on the estimates derived above, we assert that the weak-star cluster points of the sequences and and functions and verify the estimates (64)–(77), respectively. Moreover, in light of Lemma 2, it follows that , , and . Besides, by the Gagliardo–Nirenberg inequality, we havewhich combined with the continuity of in and yields , and consequently and .
Now, we can check all the conditions for the regularity of the solution . For , we define in and in . We recall that, both and are sectorial operators with the domain and , and their native operators generate two uniformly bounded and exponentially decaying, analytic semigroups and , respectively. For all , by interpolation, we have and . Thus, for every , we have and . Furthermore, by the uniqueness of the integral solution, can be represented bySince , and , by employing the maximal regularity of and (refer to [16, 17]) and unique solvability of (79), we conclude thatand verifies (34) a.e. on . Thus, by the arbitrariness of , we eventually obtain (32), (33), and (34) for a.e. . In other words, is a strong solution for the Boussinesq system (1.1).
The second part of the proof is devoted to the validation of (35)–(37) where and for has been verified in the first part. From the continuity of in , together with the estimates (29) and (78), and the method of interpolation, one can easily derive that and for all . Besides, for any , let , then analogous to the treatment of , we havewhich in turn yields , and consequently with the same estimate provided that is small enough, where . This result can also be checked for by means of interpolation.
Moreover, for , we take , then analogous to (77), we can derive thatwhich results in , and finally, with the same estimate for a sufficiently small number of .
Remark 10. Uniqueness of the strong solution for the Boussinesq system (1) comes from the unique solvability of the system (12).
Remark 11. Under the initial condition in the case of , we can check the following estimate (see [4, 5] for references):
Remark 12. By reviewing the proof of Theorem 6, we should mention that associated with the estimate (30), the upper bounds of the small number can only be taken uniformly for belonging to an bounded interval for any , but uniformly for lying in the unbounded interval .
4. Decay Rates of the Higher Order Norms
Proposition 13. Suppose that is the global strong solution of (1) obtained in Theorem 6 under the initial hypotheses of (10) and (12). Then, for and , it holds thatandwhere , , and.
Proof. We only prove (84), and (85) can be dealt in the same way. For , we consider the following decomposition:By invoking (20) and (21), we can deduce thatBy combining the abovementioned three estimates, we obtained (84).
Remark 14. It is observed that for , estimate (84) also holds on (0, 1) but only with the exponent in the power substituted by .
Proof of Theorem 8: Here, we only deal with (40), the derivation of (39) is much similar. The standard potential theory shows that for all and with . So, we first use (84) and (85) to derive the estimates for . For this purpose, we divide into four partsIt is easily deduced from (20), (30), and (37) thatfor . By employing (84) and (85) with or , we obtainfor . Here, , , , and .
As for , we use (84) with to getfor , where is taken so close to such that and .
Putting all the estimates for and together, we obtain the desired estimates for and consequently for . In addition, the continuity of in with respect to can be deduced from the decomposition (88), jointly with (84) and (85), in light of Lemma 2.
Finally, by performing the operator on both sides of the second equation in (1), we can deduce that , andwhich in turn infers the existence of the pressure of the fluid verifying and the estimatefor . Thus, the desired estimate of (40) has been reached.
Remark 15. Under some extra reasonable hypothesis, the velocity of the flow exhibits rapid decaying behavior as . Precisely, in addition to (10) and (11), if in the case or for some in the case , then for every , it holds that as (cf. [4, 5]).
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The first part of the manuscript, including introduction and preliminaries, was written by Shuokai Yan and Qinghua Zhang. The second part, including the main results and proofs, was written by Lu Wang and Qinghua Zhang. All authors reviewed the manuscript.