Research Article | Open Access
Jiafa Xu, Lishan Liu, "On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations", Journal of Mathematics, vol. 2021, Article ID 5541403, 7 pages, 2021. https://doi.org/10.1155/2021/5541403
On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations
Abstract
In this paper, we consider the Cauchy problem for the 2D inviscid Boussinesq equations with being the buoyancy frequency. It is proved that for general initial data with , the life span of the classical solutions satisfies .
1. Introduction
The 2D inviscid Boussinesq equations read as
The unknown functions stand for the temperature, the velocity field, and the scalar pressure, respectively. is the unit vector in the vertical direction. They can describe the natural convection in the 2D inviscid incompressible fluids such as the dynamics of the ocean or the atmosphere (see, e.g., [1–3]). Besides the physical significance, there is also a strong mathematical motivation for studying these equations. In fact, the 2D Boussinesq equation can be used as a model for the 3D axisymmetric Euler equations (see, e.g., [4]). Therefore, the study of equation (1) can provide us with the useful information to understand the Euler equations.
Due to the physical and strong mathematical significance of the 2D Boussinesq equations, many researchers have paid much attention on its study. For the Cauchy problem around the stationary solution , global regularity of solutions is known when the classical dissipation is present in at least one of the equations (1) (see [5, 6]) or under a variety of more general conditions on dissipation (see [7]) or under adding a damp term (see [8–11]). In contrast, the global regularity problem on the inviscid 2D Boussinesq equations (1) now is still open. Some attempts have been made for a step forward in this direction, which appears to be out of reach in spite of the progress on the local well-posedness and regularity criteria (see [12, 13]). On the other hand, for the initial boundary problem, Hu et al. [14] obtained the global well-posedness for the Boussinesq equations with non-slip boundary condition, and Lai et al. [2] and Littman [4] obtained the local well-posedness for the Boussinesq equations with slip boundary condition. Recently, some researchers started to consider the global well-posedness around the stationary solution of the strong stratification [11, 15, 16]. In particular, Elgindi and Widmayer [15] proved the long-time existence which is the life span of the associated solutions is if the initial data are of size . In this paper, we will present a new life span which is suitable for general initial data by the method of Strichartz estimate combining with a blowup criterion.
To state our result more precisely, we firstly consider the solutions of (1) around a stratified solution and rewrite equation (1). It is well known that equation (1) admits an explicit stationary solution of the formsatisfying the hydrostatic balancewhere is called the buoyancy or the Brunt–Vaisala frequency and represents the strength of stable stratification. Settingwe can reformulate (1) into
Furthermore, let us setand we getwhere and is a constant matrix given by
Since , let us introduce the extended Helmholtz projiector of the velocity onto the divergence-free vector fields which is defined bywhere denote the Riesz transforms on . Applying the operator to (7) gives the following equation:
The initial data to the above equations are given by
Now we state the main result.
Theorem 1. Let satisfy andThen, equations (10) and (11) possess a unique solution:withwhere the constant is independent of .
Remark 1. This result implies that the existence time will be larger as the buoyancy increases and the life span of the classical solutions satisfies
1.1. Plan of the Article
The paper is organized as follows. In Section 2, we first derive the explicit formula of solutions to the linearized equations of (10) and (11), and then we establish the decay estimate and Strichartz estimate of a linear propagator. In Section 3, we establish the blowup criterion of equations (10) and (11). In Section 4, we present the proof of Theorem 1.
1.2. Notations
Throughout this paper, we denote by the constants which may differ from line to line. , , and denote the Lebesgue spaces, Sobolev spaces, and the inhomogeneous Besov spaces, respectively. Let denote the Fourier transformation and inverse Fourier transformation, respectively. The Littlewood–Paley multipliers are defined bywhere such that
The low-frequency multiplier is defined by
2. Linearized Equations
In this section, we derive the representation of solutions to the linearized equations of (10) and (11) and establish the decay estimate and Strichartz estimate of the linear propagator given by
2.1. The Representation of Solutions to Linearized Equations
We study the following linearized equations associated to equations (10) and (11):
Applying the Fourier transform to (20) yieldswhere is the multiplier matrix of the operator defined bywhich is given explicitly by
Then, a direct calculation yieldsand the eigenvalues of areand the corresponding eigenvectors are
Thus, the solution of (21) is
Since , we have . Hence, we get
Settingone has
2.2. Decay Estimate
We derive the following decay estimate of the operator .
Lemma 1. It holds thatfor all , where the operator is defined byin which satisfies
Actually, the result of Lemma 1 is an immediate consequence of the following lemma.
Lemma 2. There exists a positive constant independent of such that
Next, we give the details of the proof of Lemma 2. Firstly, we recall an important lemma (see Keel and Tao [17] and Majda [18]).
Lemma 3. Let be a surface measure on a smooth surface in and let . Suppose that for all , at least of the principle curvatures are non-zero. Then, it holds that
2.2.1. The Proof of Lemma 2
By the theory of Fourier transform of measures supported on surfaces [18], we havewhere is some measure supported on the surface which is given by
By Lemma 3, the decay of is determined by the number of non-vanishing principle curvatures of the surface . Equivalently, the number of non-vanishing principle curvatures of the surface is the rank of the Hessian matrix . By some computations, we obtain
From (38), we have
(39) shows that the surface has a non-vanishing principle curvatures unless . Thus, we decomposewherewhere is a smooth function on such that on and .
For the estimate of , by Lemma 3, we have
For the estimate of , since , we see that
From (43) and the fact , we see one non-vanishing principle curvature of the surface . By Lemma 3, we have
Combining (36), (42), and (44), we obtain
For small , it is trivial that
Thus, we complete the proof of Lemma 2.
2.3. Strichartz Estimate
Firstly, we recall the following result obtained by Hu et al. [14].
Lemma 4. Let be a family of operators. Suppose that for all ,
Then, the estimateshold for all with satisfying
From Lemmas 1 and 4, the fact , and the scaling in time , we obtain the following.
Lemma 5. For satisfying we have the following Strichartz estimate:
3. Blowup Criterion
This section shows a blowup criterion of equations (10) and (11).
Lemma 6. Let be a solution of equations (10) and (11) defined on a time interval containing . Then, for any , we have the bounded estimate
Proof. For the vector variable , we obtainWe take derivatives of the first and the second equation in (53), multiply by and , respectively, and integrate over to obtainNote thatSince , we haveThe commutator-type estimate (see, e.g., [12, 15]) provides us withThus, adding (54) and (55) giveswhich implies that by the Gronwall inequality for ,The above equality implies that Lemma 6 holds.
4. Proof of Theorem 1
Applying to (10) and using the fact gives
By the Duhamel principle, we get
Due to Lemma 5 and scaling, we find that for ,
In the following, for , we are going to derive the estimates of
By (63) and the Bernstein inequality, we have
On the other hand, we have from (61) that
By (63), we have
Similar to (66), we have
Combining (67) and (70) yields
Define
We get from (71) and Lemma 6 that
Let and suppose that
We can choose sufficiently large such that
Combining (73) with (74) gives
Finally, the restriction (75) implies can choose
By the bootstrap principle, we deduce from (76) that (74) actually holds. Thus, from the classical local existence (see [12]) and the blowup criterion Lemma 6, (10) and (11) possess unique classical solutions satisfying .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by the NSFC (grant no. 11871302), China Postdoctoral Science Foundation (grant no. 2019M652348), Natural Science Foundation of Chongqing (grant no. cstc2020jcyj-msxmX0123), and Technology Research Foundation of Chongqing Educational Committee (grant nos. KJQN201900539 and KJQN202000528).
References
- J. Lu, “Local existence for Boussinesq equations with slip boundary condition in a bounded domain,” Journal of Applied Mathematics and Physics, vol. 5, Article ID 510165, 2017. View at: Publisher Site | Google Scholar
- M. J. Lai, R. Pan, and K. Zhao, “Initial boundary value problem for two-dimensional viscous Boussinesq equations,” Archive for Rational Mechanics and Analysis, vol. 199, no. 3, Article ID 7398C760, 2011. View at: Publisher Site | Google Scholar
- A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, UK, 2002.
- W. Littman, “Fourier tranforms of surface-carried measures and differentiability of surface averages,” Bulletin of the American Mathematical Society, vol. 69, pp. 766–770, 1963. View at: Google Scholar
- D. Chae, “Global regularity for the 2D Boussinesq equations with partial viscosity terms,” Advances in Mathematics, vol. 203, pp. 497–513, 2006. View at: Google Scholar
- T. Hou and C. Li, “Global well-posedness of the viscous Boussinesq equations,” Discrete and Continuous Dynamical Systems - A, vol. 12, no. 1, pp. 1–12, 2005. View at: Publisher Site | Google Scholar
- C. Cao and J. Wu, “Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation,” Archive for Rational Mechanics and Analysis, vol. 208, pp. 985–1004, 2013. View at: Google Scholar
- D. Adhikari, C. Cao, J. Wu, and X. Xu, “Small global solutions to the damped two-dimensional Boussinesq equations,” Journal of Differential Equations, vol. 256, no. 11, pp. 3594–3613, 2014. View at: Publisher Site | Google Scholar
- A. Castro, D. Cordoba, and D. Lear, “On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term,” Mathematical Models and Methods in Applied Sciences, vol. 29, pp. 1227–1277, 2019. View at: Publisher Site | Google Scholar
- T. Taniuchi, “A note on the blow-up criterion for the inviscid 2D Boussinesq equations, the Navier-Stokes equations:theory and numerical methods,” Lectures Notes in Pure and Applied Mathematics, CRC Press, Boca Raton, FL, USA, 2002. View at: Google Scholar
- R. Wan and J. Chen, “Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations,” Zeitschrift für angewandte Mathematik und Physik, vol. 67, p. 104, 2016. View at: Publisher Site | Google Scholar
- D. Chae and H.-S. Nam, “Local existence and blow up criterion for the Boussinesq equations,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 258, pp. 935–946, 1997. View at: Google Scholar
- R. Danchin, “Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics,” Proceedings of the American Mathematical Society, vol. 141, pp. 1979–1993, 2013. View at: Google Scholar
- W. Hu, I. Kukavica, and M. Ziane, “On the regularity for the Boussinesq equations in a bounded domain,” Journal of Mathematical Physics, vol. 54, Article ID 081507, 2013. View at: Publisher Site | Google Scholar
- T. Elgindi and K. Widmayer, “Shap decay estimats for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq ststems,” SIAM Journal on Mathematical Analysis, vol. 47, pp. 4672–4684, 2015. View at: Google Scholar
- R. Wan, “Global well-posedness of strong sloutions for the 2D damped Boussinesq and MHD equations with large velocity,” Communications in Mathematical Sciences, vol. 15, pp. 1617–1626, 2017. View at: Publisher Site | Google Scholar
- M. Keel and T. Tao, “Endpoint Strichartz estimates,” American Journal of Mathematics, vol. 120, pp. 955–980, 1998. View at: Google Scholar
- A. Majda, “Introducing to PDEs and waves for the atmosphere and ocean,” vol. 9, New York University, Courant Institute of Mathematical Sciences, New York, NY, USA, 2003, Courant Lecture Notes in Mathematics. View at: Google Scholar
Copyright
Copyright © 2021 Jiafa Xu and Lishan Liu. 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.