Stability on 3D Boussinesq system with mixed partial dissipation
-
Hongxia Lin
,
Sen Liu
Abstract
In the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in
1 Introduction
The Boussinesq system describes the influence of the convection (or convection-diffusion) phenomenon in a viscous or inviscid fluid, such as atmospheric fronts and ocean circulations. It plays an important role in the study of turbulence and suitably modified in atmospheric and oceanographic situations where rotation and stratification play a dominant role.
We are concerned with the three-dimensional incompressible anisotropic Boussinesq equations in the space domain
where
The objective of this article is to comprehend the stability of perturbations near the hydrostatic balance or hydrostatic equilibrium for the 3D Boussinesq equations (1.1) and to present accurate predictions of the long-term behavior. The motivations are twofold. First, many geophysical fluids are in an important equilibrium known as hydrostatic balance in physics. The hydrostatic equilibrium occurs when the force of gravity is balanced by the force of the pressure gradient. Actually, between the downward-directed force of gravity and the upward-directed force of the pressure gradient, our atmosphere is typically in a state of hydrostatic equilibrium. Gaining insight into certain severe weather phenomena may be made easier by being aware of the stability of perturbations close to the hydrostatic equilibrium [14,27]. Second, the Boussinesq equations (1.1), which only take into account partial dissipation, are used to simulate anisotropic fluids. Even though significant progress has been made on the large-time behavior of fully dissipated partial differential equation (PDE) systems [29–31], anisotropic PDE system behavior is still a hot topic in the field of partial differential equations.
Due to its physical background and mathematical features, the 2D or 3D Boussinesq equations with full or partial dissipation have attracted considerable attention from the community of mathematical fluid mechanics. In the past decades, the global regularity problem has been a topic of interest and significant progress were made. One can refer to [1,3,5,6,7,9–12,15,20–22,25,32] for partial dissipations and to [13,18,19,26,35,36,38] for fractional dissipations. Recently, the stability problem for the Boussinesq equations near the equilibrium state has also been examined. The stability of the system close to the hydrostatic equilibrium is a significant subject [2,16,17,23,28,33,36,37]. Said et al. [28] solved the stability problem for a two-dimensional Boussinesq system with only vertical dissipation and horizontal thermal diffusion in
This article investigates the Boussinesq equations (1.1) on the periodic domain
Consider the perturbations
Then
With loss of generality, let
This article attempts to achieve two goals. The first is to prove the stability of the hydrostatic equilibrium. The second goal is to establish the large-time behavior of solutions to (1.2). We shall show that
Before stating our results, we first make the symmetry assumptions for the initial data
We obtain the first result as follows.
Theorem 1.1
Assume
then the system (1.2) possesses a unique global solution
As aforementioned, when the spatial domain is the whole space
However, due to lack of horizontal thermal diffusion, it is very challenging to control the growth of the nonlinear term involving
The only vertical thermal diffusion prevents us from establishing the closed bound.
Therefore, we turn to consider the stability in the domain
which makes the bound for the integral term involving
With the fine inequality (1.8) at our proposal, several anisotropic inequalities associated with the oscillation are able to be established (Lemma 2.4). All the preparations together with the key property
We point out that it remains impossible to bound all parts of the nonlinear integral terms with
And the difficult terms involving
The procedure for the proof of Theorem 1.1 is standard. We define the energy functional
Efforts are devoted to establishing the a priori estimate
where
Our second result shows that oscillation part
Theorem 1.2
Assume that the initial data
where
We briefly outline the sketch of the proof of Theorem 1.2. The proof is split into three steps, which are devoted to the proof of (1.10), (1.11), and (1.12). First, we prove (1.10). Generally speaking, to obtain the exponential decay rates, we need to prove the following differential inequality:
for some constant
we attempt to prove the decay rates for
If we select the initial data sufficiently such that
then we assert that the decay rate holds
The rates
Second, for the decay estimate in (1.11), based on a significant observation
It is noted that when
cannot be closed if
Third, we present the proof of (1.12). That is the decay rates for
or
it suffices to deal with either of the two. With loss of generality, we consider the decay of
Compared to (1.15), the system (1.16) has more complicated forms. Therefore, to obtain a similar differential inequality to (1.13), we need more subtle estimates. A key step is to decompose the nonlinear terms in some items associated with
Then by making full use of the strong poincaré inequality (2.3) and several anisotropic inequalities on the oscillation in
The rest of this article is divided into three sections. Section 2 is devoted to providing the strong poincaré-type inequalities and some anisotropic inequalities associated with oscillation, as well as the properties of the decomposition. Section 3 proves Theorem 1.1, while Section 4 presents the proof of Theorem 1.2.
2 Preliminary
Before proving Theorems 1.1 and 1.2, we shall make some preparations in this section. We first list some properties related to the decomposition. Then the strong Poincaré-type inequalities on the oscillations and
Lemma 2.1
Let
(i) We have the following basic properties:
(ii) If f is a divergence-free vector filed, namely,
(iii) Let
Clearly,
Also,
(iv)
Proof
For the properties in (i), (ii), and (iii), it suffices to prove that for
(i) The properties can be proved via the definitions of
Similarly,
(ii) If
and
(iii) Since
then, we have
For
Therefore, we obtain
Furthermore, by the orthogonality, we can obtain
That is,
(iv) It follows from the definition of
Similarly, we have
This completes the proof of Lemma 2.1.□
Lemma 2.2
Let
where
Proof
Obviously, we only take the case
It is noted that
Then integrating in space
(2.4) is thus established.
Now taking
where we have used the fact
Finally, by the symmetry condition (1.7), which will be proved in Section 3, the facts
The lemma assesses the anisotropic inequalities in
which can be obtained by the direct estimates. One refers to a similar proof of Lemma 2.4 in [24].
Lemma 2.3
Assume that
Proof
By Hölder’s inequality, Minskowski’s inequality, (2.6), and (2.7), we can easily to obtain
Similarly,
This completes the proof of Lemma 2.4.□
If we replace
Lemma 2.4
Assume that
Proof
(2.10), (2.11), and (2.12) follow directly from (2.8), (2.9), and the Poincaré-type inequalities (2.4). For (2.13), a similar argument to (2.9) yields
Then replacing
When
Lemma 2.5
Assume that
We now present the last lemma that assesses the anisotropic upper bound on
Lemma 2.6
Assume that
Proof
First, the following Sobolev inequality in
for
By multiplying aforementioned inequality on both sides by
It then follows from (2.19), (2.7), and (2.3) that
This completes the proof of Lemma 2.6.□
3 Proof Theorem 1.1
This section is devoted to proving Theorem 1.1. To prove the global bound result (1.6), we need first show the the symmetry (1.7) for
Then by applying Poincaré inequalities and anisotropic inequalities, the
3.1 Proof of the symmetry (1.7)
The symmetry (1.7) can be obtained via the uniqueness. First, we can use the standard energy method to obtain the local well-posedness of the solution. Thus, our attentions mainly focus on the uniqueness of the solution. It is simple to achieve it. For the convenience of the readers, here, we only give the sketch of the proof.
Proposition 3.1
Let
Then, for any
Proof of Proposition 3.1
Denote
It is easily to check
Taking the
By using the anisotropic inequality (2.8),
Also, we have
Therefore, we obtain
Then by Gronwall’s inequality, for any
which together the local well-posedness of the solution on
Proposition 3.2
Suppose that
Proof of Proposition 3.2
We will utilize the uniqueness to verify proposition 3.2. Let
To prove the symmetries (1.7), it is only to verify that
We derive
Therefore,
For
which together with (3.4) asserts that
3.2 Proof of the global well-posedness (1.6)
To prove the global uniform bound of the solution, as aforementioned in Section 1, we first establish the a priori estimate for
Proposition 3.3
Let the initial datum
With the energy inequality at our proposal, we are able to prove (1.6) in Theorem 1.1.
Proof of (1.1)
We will use the bootstrapping argument to obtain the global existence of the solution and its stability. A rigorous statement of the abstract bootstrap principle can be found in Tao’s book (see [34] p. 21). More precisely, once (3.5) holds, the bootstrapping argument will imply
The argument starts with the ansatz that, for
Then it follows from (3.5) and initial data assumption (3.6) that
or
That asserts that
In what follows, we present the proof of Proposition 3.3.
Proof of Proposition 3.3
Due to the equivalence of the norms
We only need to bound
where we have used the fact that
Next we focus on the second-order derivatives estimates. We first apply the operator
Since
Since all of the derivatives in
By means of Lemma 2.1 and integration by parts, we first split
Then by Lemmas 2.1 and 2.3, invoking Hölder’s inequality and (2.4) yields
and
To deal with
where we have used
By collecting the estimates for
The bound for
where we have used
By applying the anisotropic inequality (2.11) and Hölder’s inequality along with
Similarly,
For
Due to
Thus, we obtain the following bound:
By substituting (3.9) and (3.10) into (3.8), integrating over
which is the desired inequality (3.5). We complete the proof of Proposition 3.3.□
4 Proof of Theorem 1.2
This section focuses on the proof of the decay estimates in Theorem 1.2. We will apply the inequalities obtained in Section 2 and use the global existence result (1.6) as well as the symmetry (1.7) to achieve this proof. Before the proof, we first present the stability in
Proposition 4.1
Assume the initial datum
then (1.2) has a unique global solution
Proof of Theorem 1.2
The proof is split into three steps which is devoted to the proof of (1.10), (1.11), and (1.12), respectively.
Step 1: the proof of (1.10)
It is noted that
The anisotropic inequality (2.10) together with
Applying (2.11) for
Then we derive, for some constant
Now we choose
Then employing the Poincaré inequality (2.3) together with
Furthermore, by (2.1), (2.2), (2.3), and (2.4), we infer
where we have used
The proof of (1.10) is now completed.
Step 2: the proof of (1.11)
With (4.1) and (4.2) at our proposals, we are able to present the exponential decay of
Then by subtracting (4.4) from (1.2) yields
where we have used the fact
By multiplying (4.5) by
where
Next we bound
To bound
Then, by applying Sobolev’s inequality, Poincaré inequality (2.4) and
where we have used
For
Similarly, for
We turn to bound the last terms. By integration parts, applying the anisotropic inequality (2.11) yields
where we have used
Finally, collecting all estimates for
If the initial data
where
That means
In what follows, we show the proof of the decay on
Similar to
where we have used
Also, we first split
Applying (2.11) and (2.12) again, except for
where we use
In summary,
which implies
provided that
Step 3: the proof of (1.12)
We first establish the equations of
Then from (4.8) and (1.2), we can deduce
By taking the
where
Now we estimate
Then by Hölder’s inequality, the Poincaré inequality (2.3), and the anisotropic inequality (2.17), we can derive
Similarly,
For
Then, applying Poincaré inequality (2.4) and the anisotropic inequality (2.14), (2.16) yields
Similarly,
It follows from (2.18) that
Therefore,
A similar argument to
For
We turn to bound the last two terms. By integration parts, applying the anisotropic inequality (2.14) yields
where we have used
Therefore, invoking the uniform bound (4.1), we have for some positive pure constant
which together with (4.7) leads to
Since by poincaré inequality (2.3), we have
then we can derive for
which implies that if the initial data
For
Then by the decay for
This completes the proof of Theorem 1.2.□
Acknowledgements
The authors are grateful for the reviewer’s valuable comments that improved the manuscript. We also express our gratitude to the editors for efficiently handling our manuscript.
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Funding information: Lin was partially supported by the National Natural Science Foundation of China (NNSFC) under Grant 11701049 and the Natural Science Foundation of Sichuan Province under Grant 2023NSFSC0056. Our manuscript has no associated data.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. Lin and Liu made formal analysis and the derivation of the proof. Liu and Guo prepared the original manuscript with contributions from all co-authors. You performed the review and editing.
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Conflict of interest: The authors state no conflict of interest.
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- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
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Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
