Remark on a Regularity Criterion in Terms of Pressure for the 3D Inviscid Boussinesq–Voigt Equations

We consider the three-dimensional inviscid Boussinesq–Voigt system which is a regularization model for the inviscid Boussinesq equations. We prove a regularity criterion for the weak solutions (in particular, for the time-derivative of the velocity), in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm L^p$$\end{document}-spaces, involving first derivatives of the pressure.

where u = u(x, t) , = (x, t) and = (x, t) are respectively the velocity, the temperature and the pressure of the fluid, and they depend on spatial position x ∈ 3 and on time t > 0 .Also,  > 0 is the diffusion coefficient, and e 3 = (0, 0, 1) T .The above system is supple- mented by the initial conditions (u 0 , 0 ) , with ∇ ⋅ u 0 = 0.
For it, after introducing a suitable class of weak solutions for (1), we provide a control for the time-derivative of the velocity t u in terms of the partial derivative of the pressure 3 .Our analysis moves from results presented in [28,31,33].
In recent years various regularization models have been proposed in the context of numerical simulations of turbulent flows and in the particular case of incompressible turbulent fluids (see, e.g., [11,12,14,19,20]), and among these we find the so-called -models.These models are based on a smoothing procedure obtained through the application of the inverse of the Helmholtz operator A , which is defined as follows Loosely speaking, the effect of applying the inverse of the Helmholtz operator to a vector field v (thus obtaining v ∶= (A ) −1 v ), is to gain two additional space-derivatives for the considered quantity.
There is a wide range of -approximation models for the Navier-Stokes equations and for other related systems in fluid dynamics (see, for instance, [1,9,12,17,23], see also [6]).Further details can be found in [8,9].
From (1), when = 0 , we formally recover the inviscid Boussinesq equations, which are often used for the determination of the coupled flow and temperature field in natural convection (see, e.g., [22]).These equations are also used, for instance, as a mathematical scheme to describe Newtonian fluids whenever salinity concentration or density stratification -according to the meaning of -play a significant role (see, e.g., [3][4][5]7] for some recent papers on this subject).Moreover, they are extensively employed in studying oceanographic and atmospheric phenomena (see [26,27,30]).The problem related to the uniqueness and global regularity of the weak solutions of the 3D Boussinesq equations, strictly related to the one of the 3D Navier-Stokes equations, is a relevant open issue.These facts highlight the importance of this system of equations in a number of different contexts.
The main result of the paper, i.e.Theorem 2, provides the mentioned regularity criterion for t u in L p -norm, 4 ≤ p < 9 , in terms of 3 in L -norm, = (p) (see (6) below), within reasonably low regularity requirements on the initial data (u 0 , 0 ).
We also mention that, recently, regularity criteria of the type considered in the present paper have been successfully used to study (in Sobolev and Besov spaces) 3D Navier-Stokes equations [10,32], 2D and 3D viscous MHD equations [15,16], and the 3D Ericksen-Leslie system [34].

Preliminaries
Denote by x ∶= (x 1 , x 2 , x 3 ) a generic point in ℝ 3 and L a number in ℝ + * ∶= (0, +∞) .We consider the case of periodic boundary conditions and the equations in (1) are set in a 3D torus 3 of size L: We set T 3 ∶= 2 ℤ 3 ∕L , and 3 as the torus given by the quotient We use classical Lebesgue spaces L p ∶= L p ( 3 ) , p ≥ 1 , Sobolev spaces W k,p ∶= W k,p ( 3 ) , k non-negative integer, p as before, and H k ∶= W k,2 with mean value equal to zero (essentially, for having at disposal the Poincaré inequality).We denote by ‖ ⋅ ‖ the L 2 -norm and, similarly, ‖ ⋅ ‖ p ∶= ‖ ⋅ ‖ L p denotes the L p -norm.Moreover, given X a real Banach space with norm ‖ ⋅ ‖ X , we will use the customary Bochner spaces L q (0, T;X) , q ∈ ℕ , with norm denoted by ‖ ⋅ ‖ L q (0,T;X) , and we will also make use of the spaces C([0, T]; X) and C 1 ([0, T];X).
Because we deal with divergence-free velocity field, we also define for a general exponent s ≥ 0 , the following spaces and H ∶= H 0 .Let us also recall the notation for dual spaces, i.e. (H s ) � = H −s and (H s ) � = H −s .
For v ∈ H s , we can expand such a field in Fourier series as v(x) = ∑ k∈T * 3 ŵk e ik⋅x where k ∈ T * 3 is the wave number, and the Fourier coefficients are defined by We mention only the scalar case for v ∈ H s , the notation translate accordingly in the vector case ( v ∈ H s ).The H s norms are defined by ‖v‖ �k� 2s ŵk ⋅ vk where vk denotes the complex conjugate of vk .To have real valued vector fields, we impose ŵ−k = ŵk for any k ∈ T * 3 and for any field denoted by w.Observe that, given w = ∑ k∈T * 3 ŵk e ik⋅x ∈ H s , the inverse of the Helmholtz operator G = (A ) −1 can be expressed, in terms of a Fourier series, as follows Loosely speaking, the action of G makes to gain two derivatives to w (for more details see, e.g., [2,5]).
In the sequel, we will use the same notation for scalar and vector-valued functions, as well as for related spaces, since no ambiguity occurs.Also, we will denote by c, or by C, a generic constant which may change from line to line.As a further matter of notation, for the remainder of the paper we will denote t u by u t , and t by t .

Weak Solutions
Let us recall the following definition (see [25]) For a given T > 0 , we say that (u, ) is a weak solution of the problem (1), on the interval [0, T], if it satisfies Eq. ( 1) 1 in the sense of L 2 (0, T;H 1 ) , and Eq. ( 1) 2 in the sense of L 2 (0, T;H −1 ) .Moreover, u ∈ C 1 ([0, T];H 1 ) , and The following result provides an existence criterion for the considered system.
In reference [25], this result is proved starting from initial data u 0 ∈ H 3 .However, an analogous existence result can be obtained starting from u 0 ∈ H i , i ≥ 1 (see [9], see also [1,Theorem 2.1] for the case of the 3D Euler-Voigt equations).

A Priori Estimates
In the following we proceed formally and establish a priori estimates.The procedure actually goes through the use of a suitable Galerkin approximation scheme (see, e.g., [5]), which however we do not report here for the sake of conciseness.
Multiplying the Eq. ( 1) 1 by u and the equation (1) 2 by , and integrating over 3 , we obtain and From the second equation it follows that ‖ (t)‖ 2 ≤ ‖ 0 ‖ 2 for 0 ≤ t ≤ T .Then, for the first equation, we have and the fact that u ∈ L ∞ (0, T;H 1 ) follows by a direct application of Gronwall's lemma.Now, assuming 0 ∈ L p , p ≥ 2 , and taking product of (1) 2 against | | p−2 and inte- grating over 3 , we obtain where we used the identity Therefore, we have the following bound (which is still valid when = 0 ) for , i.e. with 0 ≤ t ≤ T , and T > 0.

Regularity Criterion for the Inviscid Boussinesq-Voigt Model
This section is devoted to the proof of the following regularity result Theorem 2 Let u 0 , 0 ∈ H 1 × L p , with 4 ≤ p < 9 .Given T > 0 , let (u, ) be the weak solution of (1).If the temperature , and the partial derivative of the pressure 3 verify the condition then the solution (u, ) of the problem (1) is such that and in particular u t ∈ C([0, T];H 1 ) ∩ L p (0, T;L p ) .Moreover, for any ∈ (0, 1) , it holds that u t ∈ L p (0, T;W Remark 1 Since we already have at disposal the global regularity stated in Theorem 1 (in particular u ∈ C 1 ([0, T];H 1 ) ), the above result goes in a quite different direction than that of other papers making use of similar logarithmic criteria.In fact, here we show some further properties for u t not coming directly from the definition of weak solution, under the assumptions of initial data (u 0 , 0 ) ∈ H 1 × L p , 4 ≤ p < 9 .When p > 6 , the regularity of u t ∈ C([0, T];H 1 ) is, in principle, no longer sufficient to control the time-integral term in the inequality (7), and it does not even produce the last inclusion.
In proving Theorem 2, we will use the following estimate (see, e.g., [13,18]): where the parameters , and satisfy the relations Let us also recall, for w ∈ H p , p ≥ 2 , the Poincaré inequality

Proof of Theorem 2
Multiplying Eq. (1) 1 by |u t | p−2 u t and integrating over 3 , after an integration by parts, we have By using Hölder's and Young's inequalities for the first term on the right-hand side of the above relation, and recalling the divergence-free nature of u t , we get Here, we require p ≥ 4 .Now, we have to estimate the two integral terms in the right-hand side of (10).Let us start with I 2 .We have that where we used twice Hölder's inequality, first with exponents p = 3 and q = 3∕2 , and subsequently with exponents p = 4∕3 and q = 4 .In particular, the following inequality emerges Exploiting the embedding H 1 ↪ L 6 along with Poincaré's inequality, the last factor in the above formula can be estimated as follows By plugging such a relation into (11) and using Hölder's inequality, we reach where in the last step we used again H 1 ↪ L 6 along with Poincaré's inequality.Now, consider I 1 .We have 6 .
Hence, we reach where we employed Hölder's and Young's inequalities.Now, we can use this last inequality to make explicit the terms, involving the spacederivatives of , needed to reach (6).To do so, we follow the same line of reasoning as in the proof of [28,Theorem 2.2].Taking the operator div (i.e.divv = ∇ ⋅ v) on both sides of Eq. ( 1) 1 , we have As a consequence, we get Applying the inverse of the Helmholtz operator, i.e.G , to both sides of the above relation, we get then, taking the L q -norm, q > 1 , to both sides of ( 14), we reach Remark 2 Here, the procedure to get the estimate ( 15) is formal, but after properly selecting q = 3∕2 (see (17) below), we obtain the actual control on ∇ in ) and even more, and consequently, by using the classical regularity theory for the Poisson equation in the periodic setting, we obtain ∇ ∈ L ∞ (0, T;L 3∕2 ) (see, e.g., [1]).

3
and for = 3∕2 we find after using inequality (5) and taking C = C(‖ 0 ‖ p ) .Thus, from ( 16) and ( 17) we get and plugging such a control into (13), we obtain Remark 3 With the above choices for the positive parameters , , and , the requirements in (9) are satisfied.Indeed, for = p , with p ≥ 4 , due to (10), we have In particular, we also have where = 3∕2 as done in (17).Taking 4 ≤ p < 9 the above condition is verified.
Finally, by inserting ( 12) and ( 18) into (10), and taking = , we get and hence Now, let us rewrite and improve this control in order to get the claimed regularity criterion.As a consequence of relation ( 4) and inequality (20), we have where, in the right-hand side, to keep the notation compact we omit the explicit dependence on t for the various terms taken in space-norm.Define the auxiliary quantity L as By Young's inequality for 1 and ‖ ‖ p , with exponents p and p∕(p − 1) , we have and L can be rewritten as follows where Hence, from (21) we obtain for all t ∈ [0, T] .By Gronwall's lemma, we have ( 21) As a consequence of the regularity of u ∈ L (0, T, H 1 ) , we find and hence, up multiply both sides by e and then log them, we obtain where in the last inequality we used again the uniform control on ‖∇u‖ in [0, T] and we actually have that C(T) = C(‖u 0 ‖ 1,2 , ‖ 0 ‖, T) .Moreover, by Gronwall's lemma, we obtain where C = C(‖u 0 ‖ 1,2 , ‖ 0 ‖ p , T) .This relation, together with (6), ( 5) and (19), implies which gives (7).Now, set The above calculations, along with (20), also provide In particular, the boundedness of J p (u t ) implies that u t belongs also to a specific type of Besov space, i.e. the Nikol'skiĭ space N 2 p ,p ∶= B 2 p ,p ∞ (see [21]), and that where c > 0 is depending only on the exponent p and on the space-domain.For any ∈ (0, 1) , we have the following embeddings (see [21,29])