THE STOCHASTIC PRIMITIVE EQUATIONS WITH NON-ISOTHERMAL TURBULENT PRESSURE ANTONIO AGRESTI MATTHIAS HIEBER AMRU HUSSEIN AND MARTIN SAAL

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THE STOCHASTIC PRIMITIVE EQUATIONS WITH

NON-ISOTHERMAL TURBULENT PRESSURE

ANTONIO AGRESTI, MATTHIAS HIEBER, AMRU HUSSEIN, AND MARTIN SAAL

Abstract. In this paper, we introduce and study the primitive equations with non-isothermal

turbulent pressure and transport noise. They are derived from the Navier-Stokes equations

by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The

temperature dependence of the turbulent pressure can be seen as a consequence of an additive

noise acting on the small vertical dynamics. For such a model we prove global well-posedness in

H1where the noise is considered in both the Itˆo and Stratonovich formulations. Compared to

previous variants of the primitive equations, the one considered here presents a more intricate

coupling between the velocity ﬁeld and the temperature. The corresponding analysis is seriously

more involved than in the deterministic setting. Finally, the continuous dependence on the initial

data and the energy estimates proven here are new, even in the case of isothermal turbulent

pressure.

Contents

1. Introduction 1

2. Physical derivations 7

3. Local and global well-posedness 12

4. Proof of Theorems 3.4, 3.6 and 3.7 20

5. Basic estimates 24

6. The main intermediate estimate 28

7. Proof of Proposition 4.2 48

8. Stratonovich formulation 50

References 53

1. Introduction

In this paper, we introduce and study the stochastic primitive equation with non-isothermal

turbulent pressure and transport noise. The primitive equations are one of the fundamental

models for geophysical ﬂows used to describe oceanic and atmospheric dynamics. They are derived

from the Navier-Stokes equations on domains where the vertical scale is much smaller than the

horizontal scale by the small aspect ratio limit. Additional information for the various versions

of the deterministic primitive equations can be found, e.g. in [Ped87,Val06]. The introduction of

Date: November 4, 2024.

2010 Mathematics Subject Classiﬁcation. Primary 35Q86; Secondary 35R60, 60H15, 76M35, 76U60.

Key words and phrases. stochastic partial diﬀerential equations, primitive equations, global well-posedness,

gradient noise, stochastic maximal regularity, turbulent ﬂows, Kraichnan’s turbulence, thermal ﬂuctuations.

Antonio Agresti has received funding from the European Research Council (ERC) under the European Union’s

Horizon 2020 research and innovation programme (grant agreement No 948819) . Antonio Agresti is a

member of GNAMPA (INδAM).

Matthias Hieber gratefully acknowledges the support by the Deutsche Forschungsgemeinschaft (DFG) through the

Research Unit 5528 – project number 500072446.

Amru Hussein has been supported by Deutsche Forschungsgemeinschaft (DFG) – project number 508634462 and

by MathApp – Mathematics Applied to Real-World Problems – part of the Research Initiative of the Federal State

of Rhineland-Palatinate, Germany.

Martin Saal has been supported by Deutsche Forschungsgemeinschaft (DFG) – project number 429483464.

arXiv:2210.05973v3 [math.AP] 1 Nov 2024

2 AGRESTI, HIEBER, HUSSEIN, AND SAAL

additive and multiplicative noise into models for geophysical ﬂows can be used on the one hand

to account for numerical and empirical uncertainties and errors and on the other hand as subgrid-

scale parameterizations for data assimilation, and ensemble prediction as described in the review

articles [Del04,FOB`14,Pal19]. The primitive equations with non-isothermal turbulent pressure

introduced here present a more intricate interplay between the velocity ﬁeld and the temperature

which leads to serious mathematical complications compared to the deterministic situation, see

e.g. [CT07,HH20]. The same diﬃculties also appear when comparing previously studied stochastic

perturbations of the primitive equations (see e.g. [AHHS24,BS21,DGHT11,DGHTZ12] and the

references therein) with the one considered here. A discussion of these diﬃculties can be found in

Subsection 1.1 below. The presence of the temperature in the balance for the turbulent pressure

can be thought of as the large-scale eﬀect of thermal ﬂuctuations acting on the small vertical

dynamics. From a modelling point of view, a non-isothermal turbulent pressure may provide a

new perspective on the contribution of the temperature on geophysical ﬂows ruled by the primitive

equations. For instance, we hope that the model introduced in the current paper can be used in

the study of the inﬂuence of thermal ﬂuctuations on oceanic streams. As in [AHHS24], we also

consider dynamics driven by transport noise. The latter was ﬁrst introduced by R.H. Kraichanan

in the study of turbulent ﬂows [Kra68,Kra94], and it has been widely studied in the context

of the Navier-Stokes equations, see [HLN21,MR01,MR04] for a physical justiﬁcation and also

[AV24b,BCF91,BCF92,Fla08,HLN19,MR05] and the references therein for related mathematical

results. Let us stress that the diﬃculties arising from the non-isothermal turbulent pressure are

still present in the absence of transport noise, see Subsection 1.1 for details.

The primitive equations with non-isothermal turbulent pressure in the domain O“T2ˆp´h, 0q,

where hą0 and T2denotes the two-dimensional ﬂat torus, are given by the following system:

dv´∆vdt“”´∇HP´ pv¨∇Hqv´wB3v`Fvıdt

`ÿ

ně1”pϕn¨∇qv´∇Hr

Pn`Gv,nıdβn

(1.1a)

dθ´∆θdt“”´ pv¨∇Hqθ´wB3θ`Fθıdt`ÿ

ně1”pψn¨∇qθ`Gθ,nıdβn

t,(1.1b)

B3P`κθ “0,(1.1c)

B3r

Pn`σnθ“0,(1.1d)

divHv` B3w“0,(1.1e)

vp0,¨q “ v0, θp0,¨q “ θ0.(1.1f)

Here κ, σnand ϕn“ pϕj

nq3

j“1, ψn“ pψj

nq3

j“1are assigned maps. Moreover v“ pvkq2

k“1:r0,8q ˆ

ΩˆOÑR2denotes the horizontal component of the unknown velocity ﬁeld u“ pv, wqand

w:r0,8q ˆ ΩˆOÑRthe vertical one, P:r0,8q ˆ ΩˆOÑRthe unknown pressure,

Pn:r0,8q ˆ ΩˆOÑRthe components of the unknown turbulent pressure and θ:r0,8q ˆ

ΩˆOÑRthe unknown temperature, respectively. Finally, pβn

t:tě0qně1is a sequence of

independent standard Brownian motions on a given ﬁltered probability space pΩ,A,pFtqtě0,Pq,

and pFv, Fθ, Gv,n, Gθ,nqare given maps possibly depending on pv, θ, ∇v, ∇θq. These describe

deterministic and stochastic forces, they also take into account lower-order eﬀects like the Coriolis

force. The reader is referred to Subsection 1.5 for the unexplained notation.

The problem (1.1) is supplemented with the following boundary conditions

B3vp¨,´hq “ B3vp¨,0q “ 0 on T2,(1.2a)

B3θp¨,´hq “ B3θp¨,0q ` αθp¨,0q “ 0 on T2,(1.2b)

where αPRis given and

(1.3) wp¨,´hq “ wp¨,0q “ 0 on T2.

PRIMITIVE EQUATIONS WITH NON-ISOTHERMAL TURBULENT PRESSURE 3

Actually, in our main results, we consider a generalization of the system in (1.1), see (3.1) in

the main text. Moreover, our arguments also cover the case where the boundary conditions (1.2)

are replaced by periodic ones. Further comments are given in Remark 3.13.

The aim of this paper is to show the global well-posedness in the strong setting (both analytically

and probabilistically) of the system (1.1)-(1.3), see Theorems 3.6 and 3.7. In these results, the

noise is understood in the Itˆo-sense. In Section 8we also discuss the case of Stratonovich noise. In

stochastic ﬂuid mechanics, and in particular, for geophysical ﬂows, the Stratonovich formulation

of the noise is relevant, and it is seen as a more realistic model compared to the Itˆo one, see e.g.

[BF20,DP24,FP22,FOB`14,HL84,MR01,MR04,Wen14]. From an analytic point of view, the

Stratonovich noise is not more diﬃcult than the Itˆo one and, at least formally, one can convert

the Stratonovich formulation into the Itˆo one up to some additional corrective terms. The global

well-posedness of (1.1) in the strong setting with Stratonovich noise is proved in Section 8.

For the reader’s convenience, we state here a simpliﬁed version of the Theorems 3.6 and 3.7.

Below we write ϕjdef

“ pϕj

nqně1,ψjdef

“ pψj

nqně1and R`def

“ p0,8q.

Theorem 1.1 (Simpliﬁed version).Let κbe constant, pσnqně1Pℓ2,Gk

v,n “Gθ,n “0,Fθ“0,

and let Fv“k0pv2,´v1qfor k0PRbe the Coriolis force. For all ně1let the maps

ϕn, ψn:R`ˆΩˆOÑR3

be PbB-measurable, and let for some δą0and all jP t1,2,3ube

ϕj, ψjPL8pR`ˆΩ; H1,3`δpO;ℓ2qq.

Suppose that pϕj

n, ψj

nqare independent of x3for jP t1,2u. Furthermore, assume that there exists

νP p0,2qsuch that, a.s. for all tPR`,xPOand ξPR3the parabolicity conditions

ně1´ÿ

1ďjď3

ϕj

npt, xqξj¯2ďν|ξ|2and ÿ

ně1´ÿ

1ďjď3

ψj

npt, xqξj¯2ďν|ξ|2

hold. Then for each v0PL0

F0pΩ; H1pOqq and θ0PL0

F0pΩ; H1pOqq the following hold:

(1) There exists a unique global strong solution pv, θqto (1.1)-(1.3)satisfying

pv, θq P L2

locpr0,8q;H2

NpOq ˆ H2

RpOqq X Cpr0,8q;H1pOq ˆ H1pOqq a.s.

(2) For all TP p0,8q and all γąee,

P´sup

tPr0,T s

}vptq}2

H1`ˆT

}vptq}2

H2dtěγ¯ÀT

1`E}v0}4

H1`E}θ0}4

log log logpγq,

P´sup

tPr0,T s

}θptq}2

H1`ˆT

}θptq}2

H2dtěγ¯ÀT

1`E}v0}4

H1`E}θ0}4

log log logpγq.

(3) The assignment pv0, θ0q ÞÑ pv, θqis continuous in probability in the sense of Theorem 3.7.

The reader is referred to Subsections 1.5 and 3.1 for the deﬁnition of PbB-measurable,

F0pΩ; Xqand the notation for the function spaces. In the above, we have not speciﬁed the

unknowns w,Pand r

Pnas they are uniquely determined by vand θdue to the divergence-free

condition and the hydrostatic Helmholtz projection. The reader is referred to [AHHS24, Section

1] for comments on the relation between the regularity of the transport noise considered in this

paper and Krainchan’s noise.

Physical motivations for the independence of pϕj

n, ψj

nqon the x3-coordinate for jP t1,2uare

discussed in Remarks 2.2 and 2.3. In a nutshell, the small aspect ratio limit (i.e. the hydrostatic

approximation discussed for the deterministic setting in [FGH`20,LT19]) shows that the primitive

equations can be derived by taking the limit εÓ0 of the anisotropic Navier-Stokes equations on

a thin domain T2ˆ p´ε, 0q(see Figure 1), and therefore the variability in the vertical direction

of the coeﬃcients disappear in the limit. Hence, the independence of pϕj

n, ψj

nqon x3for jP t1,2u

is justiﬁed. In particular, the situation for geophysical ﬂows is diﬀerent from usual turbulence

models concerning Navier-Stokes equations [BE12,Tab02].

4 AGRESTI, HIEBER, HUSSEIN, AND SAAL

The logarithmic bounds of Theorem 1.1(2) seem rather weak. However, compared to the

estimates in the deterministic setting (see e.g. [CT07]), even in the absence of noise, it does not

seem possible to obtain in (2) more than a log log-decay due to three applications of Grownall’s

inequality. Moreover, it is unclear how to improve the estimates in (2) without enforcing regularity

assumptions on the noise. The reader is referred to the text below Theorem 3.6 and to Remark

3.10 for more details. The bounds in Theorem 1.1(2) remind us of the estimates obtained in

[GHKVZ14, Theorem 4.2], where the authors proved logarithmic moment bounds in H2pOqunder

additional assumptions on the noise. In particular, in [GHKVZ14], it is not possible to consider

gradient or transport type noises (in particular, this forces σn”0, cf. Subsection 1.1 below).

However, it seems that there is no direct relation between the estimates of (2) and the above-

mentioned estimate of [GHKVZ14]. In the latter, the authors used logarithmic moment bounds

to prove the existence of ergodic invariant measures in H1pOq. The extension of such result to the

system (1.1) goes beyond the scope of this manuscript. Finally, let us mention that the continuous

dependence on the initial data in (3) readily implies the Feller property for (1.1) which is a ﬁrst

step in the proof of the existence of ergodic measures, and it is based on the energy estimates in

(2). The reader is referred to Remark 3.8 for more details on the Feller property.

1.1. Novelties and description of the main diﬃculty. Compared to the results in [AHHS24],

the major novelty of the current work is the presence of σn‰0. Here we explain the main analytic

diﬃculty behind this fact. For simplicity, as in Theorem 1.1, in this subsection we assume that

pσnqně1Pℓ2is constant. Note that (1.1d) yields, for all pxH, x3q P O(here and below xHPT2

and x3P ph, 0qdenote the horizontal and vertical variables, respectively) and tPR`,

Pnpt, xH, x3q “ rpnpt, xHq ` σnˆx3

´h

θpt, xH, ζqdζ,

where rpndepends only on xHPT2(typically referred as turbulent surface pressure). Using the

above identity in (1.1a), the following gradient noise term appears in the v-dynamics:

(1.4) ÿ

ně1

σnˆx3

´h

∇Hθpt, xH, ζqdζdβn

where ∇H“ pB1,B2q. In particular, as maximal L2-regularity estimates show (see e.g. [AHHS24,

Proposition 6.8] or Lemma 4.1), to obtain a-priori L8

tpH1

xqXL2

tpH2

xq-bounds for v(and hence global

existence for (1.1)), one needs L8

tpH2

xq-bounds for θ. This is dramatically diﬀerent from the case

of isothermal turbulent pressure (i.e. σn”0), where it is suﬃcient to show L8

tpH1

xq-bounds for

θto obtain L8

tpH1

xq X L2

tpH2

xq-estimates for v(see [AHHS24, Section 5]). Since L8

tpH1

xq-bounds

for θfollow from standard energy estimates, from an analytic point of view, the proof of global

existence of strong solutions in the case σn”0 is essentially independent of the θ-dynamics, cf.

[AHHS24, Section 5]. This is not the case for (1.1) with σn‰0 where the coupling between

the evolution of vand the one of θis more subtle and vcannot be decoupled from θin the

tpH1

xq X L2

tpH2

xq-estimates. Let us remark that these diﬃculties are also present even in the

absence of transport noise in (1.1a)-(1.1b), i.e. having ϕn“ψn”0.

Before going further, let us mention some more diﬀerences compared with [AHHS24]. The

energy estimates and the continuous dependence on the initial data of Theorem 1.1(2)-(3) were

not contained in [AHHS24] and are based on the use of a recent stochastic Grownall’s lemma

proven in [AV24a, Appendix A]. Finally, due to the presence of the term (1.4) in the v-dynamics

(1.1a), we cannot allow for a strong-weak setting as in [AHHS24, Section 3], i.e. considering (1.1a)

in the strong setting (in the sense of Sobolev spaces) and (1.1b) in the weak analytic one. Hence

we only consider the strong setting, i.e. both (1.1a) and (1.1b) are understood in the strong sense.

To conclude, let us anticipate that in Theorems 3.6 and 3.7 we can even allow pσnqně1to depend

on pt, ω, xHq, but not on x3. The physical relevance of the x3-independence of σnis discussed in

Remark 2.1. As for the x3-independence of ϕj

n, ψj

nfor jP t1,2uin Theorem 1.1, the justiﬁcation

is via the hydrostatic approximation.

PRIMITIVE EQUATIONS WITH NON-ISOTHERMAL TURBULENT PRESSURE 5

1.2. On the physical derivations. Besides the symmetry of the relations (1.1c)-(1.1d), to mo-

tivate the presence of the non-isothermal balance (1.1d), in Section 2we provide two physical

derivations of (1.1). In both derivations the condition (1.1d) appears naturally. Following the

strategy used in the deterministic framework, we derive (1.1) by employing suitable stochastic

variants of the Boussinesq and the hydrostatic approximations. In both cases, the main ideas are

in the Boussinesq approximation. In ﬂuid dynamics, the Boussinesq approximation is employed

in the study of buoyancy-driven ﬂows (also referred to as natural convection), and it is typically

a good approximation in the context of oceanic ﬂows. Roughly speaking, the idea behind the

Boussinesq approximation is that, in a natural convection regime, the role of the compressibility

is negligible in the inertial and the convection terms, but not in the gravity term. More precisely,

in the compressible Navier-Stokes equations one assumes

(1.5) pρ´ρrq`BtU´ pU¨∇qU˘«0

for some reference density ρrą0. Here, Uand ρdenote the velocity and density of the ﬂuid,

respectively. In our ﬁrst approach to derive (1.1), borrowing some ideas from stochastic climate

modeling (see e.g. [MTVE01]), we replace the right hand side in (1.5) by a noisy term:

(1.6) pρ´ρrq`BtU´ pU¨∇qU˘«ÿ

ně1“pρ´ρrqkn´∇r

Qn‰9

βn

Here knPR3is given and r

Qn’s are turbulent pressures that make the modelling assumption on the

right-hand side in (1.6) compatible with the divergence-free condition which follows from assuming

ρ«ρrin the density balance, cf. (1.1e) and (2.3b).

At least formally, the right-hand side in (1.6) has zero expectation (if we interpret the noise in

the Itˆo formulation). Hence, the approximation in (1.6) is consistent with (1.5) when considering

expected values, and it can be seen as a reﬁnement of the usual Boussinesq approximation. Em-

ploying the approximation (1.6) and the hydrostatic approximation used in the deterministic case

(see e.g. [AG01,FGH`20,LT19]) one obtains (1.1) where σn“ ´λk3

nfor some λPR, where k3

nis

the third component of knPR3. The reader is referred to Subsection 2.2 for more details.

Our second derivation of (1.1) is based on a two-scale interpretation of the primitive equations.

Indeed, as the small aspect ratio limit suggests, in the context of the primitive equations the

horizontal and the vertical directions can be thought of as small and large scales, respectively.

Hence, as usual in the literature (see e.g. [BE12,DP24,FP20,MTVE01]), it is physically reasonable

to consider an additive noise (per unit of mass) on the small-scale dynamics. Eventually, such

choice and a further variant of the Boussinesq and hydrostatic approximations lead to the system

(1.1). Details on this approach can be found in Subsection 2.3.

1.3. Comments on the literature. Here we collect further references to the literature on prim-

itive equations. Since the literature is extensive, we restrict to literature particularly relevant to

this work, referring to the references in the cited works for a more extensive and complete overview.

In the deterministic setting, the primitive equations were ﬁrst studied by J. L. Lions, R. Teman,

and S. Wang in a series of articles [LTW92a,LTW92b,LTW93]. There, the authors proved the

existence of global Leray-Hopf type solutions for initial data v0PL2. As for the Navier-Stokes

equations, the uniqueness of such solutions is still open. Under additional regularity assumptions

uniqueness holds, see [Ju17]. In the deterministic setting, a breakthrough result has been proven

independently by C. Cao and E.S. Titi [CT07] and R.M. Kobelkov [Kob07] where they proved the

global well-posedness of the primitive equations via L8

tpH1

xq X L2

tpH2

xqa-priori estimates provided

v0PH1. See also [KZ07] for other boundary conditions. The results of [CT07,Kob07] have been

extended to the Lp-setting by the second author and T. Kashiwabara in [HK16]. Further results

can be found in [GGH`20a,GGH`20b,GGH`21]. See also [HH20] for an overview.

Stochastic versions of the primitive equations have been studied by several authors. Global

well-posedness for pathwise strong solutions has been established for multiplicative white noise

in time by A. Debussche, N. Glatt-Holtz and R. Temam in [DGHT11] and the same authors

with M. Ziane in [DGHTZ12]. There, the authors used a Galerkin approach to ﬁrst show the

existence of martingale solutions, and then strong existence is deduced via pathwise uniqueness

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THESTOCHASTICPRIMITIVEEQUATIONSWITHNON-ISOTHERMALTURBULENTPRESSUREANTONIOAGRESTI,MATTHIASHIEBER,AMRUHUSSEIN,ANDMARTINSAALAbstract.Inthispaper,weintroduceandstudytheprimitiveequationswithnon-isothermalturbulentpressureandtransportnoise.TheyarederivedfromtheNavier-Stokesequationsbyemployingstochasticv...

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THE STOCHASTIC PRIMITIVE EQUATIONS WITH NON-ISOTHERMAL TURBULENT PRESSURE ANTONIO AGRESTI MATTHIAS HIEBER AMRU HUSSEIN AND MARTIN SAAL

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