Stationary solutions in thermodynamics of stochastically forced fluids
aa r X i v : . [ m a t h . A P ] F e b STATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLYFORCED FLUIDS
DOMINIC BREIT, EDUARD FEIREISL, AND MARTINA HOFMANOV´A
Abstract.
We study the full Navier–Stokes–Fourier system governing the motion of a gen-eral viscous, heat-conducting, and compressible fluid subject to stochastic perturbation.The system is supplemented with non-homogeneous Neumann boundary conditions for thetemperature and hence energetically open. We show that, in contrast with the energeticallyclosed system, there exists a stationary solution. Our approach is based on new global-in-time estimates which rely on the non-homogeneous boundary conditions combined withestimates for the pressure. Introduction
It is a common believe that the behaviour of turbulent fluid flows can be fully characterizedby a steady state of the system (driven by a suitable stochastic forcing to substitute for possibleperturbations due to changes in the boundary data), which is approached asymptotically forlarge times. Mathematically speaking this gives rise to an invariant measure of the underlyingsystem. This is well-understood for the 2D incompressible stochastic Navier–Stokes equations,cf. [10, 12, 19, 16], where uniqueness is well-known.If uniqueness is not at hand, even the definition of an invariant measure becomes ambiguous,and one rather studies stationary solutions of the dynamics: solutions with a probability lawwhich does not change in time. This law serves as a substitute for an invariant measure. Theexistence of stationary solutions to the 3D incompressible stochastic Navier–Stokes equationsis a nowadays classical result from [11]. More recently a counterpart for the compressiblestochastic Navier–Stokes equations has been established in [4]. It is interesting to note that inboth cases stationarity provides a certain regularising effect on the solutions (see also [13] inconnection with this).One may think that adding further physical principles such as the possibility of heat transfer completes the picture. The stochastic Navier–Stokes–Fourier equations haven been studied in[1] and the existence of weak martingale solutions has been shown. They describe the motionof a general viscous, heat-conducting, and compressible fluid subject to stochastic perturbationbased on the
Second Law of Thermodynamics via an entropy balance as in [8] (see also [20]for an alternative approach based on the internal energy balance due to [7]). Supplementedwith homogeneous Neumann boundary conditions for the temperature this is an energeticallyclosed system. The mechanical energy which is lost as dissipation is transfered into heat and,different to the incompressible or the isentropic Navier–Stokes equations, weak solutions areknown to satisfy an energy equality. The latter one shows that the noise is constantly addingenergy to the system such that it can never reach a steady state and, as shown in [1, Section7], stationary solutions do not exist. Since this is physically not acceptable we are looking for
Date : February 9, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Compressible fluids, stochastic Navier–Stokes–Fourier system, stationary solution.The research of E.F. leading to these results has received funding from the Czech Sciences Foundation(GAˇCR), Grant Agreement 21–02411S. The Institute of Mathematics of the Academy of Sciences of the CzechRepublic is supported by RVO:67985840. The stay of E.F. at TU Berlin is supported by Einstein Foundation,Berlin.M.H. gratefully acknowledges the financial support by the German Science Foundation DFG via the Col-laborative Research Center SFB1283. a physical principle which can counteract the energy creation by the noise.Different to [1] we consider in this paper an energetically open version of the stochasticNavier–Stokes–Fourier equations, where heat can drain through the boundary, see (1.5) below.The time evolution of the fluid in the reference physical domain Q ⊂ R is governed by thefollowing set of equations: d ̺ + div( ̺ u ) d t = 0 , (1.1a) d( ̺ u ) + [div( ρ u ⊗ u ) + ∇ p ( ̺, ϑ )] d t = div S ( ϑ, ∇ u ) d t + ̺ F ( ̺, ϑ, u ) d W, (1.1b) d( ̺e ( ̺, ϑ )) + (cid:2) div( ̺e ( ̺, ϑ ) u ) + div q ( ϑ, ∇ ϑ ) (cid:3) d t (1.1c) = (cid:2) S ( ϑ, ∇ u ) : ∇ u − p ( ̺, ϑ ) div u (cid:3) d t, where W is a cylindrical Wiener process and the diffusion coefficient F can be identified witha sequence ( F k ) k ≥ satisfying a suitable Hilbert-Schmidt assumption, see Section 2 for theprecise definitions. Here ̺ denotes the density of the fluid, ϑ the absolute temperature and u the velocity field. For the viscous stress tensor we suppose Newton’s rheological law S = S ( ϑ, ∇ u ) = µ ( ϑ ) (cid:16) ∇ u + ∇ u T −
23 div u I (cid:17) + η ( ϑ ) div u I . (1.2)The internal energy (heat) flux is determined by Fourier’s law q = q ( ϑ, ∇ ϑ ) = − κ ( ϑ ) ∇ ϑ = −∇K ( ϑ ) , K ( ϑ ) = Z ϑ κ ( z ) d z. (1.3)The thermodynamic functions p and e are related to the (specific) entropy s = s ( ̺, ϑ ) throughGibbs’ equation(1.4) ϑDs ( ̺, ϑ ) = De ( ̺, ϑ ) + p ( ̺, ϑ ) D (cid:16) ̺ (cid:17) for all ̺, ϑ > , where D denotes the total derivative with respect to ( ̺, ϑ ). We supplement (1.1)–(1.4) withthe boundary conditions (see also [9])(1.5) u | ∂Q = 0 , q · n | ∂Q = d ( ϑ )( ϑ − Θ ) , and fix the total mass Z Q ̺ d x = M , where Θ ∈ L ( ∂Q ) is strictly positive, M > d, d > dϑ ≤ d ( x, ϑ ) ≤ dϑ for all ( x, ϑ ) ∈ ∂Q × [0 , ∞ ) . (1.6)In view of Gibb’s relation (1.4), the internal energy equation (1.1c) can be rewritten in theform of the entropy balanced( ̺s ) + h div( ̺s u ) + div (cid:16) q ϑ (cid:17)i d t = σ d t (1.7)with the entropy production rate σ = 1 ϑ (cid:16) S : ∇ u − q · ∇ ϑϑ (cid:17) . (1.8)In view of possible singularities, it is convenient to relax the equality sign in (1.8) to theinequality σ ≥ ϑ (cid:16) S : ∇ u − q · ∇ ϑϑ (cid:17) . (1.9)The system is augmented by the total energy balance(1.10) d Z Q (cid:20) ̺ | u | + ̺e (cid:21) d x = Z Q ̺ F · u d W + X k ≥ Z Q ̺ | F k | d x d t − Z ∂Q d ( ϑ )( ϑ − Θ ) d H d t, TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 3 cf. [8, Chapter 2]. In case of a stationary solution applying expectations to (1.10) clearly yields X k ≥ E Z Q ̺ | F k | d x d t = E Z ∂Q d ( ϑ )( ϑ − Θ ) d H d t, meaning energy created by the stochastic forcing can leave through the boundary. The exis-tence theory from [1], which leans on the analysis of the isentropic stochastic Navier–Stokesequations from [5] and the deterministic Navier–Stokes–Fourier equations from [8], can be ap-plied to (1.1)–(1.5) without essential differences. In case of the initial value problem an energyestimate can be derived in terms of the initial data. Looking for stationary solutions, theinitial data is not known and one has to use stationarity instead. In [4] stationarity is usedin combination with pressure estimates to obtain a corresponding estimate for the isentropicproblem. When applying the same strategy to the non-isentropic problem (1.1)–(1.4), sup-plemented with homogeneous boundary conditions for the temperature flux, the temperatureis deemed to grow unboundedly due to the irreversible transfer of the mechanical energy intoheat.Assuming the non-homogeneous boundary conditions (1.5) instead we are able to derive newglobal-in-time energy estimates, see (4.4). The main task is to control the radiation energygiven by aϑ without an information on the initial data. In the case of homogeneous boundaryconditions one can only obtain informations on the temperature gradient which is not enoughto even get estimates for ϑ in L . For the non-homogeneous problem we benefit from theboundary term in the energy balance (1.10). A suitable application of Itˆo’s formula combinedwith Sobolev’s embedding and an interpolation argument allows to control a higher power ofthe temperature in terms of the energy, see (4.4). Finally, we derive some pressure estimateby means of the Bogovskii operator in (4.6) and (4.16) to close the argument and to obtainuniform-in-time estimates for the total energy. This leads to our main result which is theexistence of stationary martingale solutions to (1.1)–(1.5), see Theorem 2.1 for the precisestatement.In order to make the ideas just explained rigorous one has to regularise the system byadding artificial viscosity to the continuity equation (1.1a) ( ε -layer) and add a high power ofthe pressure in the momentum equation (1.1b) ( δ -layer). The resulting system has been solvedin [1] by adding three additional layers. The same tedious strategy has been applied [4] in theconstruction of stationary solutions to the isentropic system. Here, we follow a different strategywith a much simpler proof. Namely, inspired by the approach due to Itˆo-Nisio [17] which werecently also applied to the isentropic system with hard sphere pressure [3], we constructstationary solutions directly on the ε -level. The first step is to show uniform-in-time estimatesfor martingale solutions to the initial value problem. In a second step stationary solutionscan be constructed by the Krylov-Bogoliubov method as the narrow limit of time-averages. Astriking feature of this approach is that stationary solutions are sitting on the trajectory spaceand are approached asymptotically in time by any solution starting with bounded initial data ofcertain moments. With a stationary solution to the approximate system at hand one can proveestimate (4.4) which is uniform in time, ε and δ . It has to be combined with pressure estimateswhich differ on both levels, see (4.6) and (4.16), before one can pass to the limit (both limitshave to be done independently). The limit passage can be performed as in previous papersand stationarity is preserved in the limit.2. Mathematical framework and the main result
Stochastic forcing.
The process W is a cylindrical Wiener process on a separable Hilbertspace U , that is, W ( t ) = P k ≥ β k ( t ) e k with ( β k ) k ≥ being mutually independent real-valuedstandard Wiener processes relative to ( F t ) t ≥ . Here ( e k ) k ≥ denotes a complete orthonormalsystem in U . In addition, we introduce an auxiliary space U ⊃ U via U = (cid:26) v = X k ≥ α k e k ; X k ≥ α k k < ∞ (cid:27) , DOMINIC BREIT, EDUARD FEIREISL, AND MARTINA HOFMANOV´A endowed with the norm k v k U = X k ≥ α k k , v = X k ≥ α k e k . Note that the embedding U ֒ → U is Hilbert-Schmidt. Moreover, trajectories of W are P -a.s.in C ([0 , T ]; U ) (see [6]).Choosing U = ℓ we may identify the diffusion coefficients ( F e k ) k ≥ with a sequence of realfunctions ( F k ) k ≥ , ̺ F ( ̺, ϑ, u )d W = ∞ X k =1 ̺ F k ( x, ̺, ϑ, u )d β k . We suppose that F k are smooth in their arguments, specifically, F k ∈ C ( Q × [0 , ∞ ) × R ; R ) , where(2.1) k F k k L ∞ + k∇ x,̺,ϑ, u F k k L ∞ ≤ f k , ∞ X k =1 f k < ∞ . We easily deduce from (2.1) the following bound k ̺ F k ( ̺, ϑ, u ) k W − k, ( Q ; R ) < ∼ k ̺ F k ( ̺, ϑ, u ) k L ( Q ; R ) < ∼ f k k ̺ k L ( Q ) whenever k > . Accordingly, the stochastic integral Z τ ̺ F d W = ∞ X k =1 Z τ ̺ F k ( ̺, ϑ, u ) d β k can be identified with an element of the Banach space space C ([0 , T ]; W − k, ( Q )), Z Q (cid:18)Z τ ̺ F ( ̺, ϑ, u )d W · ϕ (cid:19) d x = ∞ X k =1 Z τ (cid:18)Z Q ̺ F k ( x, ̺, ϑ, u ) · ϕ d x (cid:19) d β k , ϕ ∈ W k, ( Q ; R ) , k > . Structural and constitutive assumptions.
Besides Gibbs’ equation (1.4), we impose severalrestrictions on the specific shape of the thermodynamic functions p = p ( ̺, ϑ ), e = e ( ̺, ϑ )and s = s ( ̺, ϑ ). They are borrowed from [8, Chapter 1], to which we refer for the physicalbackground and the relevant discussion.We consider the pressure p in the form(2.2) p ( ̺, ϑ ) = p M ( ̺, ϑ ) + a ϑ , a > , p M ( ̺, ϑ ) = ϑ / P (cid:16) ̺ϑ / (cid:17) , (2.3) e ( ̺, ϑ ) = e M ( ̺, ϑ ) + a ϑ ̺ , e M ( ̺, ϑ ) = 32 p M ( ̺, ϑ ) ̺ = 32 ϑ / ̺ P (cid:16) ̺ϑ / (cid:17) , (2.4) s ( ̺, ϑ ) = s M ( ̺, ϑ ) + 4 a ϑ ̺ , s M ( ̺, ϑ ) = S (cid:16) ̺ϑ / (cid:17) , (2.5) S = S ( Z ) , S ′ ( Z ) = − P ( Z ) − ZP ′ ( Z ) Z < , lim Z →∞ S ( Z ) = 0 , where(2.6) P ∈ C [0 , ∞ ) ∩ C (0 , ∞ ) , P (0) = 0 , P ′ ( Z ) > , for all Z ≥ , (2.7) 0 < P ( Z ) − ZP ′ ( Z ) Z < c, for all
Z > , TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 5 and(2.8) lim Z →∞ P ( Z ) Z / = p ∞ > . As shown in [8, Section 3.2] the assumptions above imply that there is c > c − ̺ / ≤ p M ( ̺, ϑ ) ≤ c ( ̺ / + ̺ϑ ) , (2.9) 3 p ∞ ̺ / + aϑ ≤ ̺e ( ̺, ϑ ) , (2.10) 0 ≤ e M ( ̺, ϑ ) ≤ c ( ̺ / + ϑ ) , (2.11)for all ϑ, ̺ >
0. Moreover, there is s ∞ > ≤ s M ( ̺, ϑ ) ≤ s ∞ (1 + | log( ̺ ) | + [log( ϑ )] + ) . (2.12)Finally, for ϑ > ballistic free energy given by H ϑ ( ̺, ϑ ) = ̺ (cid:0) e ( ̺, ϑ ) − ϑs ( ̺, ϑ ) (cid:1) , which satisfies − c ( ̺ + 1) + 14 (cid:0) ̺e ( ̺, ϑ ) + ϑ | s ( ̺, ϑ ) | (cid:1) ≤ H ϑ ( ̺, ϑ ) ≤ c (cid:0) ̺ / + ϑ + 1 (cid:1) (2.13)on account of (2.11), (2.12) and [8, Prop. 3.2]. The viscosity coefficients µ , η are continu-ously differentiable functions of the absolute temperature ϑ , more precisely µ, λ ∈ C [0 , ∞ ),satisfying(2.14) 0 < µ (1 + ϑ ) ≤ µ ( ϑ ) ≤ µ (1 + ϑ ) , (2.15) sup ϑ ∈ [0 , ∞ ) (cid:0) | µ ′ ( ϑ ) | + | λ ′ ( ϑ ) | (cid:1) ≤ m, (2.16) 0 ≤ λ ( ϑ ) ≤ λ (1 + ϑ ) . The heat conductivity coefficient κ ∈ C [0 , ∞ ) satisfies(2.17) 0 < κ (1 + ϑ ) ≤ κ ( ϑ ) ≤ κ (1 + ϑ ) . Finally, we introduce certain regularised versions of p, e, s and κ for fixed δ > p δ ( ̺, ϑ ) = p ( ̺, ϑ ) + δ ( ̺ + ̺ Γ ) ,e M,δ ( ̺, ϑ ) = e M ( ̺, ϑ ) + δϑ, e δ ( ̺, ϑ ) = e ( ̺, ϑ ) + δϑ,s M,δ ( ϑ, ̺ ) = s M ( ϑ, ̺ ) + δ log ϑ, s δ ( ̺, ϑ ) = s ( ̺, ϑ ) + δ log( ϑ ) ,κ δ ( ϑ ) = κ ( ϑ ) + δ (cid:16) ϑ Γ + 1 ϑ (cid:17) , K δ ( ϑ ) = Z ϑ κ δ ( z ) d z. (2.18)2.3. Martingale & stationary solutions.
We start with a rigorous definition of (weak) martingalesolution to problem (1.1)–(1.5) as given in [1], where also the existence of a solution to thecorresponding initial value problem is proved.
Definition 2.1 (Martingale solution) . Let Q ⊂ R be a bounded domain of class C ν , ν > .Then (cid:0) (Ω , F , ( F t ) , P ) , ̺, ϑ, u , W ) is called (weak) martingale solution to problem (1.1) –(1.5) provided the following holds. (a) (Ω , F , ( F t ) , P ) is a stochastic basis with a complete right-continuous filtration; (b) W is an ( F t ) -cylindrical Wiener process; DOMINIC BREIT, EDUARD FEIREISL, AND MARTINA HOFMANOV´A (c) the random variables ̺ ∈ L ([0 , ∞ ); L ( Q )) , ϑ ∈ L ([0 , ∞ ); L ( Q )) , u ∈ L ([0 , ∞ ); W , ( Q ; R )) are ( F t ) -progressively measurable , ̺ ≥ , ϑ > P -a.s.; (d) the equation of continuity Z ∞ Z Q [ ̺∂ t ψ + ̺ u · ∇ ψ ] d x d t = 0;(2.19) holds for all ψ ∈ C ∞ c ((0 , ∞ ) × R ) P -a.s.; (e) the momentum equation Z ∞ ∂ t ψ Z Q ̺ u · ϕ d x d t + Z ∞ ψ Z Q ̺ u ⊗ u : ∇ ϕ d x d t − Z T ψ Z Q S ( ϑ, ∇ u ) : ∇ ϕ d x d t + Z ∞ ψ Z Q p ( ̺, ϑ ) div ϕ d x d t + Z ∞ ψ Z Q ̺ F ( ̺, ϑ, u ) · ϕ d x d W = 0;(2.20) holds for all ψ ∈ C ∞ c (0 , ∞ ) , ϕ ∈ C ∞ c ( Q ; R ) P -a.s. (f) the entropy balance − Z ∞ Z Q [ ̺s ( ̺, ϑ ) ∂ t ψ + ̺s ( ̺, ϑ ) u · ∇ ψ ] d x d t ≥ Z ∞ Z Q ϑ h S ( ϑ, ∇ u ) : ∇ u + κ ( ϑ ) ϑ |∇ ϑ | i ψ d x d t − Z ∞ Z Q κ ( ϑ ) ∇ ϑϑ · ∇ ψ d x d t − Z ∞ Z ∂Q ψ d ( ϑ ) ϑ ( ϑ − Θ ) d H d t (2.21) holds for all ψ ∈ C ∞ c ((0 , ∞ ) × R ) , ψ ≥ P -a.s.; (g) the total energy balance − Z ∞ ∂ t ψ (cid:18)Z Q E ( ̺, ϑ, u ) d x (cid:19) d t = − Z ∞ ψ Z ∂Q d ( ϑ − Θ ) d H d t + Z ∞ ψ Z Q ̺ F ( ̺, ϑ, u ) · u d x d W d x + 12 Z ∞ ψ (cid:18) Z Q X k ≥ ̺ | F k ( ̺, ϑ, u ) | d x (cid:19) d t (2.22) holds for any ψ ∈ C ∞ c (0 , ∞ ) P -a.s. Here, we abbreviated E ( ̺, ϑ, u ) = 12 ̺ | u | + ̺e ( ̺, ϑ ) . In the following we are going to introduce the concept of stationary martingale solutions. Westart with a standard definition of stationarity for stochastic processes with values in Sobolevspaces.
Definition 2.1 (Classical stationarity) . Let ˜ u = { ˜ u ( t ); t ∈ [0 , ∞ ) } be an W k,p ( Q ) -valued mea-surable stochastic process, where k ∈ N and p ∈ [1 , ∞ ) . We say that ˜ u is stationary on W k,p ( Q ) provided the joint laws L (˜ u ( t + τ ) , . . . , ˜ u ( t n + τ )) , L (˜ u ( t ) , . . . , ˜ u ( t n )) on [ W k,p ( Q )] n coincide for all τ ≥ , for all t , . . . , t n ∈ [0 , ∞ ) . As can be seen from Definition 2.1, the velocity u and the temperature ϑ are not stationaryin the sense of Definition 2.1 as they are only equivalence classes in time. Therefore we usethe following definition of stationarity which has been introduced in [4], and applies to randomvariables ranging in the space L q loc ([0 , ∞ ); W k,p ( Q )). The progressive measurability is understood in the sense of random distributions as introduced in [2, Section2.2].
TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 7
Definition 2.2 (Weak stationarity) . Let ˜ u be an L q loc ([0 , ∞ ); W k,p ( Q )) -valued random variable,where Let k ∈ N and p, q ∈ [1 , ∞ ) . Let S τ be the time shift on the space of trajectories givenby S τ ˜ u ( t ) = ˜ u ( t + τ ) . We say that ˜ u is stationary on L q loc ([0 , ∞ ); W k,p ( Q )) provided the laws L ( S τ ˜ u ) , L (˜ u ) on L q loc ([0 , ∞ ); W k,p ( Q )) coincide for all τ ≥ . Definition 2.1 and Definition 2.2 are equivalent as soon as the stochastic process in questionis continuous in time; or alternatively, if it is weakly continuous and satisfies a suitable uniformbound, cf. [4, Lemma A.2 and Corollary A.3]. Furthermore, it can be shown that both notionsof stationarity are stable under weak convergence as can be seen from the following two lemmas(the proofs of which can be found in [4, Appendix]).
Lemma 2.1.
Let k ∈ N , p, q ∈ [1 , ∞ ) and let (˜ u m ) be a sequence of random variables takingvalues in L q loc ([0 , ∞ ); W k,p ( Q ))) . If, for all m ∈ N , ˜ u m is stationary on L q loc ([0 , ∞ ); W k,p ( Q )) in the sense of Definition 2.2 and ˜ u m ⇀ ˜ u in L q loc ([0 , ∞ ); W k,p ( Q )) P -a.s.,then ˜ u is stationary on L q loc ([0 , ∞ ); W k,p ( Q )) . Lemma 2.2.
Let k ∈ N , p ∈ [1 , ∞ ) and let (˜ u m ) be a sequence of W k,p ( Q ) -valued stochasticprocesses which are stationary on W k,p ( Q ) in the sense of Definition 2.1. If for all T > m ∈ N E " sup t ∈ [0 ,T ] k ˜ u m k W k,p ( Q ) < ∞ and ˜ u m → ˜ u in C loc ([0 , ∞ ); ( W k,p ( Q ) , w )) P -a.s.,then ˜ u is stationary on W k,p ( Q ) . In the following we define a stationary martingale solution to (1.1)–(1.5).
Definition 2.3.
A weak martingale solution [ ̺, ϑ, u , W ] to (1.1) – (1.5) is called stationary pro-vided the joint law of the time shift [ S τ ̺, S τ ϑ, S τ u , S τ W − W ( τ )] on L ([0 , ∞ ); L γ ( T )) × L ([0 , ∞ ); W , ( Q )) × L ([0 , ∞ ); W , ( Q ; R )) × C ([0 , ∞ ); U ) is independent of τ ≥ . We now state our main result concerning the existence of a stationary martingale solutionto (1.1)–(1.5).
Theorem 2.1.
Let M ∈ (0 , ∞ ) be given. Suppose that the structural assumptions (2.2) – (2.17) are in force and that the diffusion coefficient satisfies F satisfies (2.1) . Then problem (1.1) – (1.5) admits a stationary martingale solution in the sense of Definition 2.3. The proof of Theorem 2.1 is split into several parts. In the next section we study theapproximate system with regularisation parameters ε and δ . The proof will be completed inSection 4 after passing to the limit in ε and δ .3. The viscous approximation
In this section we study the viscous approximation to (1.1)–(1.5), where the continuityequation contains an artificial diffusion ( ε -layer) and the pressure is stabilised by an artificialhigh power to the density ( δ -layer). In addition to the common terms we add additionalstabilising quantities in the continuity equations as in [4], see (3.1) below. DOMINIC BREIT, EDUARD FEIREISL, AND MARTINA HOFMANOV´A
Martingale solutions.
In this subsection we give a precise formulation of the approximatedproblem. For this purpose we introduce a cut-off function χ ∈ C ∞ ( R ) , χ ( z ) = z ≤ ,χ ′ ( z ) ≤ < z < ,χ ( z ) = 0 for z ≥ . We denote by M ε the unique solution to the equation 2 εz = χ ( z/M ) which obviously satisfies M ε ≤ M . Finally, the diffusion coefficients are regularised by replacing F by F ε , F ε = ( F k,ε ) k ≥ , F k,ε ( x, ̺, ϑ, u ) = χ (cid:18) ε̺ − (cid:19) χ (cid:18) | u | − ε (cid:19) F k ( x, ̺, ϑ, u ) . Let us start with a precise formulation of the problem. • Regularized equation of continuity. Z ∞ Z Q [ ̺∂ t ϕ + ̺ u · ∇ ϕ ] d x d t = ε Z ∞ Z Q [ ∇ ̺ · ∇ ϕ − ̺ϕ ] d x d t − ε Z ∞ Z Q M ε ϕ d x d t (3.1) for any ϕ ∈ C ∞ c ((0 , ∞ ) × Q ) P -a.s. • Regularized momentum equation. Z ∞ ∂ t ψ Z Q ̺ u · ϕ d x d t + Z ∞ ψ Z Q ̺ u ⊗ u : ∇ ϕ d x d t + Z ∞ ψ Z Q p δ ( ϑ, ̺ ) div ϕ d x d t − Z ∞ ψ Z Q S δ ( ϑ, ∇ u ) : ∇ ϕ d x d t − ε Z ∞ ψ Z Q ̺ u · ∆ ϕ d x d t − ε Z ∞ ψ Z Q ̺ u · ϕ d x d t = − Z ∞ ψ Z Q ̺ F ε ( ̺, ϑ, u ) · ϕ d x d W (3.2) for any ψ ∈ C ∞ c ((0 , ∞ )), ϕ ∈ C ∞ ( Q ; R ) P -a.s. • Regularized entropy balance. (3.3) − Z ∞ Z Q [ ̺s δ ( ̺, ϑ ) ∂ t ψ + ̺s δ ( ̺, ϑ ) u · ∇ ψ ] ϕ d x d t ≥ Z ∞ Z Q ϑ h S ( ϑ, ∇ u ) : ∇ u + κ δ ( ϑ ) ϑ |∇ ϑ | + δ ϑ i ψ d x d t + Z ∞ ψ Z Q κ δ ( ϑ ) ∇ ϑϑ · ∇ ϕ d x d t − Z ∞ ψ Z ∂Q ϕ d ( ϑ ) ϑ ( ϑ − Θ ) d H d t − ε Z ∞ ψ Z Q (cid:20)(cid:18) ϑs M,δ ( ̺, ϑ ) − e M,δ ( ̺, ϑ ) − p M ( ̺, ϑ ) ̺ (cid:19) ∇ ̺ϑ (cid:21) · ∇ ϕ d x d t + Z ∞ ψ Z Q (cid:20) εδ ϑ ( β̺ β − + 2) |∇ ̺ | + ε ̺ϑ ∂p M ∂̺ ( ̺, ϑ ) |∇ ̺ | − εϑ (cid:21) ϕ d x d t + Z ∞ ψ Z Q (cid:0) − ε̺ + 2 εM ε (cid:1) ϑ (cid:16) ϑs M,δ ( ̺, ϑ ) − e M,δ ( ̺, ϑ ) − p M ( ̺, ϑ ) ̺ (cid:17) ϕ d x d t for any ψ ∈ C ∞ c ((0 , ∞ )), ϕ ∈ C ∞ ( Q ; R ) P -a.s. TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 9 • Regularized total energy balance. − Z ∞ ∂ t ψ (cid:18) Z Q E δ ( ̺, ϑ, u ) d x (cid:19) d t + Z ∞ ψ Z Q εϑ d t + 2 ε Z T ψ Z Q (cid:20) δ̺ + δ ΓΓ − ̺ Γ + 12 ̺ | u | (cid:21) d x d t + Z ∞ ψ Z ∂Q d ( ϑ − Θ ) d H d t = Z ∞ Z Q δϑ ψ d x d t + Z ∞ ψεM ε Z Q (cid:18) δ̺ + δ ΓΓ − ̺ Γ − + 12 | u | (cid:19) d x d t + 12 Z T ψ (cid:18) Z Q X k ≥ ̺ | F k,ε ( ̺, ϑ, u ) | d x (cid:19) d t + Z ∞ ψ Z Q ̺ F ε ( ̺, ϑ, u ) · u d W d x (3.4) holds for any ψ ∈ C ∞ c (0 , ∞ ) P -a.s., where we have set E δ = 12 ̺ | u | + ̺e δ ( ̺, ϑ ) + δ (cid:16) ̺ + 1Γ − ̺ Γ (cid:17) . we have the following result. Proposition 3.1.
Let ε, δ > be given. Then there exists a weak martingale solution [ ̺ ε , ϑ ε , u ε ] to (3.1) – (3.4) . In addition, for n ∈ N and every ψ ∈ C ∞ c ((0 , ∞ )) , ψ ≥ , the followinggeneralized energy inequality holds true − Z ∞ ∂ t ψ h Z Q E δ,ϑH ( ̺, ϑ ) d x i n d t + nϑ Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q σ ε,δ d x d t + n Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q εϑ d t + n Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z ∂Q d ( ϑ ) ϑ (cid:16) ϑ − ϑ (cid:17) ( ϑ − Θ ) d H d t + 2 εnϑ E Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:20) δ̺ + δ ΓΓ − ̺ Γ + 12 ̺ | u | (cid:21) d x d t + 2 εn Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q ̺ (cid:16) p M ( ̺, ϑ ) ̺ϑ + e M,δ ( ̺, ϑ ) ϑ − s M,δ ( ̺, ϑ ) (cid:17) d x d t + nε Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q ϑϑ (cid:18) e M,δ ( ̺, ϑ ) + ̺ ∂e M ∂̺ ( ̺, ϑ ) (cid:19) ∇ ̺ · ∇ ϑ d x d t + n Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q ̺ F ε ( ̺, ϑ, u ) · u d W d x + n Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − (cid:18) Z Q X k ≥ ̺ | F k,ε ( ̺, ϑ, u ) | d x (cid:19) d t + nεM ε Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:18) δ̺ + δ ΓΓ − ̺ Γ − + 12 | u | (cid:19) d x d t + nϑεM ε Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:18) p M ( ̺, ϑ ) ̺ϑ + e M,δ ( ̺, ϑ ) ϑ − s M,δ ( ̺, ϑ ) (cid:19) d x d t + n ( n − Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − ∞ X k =1 Z τ τ (cid:18) Z Q ̺ F k,ε ( ̺, ϑ, u ) · u d x (cid:19) d t. (3.5) Here we abbreviated σ ε,δ = 1 ϑ h S ( ϑ, ∇ u ) : ∇ u + κ ( ϑ ) ϑ |∇ ϑ | + δ (cid:16) ϑ Γ − + 1 ϑ (cid:17) |∇ ϑ | + δ ϑ i + εδ ϑ (cid:0) β̺ Γ − + 2 (cid:1) |∇ ̺ | + ε ∂p M ∂̺ ( ̺, ϑ ) |∇ ̺ | ̺ϑ + ε ̺ϑ |∇ u | , and E δ,ϑH = ̺ | u | + H ϑ ( ̺, ϑ ) + δ (cid:0) ̺ + − ̺ Γ (cid:1) , where H δ,ϑ ( ̺, ϑ ) = ̺ (cid:0) e δ ( ̺, ϑ ) − ϑs δ ( ̺, ϑ ) (cid:1) = H ϑ ( ̺, ϑ ) + δ̺ϑ − ϑ̺ log( ϑ )(3.6) with H ϑ ( ̺, ϑ ) introduced in (2.2) .Proof. Although there are some differences to system (4.24)–(4.27) from [1] the method stillapplies (in particular, it is possible to allow an unbounded time interval by working withspaces of the from L q loc ([0 , ∞ ); X ) and C loc ([ , , ∞ ); X ) for Banach spaces X ) and we obtainthe existence of a weak martingale solution to (3.1)–(3.4). We remark, in particular, that thesolution in [1] is constructed with respect to some initial law which does not play any role inour analysis. For simplicity we choose ̺ = 1 , ϑ = 1 , u = 0 , which satisfies all the required assumptions.As far as the energy inequality is concerned, the required version can be derived on the basicapproximate level (even with equality) and it is preserved in the limit. In fact, one can argueas in [1, Section 4.1] to derive the version for n = 1, while the case n ≥ | u | and 2 δ̺ + δ ΓΓ − ̺ Γ gives rise to the terms2 εnϑ E Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:20) δ̺ + δ ΓΓ − ̺ Γ + 12 ̺ | u | (cid:21) d x d t and nεM ε Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:18) δ̺ + δ ΓΓ − ̺ Γ − + 12 | u | (cid:19) d x d t in (3.5) which are new in comparison to [1]. Also, the term nϑεM ε Z ∞ ψ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:18) p M ( ̺, ϑ ) ̺ϑ + e M,δ ( ̺, ϑ ) ϑ − s M,δ ( ̺, ϑ ) (cid:19) d x d t, which arises due to the last line in (3.3), does not appear in [1]. Finally, as in (1.10) we havethe boundary term due to non-homogeneous boundary conditions being incorporated alreadyin (3.3). (cid:3) Uniform-in-time estimates.
The first step is now to derive estimates which are uniform intime.
Proposition 3.2.
Let ( ̺, ϑ, u ) be a weak martingale solution to (3.1) – (3.4) . Assume that (3.7) ess lim sup t → E h Z Q E δ,ϑH ( ̺, ϑ )( t, · ) d x i n < ∞ for some n ∈ N . Then for any ϑ > and ε ≤ ε there is E ∞ = E ∞ ( n, ε, δ, ϑ ) such that E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n ≤ E ∞ , (3.8) as well as E Z τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − (cid:18) Z Q σ ε,δ d x + Z Q εϑ d x (cid:19) d t ≤ E ∞ (1 + τ )(3.9) for any τ > . TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 11
Proof.
The energy inequality in (3.5) yields for any 0 ≤ τ < τ h Z Q E δ,ϑH ( ̺, ϑ )( τ , · ) d x i n + nϑ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q σ ε,δ d x + n Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − (cid:18) Z Q εϑ d x + Z ∂Q d ( ϑ ) ϑ (cid:16) ϑ − ϑ (cid:17) ( ϑ − Θ ) d H (cid:19) d t + 2 εnϑ E Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:20) δ̺ + δ ΓΓ − ̺ Γ + 12 ̺ | u | (cid:21) d x d t + 2 εn Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q ̺ (cid:16) p M ( ̺, ϑ ) ̺ϑ + e M,δ ( ̺, ϑ ) ϑ − s M,δ ( ̺, ϑ ) (cid:17) d x d t = h Z Q E δ,ϑH ( ̺, ϑ )( τ , · ) d x i n + nε Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q ϑϑ (cid:18) e M,δ ( ̺, ϑ ) + ̺ ∂e M ∂̺ ( ̺, ϑ ) (cid:19) ∇ ̺ · ∇ ϑ d x d t + n Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:18) δϑ + εϑϑ (cid:19) d x d t + n Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q ̺ F ε ( ̺, ϑ, u ) · u d W d x + n Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − (cid:18) Z Q X k ≥ ̺ | F k,ε ( ̺, ϑ, u ) | d x (cid:19) d t + εM ε n Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:18) δ̺ + δ ΓΓ − ̺ Γ − + 12 | u | (cid:19) d x d t + εM ε nϑ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:18) p M ( ̺, ϑ ) ̺ϑ + e M,δ ( ̺, ϑ ) ϑ − s M,δ ( ̺, ϑ ) (cid:19) d x d t + n ( n − Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − ∞ X k =1 (cid:18) Z Q ̺ F k,ε ( ̺, ϑ, u ) · u d x (cid:19) d t =: ( I ) + ( II ) + · · · + ( V III ) . (3.10)Let us first consider the terms on the left-hand side. We have n Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z ∂Q d ( ϑ ) ϑ (cid:16) ϑ − ϑ (cid:17) ( ϑ − Θ ) d H d t ≥ n Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z ∂Q ϑ d H d t − cn Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − ≥ n Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z ∂Q ϑ d H d t − κn Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n d t − c κ ( τ − τ )for all κ > Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:20) δ̺ + δ ΓΓ − ̺ Γ + 12 ̺ | u | (cid:21) d x d t ≥ c Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n d t − c ( τ − τ ) − c Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:0) ϑ + ϑ̺ log( ϑ ) (cid:1) d t ≥ c Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n d t − c κ ( τ − τ ) − κ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:16) εϑ + δ̺ + δ ϑ (cid:17) d t due to (2.13). Finally, due to (2.9)–(2.12), p M ( ̺, ϑ ) ϑ + ̺e M,δ ( ̺, ϑ ) ϑ − ̺s δ,M ( ̺, ϑ )is bounded from below by a negative constant, such that the corresponding term can bebounded from below by − c ( τ − τ ).Using (2.1), R Q ̺ d x = M ε ≤ M and (2.13) the terms ( V ) and ( V III ) can be bounded by Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − + Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q ̺ | u | d x d t ≤ c Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − + c ( τ − τ )+ c Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q ϑ̺ log( ϑ ) d t ≤ κ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n + c κ ( τ − τ )+ κ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:16) εϑ + δ̺ + δ ϑ (cid:17) d t, where κ > IV ) vanishes after taking expectations. On account of(2.9)–(2.12) we have p M ( ̺, ϑ ) ̺ϑ + e M,δ ( ̺, ϑ ) ϑ − s M,δ ( ̺, ϑ ) . ̺ / ϑ ≤ κ̺ Γ + κ ϑ + c κ . (3.11)Consequently, the estimate for ( V II ) is analogous to one for ( V ) and ( V III ) above. We quotefrom [8, equ. (3.107)]( II ) ≤ ε Z Q ϑ (cid:12)(cid:12)(cid:12) e M ( ̺, ϑ ) + ̺ ∂e M ( ̺, ϑ ) ∂̺ (cid:12)(cid:12)(cid:12) |∇ ̺ ||∇ ϑ | d x ≤ Z Q h δ (cid:16) ϑ Γ − + 1 ϑ (cid:17) |∇ ϑ | + εδϑ (cid:16) Γ ̺ Γ − + 2 (cid:17) |∇ ̺ | i d x (3.12)provided we choose ε = ε ( δ ) > III ) ≤ κ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:16) εϑ + δ ϑ (cid:17) d t + c κ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − d t ≤ κ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q (cid:16) εϑ + δ ϑ (cid:17) d x d t + κ Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n d t + c κ ( τ − τ ) . TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 13
Combing everything and choosing κ small enough and noticing that δ ϑ ≤ σ ε,δ yields E h Z Q E δ,ϑH ( ̺, ϑ )( τ , · ) d x i n + D E Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − Z Q σ ε,δ d x + D E Z τ τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − (cid:18) Z Q εϑ d x + Z ∂Q ϑ d H (cid:19) d t ≤ E h Z Q E δ,ϑH ( ̺, ϑ )( τ , · ) d x i n + c ( τ − τ )(3.13)for all 0 ≤ τ < τ with some D >
0. We obtain in particular E h Z Q E δ,ϑH ( ̺, ϑ )( τ , · ) d x i n + D Z τ τ E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n d x ≤ E h Z Q E δ,ϑH ( ̺, ϑ )( τ , · ) d x i n + C ( τ − τ ) . Applying the Gronwall lemma from [3, Lemma 3.1] and recalling hypothesis (3.7) we obtain E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n ≤ exp( − Dt ) (cid:18) E h Z Q E δ,ϑH ( ̺, ϑ )(0 , · ) d x i n − CD (cid:19) + CD ≤ E ∞ uniformly in τ >
0. Using this in (3.13) shows E Z τ h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i n − (cid:18) Z Q σ ε,δ d x + Z Q εϑ d x + Z ∂Q ϑ d H (cid:19) d t ≤ E ∞ (1 + τ )by possibly enlarging E ∞ . (cid:3) Stationary solutions.
Based on Proposition 3.2 the method from [17] becomes availableand we can construct a stationary solution to (3.1)–(3.4) following the ideas from [3] to whichwe refer for further details. Different to Section 2.3 we consider stationary solutions sitting onthe space of trajectories that are defined on the real line R rather than the interval [0 , ∞ ). Wewill call them entire stationary solutions . This construction is clearly stronger and hence weobtain also stationary solutions in the sense of Definitions 2.1.Clearly, Definition 2.1 can be easily modified for solutions (cid:0) (Ω , F , ( F t ) t ≥− T , P ) , ̺, ϑ, u , W )being defined on [ − T, ∞ ) for some T >
0. An entire solution is than an object (cid:0) (Ω , F , ( F t ) t ∈ R , P ) , ̺, ϑ, u , W )which is a solution on [ − T, ∞ ) for any T >
0. It takes values in the trajectory space T = T ̺ × T ϑ × T u × T W , T ̺ = (cid:0) L ( R ; W , ( Q ; R d )) , w (cid:1) ∩ C weak , loc ( R ; L Γ ( Q )) , T ϑ = (cid:0) L ( R ; W , ( Q ; R d )) , w (cid:1) ∩ (cid:0) L ∞ loc ( R ; L ( Q )) , w ∗ (cid:1) T u = (cid:16) L ( R ; W , ( Q ; R d )) , w (cid:17) , T W = C loc , ( R ; U ) , where C loc , denotes the space of continuous functions vanishing at 0. We denote by P ( T ) theset of Borel probability measures on T .We say that an entire solution to (3.1)-(3.4) of the problem (3.1)–(3.4) is stationary if itslaw L T [ ̺, ϑ, u , W ] is shift invariant in the trajectory space T , meaning L T [ S τ [ ̺, ϑ, u , W ]] = L T [ ̺, ϑ, u , W ] for any τ ∈ R, with the time shift operator S τ [ ̺, ϑ, u , W ]( t ) = [ ̺ ( t + τ ) , ϑ ( t + τ ) , u ( t + τ ) , W ( t + τ ) − W ( τ )] , t ∈ R, τ ∈ R. Proposition 3.3.
Let the assumptions of Theorem 2.1 be valid and let ε ≤ ε and δϑ > begiven. Let (cid:0) (Ω , F , ( F t ) t ≥ , P ) , ̺, ϑ, u , W ) be a dissipative martingale solution of the problem (3.1) – (3.4) (in the sense of Definition 2.1with the obvious modifications) such that (3.14) ess lim sup t → E h E δ,ϑH ( t ) i < ∞ . Then there is a sequence T n → ∞ and an entire stationary solution (cid:0) ( ˜Ω , ˜ F , (˜ F t ) t ∈ R , ˜ P ) , ˜ ̺, ϑ, ˜ u , ˜ W ) such that T n Z T n L T [ S t [ ̺, ϑ, u , W ]] d t → L T h ˜ ̺, ˜ u , ˜ ϑ, ˜ W i narrowly as n → ∞ . Proof.
Let [ ̺, u , W ] be a dissipative martingale solution on [0 , ∞ ) defined on some stochasticbasis (Ω , F , ( F t ) t ≥ , P ) and satisfying (3.14). We define the probability measures(3.15) ν S ≡ S Z S L T ( S t [ ̺, ϑ, u , W ]) d t ∈ P ( T ) . We tacitly regard functions defined on time intervals [ − t, ∞ ) as trajectories on R by extendingthem to s ≤ − t by the value at − t . As in [3, Prop. 5.1] we can show that the family ofmeasures { ν S ; S > } is tight on T . In fact, Proposition 3.2 yields E " sup s ∈ [ − T,T ] E mδ ( s + t ) + E "Z T − T ∨− t Z Q (cid:0) |∇ ̺ | + |∇ ϑ | + |∇ u | (cid:1) ( s + t ) d x d s . E [ E m (0)] + c. This gives the same bounds on ̺ and u as in [3] and we control additionally E " sup s ∈ [ − T,T ] (cid:18) Z Q ϑ d x (cid:19) m ( s + t ) + E "Z T − T ∨− t Z Q |∇ ϑ | ( s + t ) d x which implies tightness of S R S L T ( S t [ ϑ ]) d t . Note also that we have control of ∇ ̺ due to ε > ν τ,S n ≡ S n Z S n L ( S t + τ [ ̺, u , W ]) d t in P ( T ) as n → ∞ exists for some τ = τ ∈ R then it exists for all τ ∈ R and is independentof the choice of τ . Applying Jakubowski–Skorokhod’s theorem [18], we infer the existence ofa sequence S n → ∞ and ν ∈ P ( T ) so that ν ,S n → ν narrowly in P ( T ) as well as ν τ,S n → ν narrowly for all τ ∈ R . Accordingly, the limit measure ν is shift invariant in the sense thatfor every G ∈ BC ( T ) and every τ ∈ R we have ν ( G ◦ S τ ) = ν ( G ) . To conclude the proof ofTheorem 3.3, it remains to show that ν is a law of an entire solution to (3.1)–(3.4).First of all, we can argue as in [3, Prop. 5.3] to show that for any S > ν S ≡ S Z S L ( S t [ ̺, u , W ]) d t ∈ P ( T )is a dissipative martingale solution on ( − T, ∞ ), provided ( ̺, ϑ, u , W ) is a dissipative martingalesolution on ( − T, ∞ ) defined on some probability space (Ω , F , P ). The idea is to use that (3.1),(3.2) and (3.4) can be written as measurable mappings on the paths space (see the proof of[3, Prop. 5.3] for how to include the stochastic integral). Unfortunately, this is not true forthe quantities hidden in σ ε,δ appearing in(3.4) and (3.13). However, they are measurable ona subset, where the laws L ( S t [ ̺, u , W ]) are supported. Recall from (3.9) that σ ε,δ belongsa.s. to L in space and locally in time for any solution. This is enough to arrive at the sameconclusion. TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 15
To finish the proof we argue that the limit ν is the law of an entire solution to (3.1)–(3.4).Now we consider the measures ν τ,S n − τ , n = 1 , , . . . , and τ >
0. According to the previousconsiderations, ν τ,S n − τ is a dissipative martingale solution to (3.1)–(3.4) on [ − τ, ∞ ) and thenarrow limit as n → ∞ exists and equals to ν . Now we take a sequence τ m → ∞ and choose adiagonal sequence such that ν τ m ,S n ( m ) − τ m → ν a s m → ∞ . Applying Jakubowski–Skorokhod’s theorem, we obtain a sequence of approximate processes[˜ ̺ m , ˜ u m , ˜ W m ] converging a.s. to a process [˜ ̺, ˜ u , ˜ W ] in the topology of T . Moreover, the lawof [˜ ̺ m , ˜ u m , ˜ W m ] is ν τ m ,S n ( m ) − τ m and necessarily the law of [˜ ̺, ˜ u , ˜ W ] is ν . By [2, Thm. 2.9.1]it follows that equations (3.1)–(3.4) as well as (3.5) also hold on the new probability space.The limit procedure on this level is quite easy due to the artificial viscosity: By definition of T ̺ the sequence ˜ ̺ n is compact on L Γ . Moreover, the strong convergence of ˜ ϑ n can be provedexactly as in the deterministic existence theory (see [8, Sec. 3.5.3]). This is enough to pass tothe limit in all nonlinearities in (3.1), (3.2) and (3.4). The terms in (3.3) and (3.5) which arenot compact (those related to the quantity σ ε,δ ) are convex functions and hence can be dealtwith by lower-semicontinuity. (cid:3) Asymptotic limit
In this section we pass to the limit and the artificial viscosity and the artificial pressurerespectively. They crucial point is a uniform-in-time estimate, see (4.7) and (4.8) below, whichpreserves stationarity in the limit. It has to be combined with pressure estimates which differ onboth levels. The key ingredient for estimates (4.7) and (4.8) is the non-homogeneous boundarycondition for the temperature, cf. (1.5).4.1.
The vanishing viscosity limit.
In this section we start with a stationary solution ( ̺, ϑ, u )to (3.1)–(3.4) existence of which is guaranteed by Proposition 3.3. We prove uniform-in-timeestimates and pass subsequently to the limits in ε and δ .The entropy balance (3.3) yields after taking expectations and using stationarity E Z Q ϑ h S ( ϑ, ∇ u ) : ∇ u + κ δ ( ϑ ) ϑ |∇ ϑ | + δ ϑ i d x ≤ E Z Q (cid:0) − ε̺ + 2 εM ε (cid:1) ϑ (cid:16) p M ( ̺, ϑ ) ̺ + e M,δ ( ̺, ϑ ) − ϑs M,δ ( ̺, ϑ ) (cid:17) ϕ d x + E Z Q εϑ d x + E Z ∂Q d ( ϑ ) ϑ ( ϑ − Θ ) d H . On account of (3.11) the first two terms can be bounded by c E h Z Q δ̺ Γ d x + 1 i + 14 E Z Q δ ϑ d x. The estimate is independent of δ if we choose ε ≤ δ . Similarly, we obtain E Z ∂Q d ( ϑ ) ϑ ( ϑ − Θ ) d H ≤ c E Z ∂Q (cid:0) ϑ + |∇ ϑ | (cid:1) + c ≤ c E h Z Q ϑ d x + 1 i + 14 E Z Q κ δ ( ϑ ) ϑ |∇ ϑ | d x using (1.6) and the trace theorem. In conclusion, E Z Q ϑ h S ( ϑ, ∇ u ) : ∇ u + κ δ ( ϑ ) ϑ |∇ ϑ | + δ ϑ i d x . E h Z Q (cid:0) δ̺ Γ + ϑ (cid:1) d x i + 1 . E h Z Q E δ,ϑH ( ̺, ϑ ) d x i + 1(4.1) independently of ε and δ recalling also (2.13) and R Q ̺ d x ≤ M ε ≤ M . By (2.13) this impliesfor any ξ > E h Z Q E δ,ϑH ( ̺, ϑ ) d x i ≤ c E h Z Q (cid:0) δ̺ Γ + ̺ / + ϑ + δ̺ | log( ϑ ) | (cid:1) d x + 1 i ≤ c E h Z Q (cid:0) δ̺ Γ + ̺ / + ϑ (cid:1) d x i + ξ E h Z Q δ ϑ d x i + c ξ ≤ c E h Z Q (cid:0) δ̺ Γ + ̺ / + ϑ (cid:1) d x i + cξ E h Z Q E δ,ϑH ( ̺, ϑ ) d x i + c ξ such that E h Z Q E δ,ϑH ( ̺, ϑ ) d x i . E h Z Q (cid:0) δ̺ Γ + ̺ / + ϑ (cid:1) d x i + 1(4.2)independently of ε and δ .In (3.5) we choose n = 2, apply expectations and use stationarity to obtain2 ϑ E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q σ ε,δ d x + 2 E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q εϑ d x + 2 E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z ∂Q dϑ (cid:16) ϑ − ϑ (cid:17) ( ϑ − Θ ) d H + 4 εϑ E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q (cid:20) δ̺ + δ ΓΓ − ̺ Γ + 12 ̺ | u | (cid:21) d x + 4 ε E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q ̺ (cid:16) p ( ̺, ϑ ) ̺ϑ + e δ ( ̺, ϑ ) ϑ − s δ ( ̺, ϑ ) (cid:17) d x ≤ ε E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q ϑϑ (cid:18) e M,δ ( ̺, ϑ ) + ̺ ∂e M ∂̺ ( ̺, ϑ ) (cid:19) ∇ ̺ · ∇ ϑ d x + E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i(cid:18) Z Q X k ≥ ̺ | F k,ε ( ̺, ϑ, u ) | d x (cid:19) + 2 εM ε E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q (cid:18) δ̺ + δ ΓΓ − ̺ Γ − + 12 | u | (cid:19) d x + 2 εM ε ϑ E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q (cid:18) p ( ̺, ϑ ) ̺ϑ + e δ ( ̺, ϑ ) ϑ − s δ ( ̺, ϑ ) (cid:19) d x + ∞ X k =1 E (cid:18) Z Q ̺ F k,ε ( ̺, ϑ, u ) · u d x (cid:19) =: ( I ) + ( II ) + ( III ) + ( IV ) + ( V ) . (4.3)Arguing as in the proof of Proposition 3.2 but paying attention to the ε - and δ -dependence wehave ( I ) ≤ E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q σ ε,δ d x, ( II ) ≤ c E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i , ( III ) ≤ εϑ E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q (cid:20) δ̺ + δ ΓΓ − ̺ Γ (cid:21) d x + c E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i + 14 E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q σ ε,δ d x, TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 17 ( IV ) ≤ εϑ E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q δ ΓΓ − ̺ Γ d x + c + 14 E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i Z Q σ ε,δ d x, ( V ) ≤ c E h Z Q E δ,ϑH ( ̺, ϑ )( τ, · ) d x i . Again these estimates are also uniform in δ if we choose ε small enough compared to δ . Recalling(2.13) and R Q ̺ d x ≤ M ε ≤ M we thus obtain k ϑ k L ( Q ) < ∼ k ϑ k W , ( Q ) < ∼ Z Q ϑϑ κ ( ϑ ) ϑ |∇ ϑ | d x + Z ∂Q ϑ d H , E (cid:20)Z Q E δ,ϑH ( ̺, ϑ, u ) d x k ϑ k L ( Q ) (cid:21) > ∼ E h k ϑ k L ( Q ) k ϑ k L ( Q ) i − E (cid:2) k ϑ k L ( Q ) (cid:3) ≥ E h k ϑ k L / ( Q ) i − E (cid:2) k ϑ k L ( Q ) (cid:3) . This, inserted in left-hand side of (4.3) yields E h k ϑ k / L / ( Q ) i + E (cid:20)Z Q E δ,ϑH ( ̺, ϑ, u ) d x Z Q σ ε,δ d x (cid:21) < ∼ E h Z Q E δ,ϑH ( ̺, ϑ ) d x i + 1 . (4.4)independently of ε and δ using also (4.1).In order to close the estimate why apply pressure estimates which now can depend on δ .Let us introduce the so–called Bogovskii operator B enjoying the following properties: B : L q ( Q ) ≡ (cid:26) f ∈ L q ( Q ) (cid:12)(cid:12)(cid:12) Z Q f d x = 0 (cid:27) → W ,q ( Q, R d ) , < q < ∞ , div B [ f ] = f, kB [ f ] k L r ( Q ) < ∼ k g k L r ( Q ; R d ) if f = div g , g · n | ∂Q = 0 , < r < ∞ , (4.5)see [14, Chapter 3] or [15]. Arguing as in [4, Section 5] (but replacing ∇ ∆ − by the Bogovskiioperator B ) we obtain E (cid:20)Z Q (cid:20) p ( ̺, ϑ ) ̺ + 13 ̺ | u | (cid:21) d x (cid:21) = c ( M ) E (cid:20)Z Q (cid:16) p ( ̺, ϑ ) + 13 ̺ | u | (cid:17) d x (cid:21) − E (cid:20)Z Q (cid:16) ̺ u ⊗ u − ̺ | u | I (cid:17) : ∇B ( ̺ − M ε ) d x (cid:21) + E (cid:20)Z Q (cid:16) µ ( ϑ ) + η ( ϑ ) (cid:17) div u ̺ d x d t (cid:21) + E (cid:20)Z Q ̺ u · B div( ̺ u ) d x (cid:21) + 2 ε E (cid:20)Z Q ̺ ε u ε · B [ ̺ ε − M ε ] d x (cid:21) + ε E (cid:20)Z Q ̺ ε div u ε d x (cid:21) =: ( I ) + ( II ) + ( III ) + ( IV ) + ( V ) + ( V I ) . The terms (II) and (IV)–(VI) can be estimated as in [4] (note that they don’t contain ϑ ). Infact, we have by (2.13) and the continuity of ∇B ( II ) . E k√ ̺ u k L x k u k L x k√ ̺ ∇B ( ̺ − M ε ) k L x . E Z Q E H ( ̺, ϑ, u ) d x k∇ u k L x + E k√ ̺ ∇B ( ̺ − M ε ) k L x . E Z Q E δ,ϑH ( ̺, ϑ, u ) d x provided Γ is large enough. Furthermore, we obtain for some α ∈ (0 , IV ) . E k ̺ u k L x . E k ̺ k L x k u k L x . E k ̺ k L x k∇ u k L x . E k ̺ k αL x k ̺ k − α ) L Γ k∇ u k L x . E k ̺ k − α ) L Γ k∇ u k L x . E Z Q E δ,ϑH ( ̺, ϑ, u ) d x k∇ u k L x + E k∇ u k L x . E Z Q E δ,ϑH ( ̺, ϑ, u ) d x using again (2.13), R Q ̺ d x ≤ M ε ≤ M , (4.4) as well as (4.1). We can estimate ( V ) and ( V I )along the same lines. Using (2.14) and (2.16) we have for Γ large enough(
III ) . E (cid:20)Z Q (cid:0) ϑ + ̺ + |∇ u | (cid:1) d x (cid:21) . E (cid:20) (cid:18)Z Q E δ,ϑH ( ̺, ϑ, u ) d x (cid:19)(cid:21) due to (2.13) (4.1). Obviously, the same estimate holds for (I) such that we can conclude E (cid:20)Z Q (cid:0) ̺ Γ+1 + ϑ ̺ + ̺ | u | (cid:1) d x (cid:21) . E (cid:20) Z Q (cid:0) ̺ Γ + ϑ + ̺ | u | (cid:1) d x (cid:21) , (4.6)using also (4.2). Obviously, we have Z Q (cid:0) ̺ Γ + ϑ (cid:1) d x ≤ ξ Z Q (cid:0) ̺ Γ+1 + ϑ / (cid:1) d x + c ( ξ ) , E (cid:20)Z Q ̺ | u | d x (cid:21) ≤ ˜ ξ E (cid:20)Z Q ̺ | u | d x (cid:21) + c ( ˜ ξ ) E (cid:20)Z Q |∇ u | d x (cid:21) ≤ ˜ ξ E (cid:20)Z Q ̺ | u | d x (cid:21) + c ( ˜ ξ ) Z Q (cid:0) ̺ Γ + ϑ + 1 (cid:1) d x for any ξ, ˜ ξ > E (cid:20)Z Q h ̺ Γ+1 + ϑ ̺ + ϑ / + ̺ | u | i d x (cid:21) ≤ c as well as E (cid:20)(cid:18)Z Q E δ,ϑH ( ̺, ϑ, u ) d x (cid:19) Z Q σ ε,δ d x (cid:21) ≤ c (4.8)using (4.1).Estimates (4.7) and (4.8) are sufficient to pass to the limit in (3.1)–(3.4) arguing as in [1,Section 5] (in fact, one has to combine ideas from [2] and [8]). In the limit ε →
0, we obtainthe following system. • Equation of continuity. Z ∞ Z Q [ ̺∂ t ϕ + ̺ u · ∇ ϕ ] d x d t = 0(4.9) for any ϕ ∈ C ∞ c ((0 , ∞ ) × Q ) P -a.s. • Momentum equation. Z ∞ ∂ t ψ Z Q ̺ u · ϕ d x d t + Z ∞ ψ Z Q ̺ u ⊗ u : ∇ ϕ d x d t + Z ∞ ψ Z Q p δ ( ϑ, ̺ ) div ϕ d x d t − Z ∞ ψ Z Q S δ ( ϑ, ∇ u ) : ∇ ϕ d x d t = − Z ∞ ψ Z Q ̺ F ( ̺, ϑ, u ) · ϕ d x d W (4.10) TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 19 for any ψ ∈ C ∞ c ((0 , ∞ )), ϕ ∈ C ∞ ( Q ; R ) P -a.s. • Entropy balance. (4.11) − Z ∞ Z Q [ ̺s δ ( ̺, ϑ ) ∂ t ψ + ̺s δ ( ̺, ϑ ) u · ∇ ψ ] ϕ d x d t ≥ Z ∞ Z Q ϑ h S δ ( ϑ, ∇ u ) : ∇ u + κ δ ( ϑ ) ϑ |∇ ϑ | + δ ϑ i ψ d x d t + Z ∞ Z Q κ δ ( ϑ ) ∇ ϑϑ · ∇ ψ d x d t − Z ∞ ψ Z ∂Q dϑ ( ϑ − Θ ) d H d t for any ψ ∈ C ∞ c ((0 , ∞ )), ϕ ∈ C ∞ ( Q ; R ) P -a.s. • Total energy balance. − Z T ∂ t ψ (cid:18) Z Q E δ ( ̺, ϑ, u ) d x (cid:19) d t + Z ∞ ψ Z ∂Q d ( ϑ − Θ ) d H d t = ψ (0) Z Q E δ ( ̺ , ϑ , u ) d x + Z T Z Q δϑ ψ d x d t + 12 Z T ψ (cid:18) Z Q X k ≥ ̺ | F k ( ̺, ϑ, u ) | d x (cid:19) d t + Z T ψ Z Q ̺ F ( ̺, ϑ, u ) · u d W d x (4.12) for any ψ ∈ C ∞ c (0 , ∞ ) P -a.s.To summarize, we deduce the following. Proposition 4.1.
Let δ > be given. Then there exists a stationary weak martingale solution [ ̺ δ , ϑ δ , u δ ] to (4.9) – (4.12) . Moreover, we have the estimates E h k ϑ k / L / ( Q ) i + E (cid:20)Z Q E δ,ϑH ( ̺, ϑ, u ) d x Z Q σ δ d x (cid:21) (4.13) < ∼ E h Z Q E δ,ϑH ( ̺, ϑ ) d x i + 1 , E Z Q σ δ d x . E Z Q E δ,ϑH ( ̺, ϑ, u ) d x, (4.14) uniformly in δ , where σ δ = 1 ϑ h S ( ϑ, ∇ u ) : ∇ u + κ ( ϑ ) ϑ |∇ ϑ | + δ (cid:16) ϑ Γ − + 1 ϑ (cid:17) |∇ ϑ | + δ ϑ i . Corollary 4.1.
The solution from Proposition 4.1 satisfies the equation of continuity in therenormalised sense.
The vanishing artificial pressure limit.
Though (4.13) and (4.14) are uniform in δ , thefinal estimates (4.7) and (4.8) are not. Again we have to close the estimate by some pressurebounds. Let ( ̺, ϑ, u ) be a stationary solution to (4.9)–(4.12) as obtained in Proposition 4.1. Arguing as in [4, Section 6] (replacing again ∇ ∆ − by the Bogovskii operator B ) we have E (cid:20)Z Q (cid:20) p δ ( ̺, ϑ ) ̺ α + ̺ αδ | u | (cid:21) d x (cid:21) ≤ c ( M ) (cid:18) E (cid:20)Z Q (cid:20) ̺ | u | + p δ ( ̺, ϑ ) (cid:21) d x (cid:21) + 1 (cid:19) + E (cid:20)Z Q (cid:18) µ ( ϑ ) + η ( ϑ ) (cid:19) div u ̺ α d x (cid:21) + E (cid:20)Z Q (cid:18) ̺ u ⊗ u − ̺ | u | I (cid:19) : ∇B [ ̺ α ] d x (cid:21) + E (cid:20)Z Q ̺ u · B [div( ̺ α u ) + ( α − ̺ α div u ] d x (cid:21) =: ( I ) + ( II ) + ( III ) + ( IV ) , (4.15)where α > I ) + ( III ) + ( IV ) . E (cid:20)Z Q E δ,ϑH ( ̺, ϑ, u ) d x Z Q |∇ u | d x (cid:21) + E (cid:20)Z Q E δ,ϑH ( ̺, ϑ, u ) d x (cid:21) + 1 . Also we see that (
III ) . E (cid:20)Z Q |∇ u | d x (cid:21) + E (cid:20)Z Q (cid:0) ̺ γ + ϑ (cid:1) d x (cid:21) + 1 . E (cid:20)Z Q |∇ u | d x (cid:21) + E (cid:20)Z Q E δ,ϑH ( ̺, ϑ, u ) d x (cid:21) + 1choosing α small enough and using (2.14) and (2.16). Combining these estimate with (4.13)and (4.14) we conclude E (cid:20) Z Q (cid:0) δ̺ Γ+ α + ̺ γ + α + ϑ ̺ α + ̺ α | u | (cid:1) d x (cid:21) . E (cid:20) Z Q (cid:0) δ̺ Γ + ̺ γ + ϑ + ̺ | u | (cid:1) d x (cid:21) , (4.16)recalling also (4.2). As in the proof of (4.7) and (4.8) we deduce E (cid:20)Z Q h ̺ γ +1 + ϑ ̺ + ϑ / + ̺ | u | i d x (cid:21) ≤ c (4.17) E (cid:20)(cid:18)Z Q E δ,ϑH ( ̺, ϑ, u ) d x (cid:19) Z Q |∇ u | d x (cid:21) ≤ c, (4.18) E (cid:20)(cid:18)Z Q E δ,ϑH ( ̺, ϑ, u ) d x (cid:19) Z Q σ δ d x (cid:21) ≤ c, (4.19)using (4.14). With estimates (4.17) and (4.18) at hand we can follow the lines of [1, Section 6]to pass to the limit δ → Compliance with Ethical Standards
Funding . The research of E.F. leading to these results has received funding from the CzechSciences Foundation (GAˇCR), Grant Agreement 21–02411S. The Institute of Mathematics ofthe Academy of Sciences of the Czech Republic is supported by RVO:67985840.
Conflict of Interest . The authors declare that they have no conflict of interest.
Data Availability . Data sharing is not applicable to this article as no datasets were generatedor analysed during the current study.
TATIONARY SOLUTIONS IN THERMODYNAMICS OF STOCHASTICALLY FORCED FLUIDS 21
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Department of Mathematics, Heriot-Watt University, Riccarton Edinburgh EH14 4AS,UK
Email address : [email protected] (E. Feireisl) Institute of Mathematics AS CR, ˇZitn´a 25, 115 67 Praha 1, Czech Republic and Insti-tute of Mathematics, TU Berlin, Strasse des 17.Juni, Berlin, Germany
Email address : [email protected] (M. Hofmanov´a) Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, D-33501 Bielefeld, Germany
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