# Perturbation of Stochastic Boussinesq Equations with Multiplicative White Noise

Authors: Chunde Yang; Xin Zhao
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper studies the Boussinesq equations perturbed by multiplicative white noise and shows the existence and uniqueness of the global solution. It also gets some regularity results for the unique solution.


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## Body

## 1. Introduction

The Boussinesq equation is a mathematics model of thermohydraulics, which consists of equations of fluid and temperature in the Boussinesq approximation. The deterministic case has been studied systematically by many authors (e.g., see [1–3]). However, in many practical circumstances, small irregularity has to be taken into account. Thus, it is necessary to add to the equation a random force, which is in general a space-time white noise, as considered recently by many authors for other equations (see [4–11]). The random attractors of boussinesq equations with multiplicative noise have been investigated by [12]. In this paper, We will study the perturbation of stochastic boussinesq equations with multiplicative white noise.We will consider the following stochastic two-dimension a Boussinesq equations perturbed by a multiplicative white noise of Stratonovich form:(1)dv+[(v·∇)v-vΔv+∇p]dt=e2(T-T1)dt+σv∘dw(t),dT+[(v·∇)T-λΔT]dt=0,divv=0.The domain occupied by the fluid isD=(0,1)×(0,1), and e1, e2 is the canonical basis of ℝ2. The unknown v=(v1,v2), T, and p stand for the velocity vector, temperature, and pressure, respectively. T1 is the temperature at the top, x2=1, while T0=T1+1 is the temperature at the boundary below, x2=0. The constant numbers λ>0, β>0, and  σ>0 are related to the usual Prandtl, Grashof, and Rayleigh numbers.W
(
t
) is two-sided Wiener processes on the probability space (Ω,𝔽,P), where Ω={ω∈C(ℝ,ℝ):ω(0)=0}, 𝔽 is the Borel sigma-algebra induced by the compact-open topology of Ω, and P is a Wiener measure.We supplement (1) with the following boundary condition:
(2)v=0atx2=0,x2=1,T=T0atx2=0,T=T1=T0-1atx2=1,ψ∣x1=0=ψ∣x1=1forψ=v,T,p,∂v∂x1,∂T∂x1.When an initial-valued problem is considered, we supplement these equations with(3)v(x,0)=v0(x),T(x,0)=T0(x)forx∈D.The existence of a compact random attractor and its Hausdorff, fractal dimension estimates have been investigated by [12]. We will solve pathwise (1)–(3). By using the Faedo-Galerkin approximation and a priori estimates, we prove the existence and uniqueness of the global solution and show that the solution continuously depends on the initial value. We also get some regularity results of the solutions.


## 2. Mathematical Setting and Basic Estimates

Let(4)η:=T-T0+x2,
and change p to p-x2+x22/2; then (1) can be  rewritten as
(5)dv+[(v·∇)v-βΔv+∇p]dt=e2ηdt+σv∘dW,dη+[(v·∇)η]dt=v2dt,divv=0.
Let the process be
(6)α(t):=e-σW(t).
Then dα=-σα∘dW, and if we let
(7)ξ:=αv,
we get the new equations (no stochastic differential appears here)
(8)dξdt+α-1(ξ·∇)ξ-βΔξ+α∇p=αe2η,(9)dηdt+α-1(ξ·∇)η-λΔη=α-1ξ2,(10)divξ=0,
with the boundary conditions
(11)ξ=0atx2=0,x2=1,η=0atx2=0,x2=1,ψ|x1=0=ψ|x1=1forψ=ξ,η,p,∂ξ∂x1,∂η∂x1
and the initial value conditions
(12)ξ(0)=ξ0,η(0)=η0.To solve (8)–(12), we consider the Hilbert space H=H1×H2 with the scalar products (·,·) and norms |·|, whereH2=L2(D) and
(13)H1={ξ∈L2(D):divξ=0,ξi|xi=0=ξi|xi=1,i=1,2}.
We also consider the subspace V=V1×V2 of H, where V2 is the space of functions in H1(D) vanishing at x2=0 and x2=1 and periodic in the direction of x1.V2 is a Hilbert space for the scalar product and the norm
(14)((η1,η2))=∫Dgradη1gradη2dx,∥η∥=((η,η))1/2,
and V1={ξ∈V22:divξ=0}. We also denote by ((·,·)) and ∥·∥ the canonical scalar product and norm in V1 and V.The bilinear form(15)μ(u1,u2)=β((ξ1,ξ2))+λ((η1,η2)),∀{ξi,ηi}∈V,i=1,2,
determines a linear isomorphism A from D(A) into H and from V into the dualspace V′, defined by
(16)(Au1,u2)=μ(u1,u2),∀ui={ξi,ηi}∈V,i=1,2,
with D(A)=D(A1)×D(A2),  where
(17)D(A1)={ξ∈V1∩H2(D)2:∂ξ∂x1|x1=0=∂ξ∂x1|x1=1}D(A2)={η∈V2∩H2(D):∂η∂x1|x1=0=∂η∂x1|x1=1}.
Four spaces D(A), V, H, and V′ satisfy
(18)D(A)⊂V⊂H⊂V′
and all embedding injections are densely continuous. It is well known that A:D(A)→H is self-adjoint and positive and A-1 is a compact self-adjoint in H.We also consider the trilinear formsγ on V defined by
(19)γ(u1,u2,u3)=((ξ1·∇)ξ2,ξ3)+((ξ1·∇)η2,η3),∀ui={ξi,ηi}∈V,i=1,2,3.
The trilinear form γ is continuous on V or even onH1(D)2×H1(D). We associate with the form γ the bilinear continuous operator B which map V×V into V′ and D(A)×D(A) into H, defined by
(20)(B(u1,u2),u3)=γ(u1,u2,u3),∀ui={ξi,ηi}∈V,i=1,2,3.
Finally, we define the continuous operators R(t) in H(21)R(t):u={ξ,η}⟶Ru={α(t)e2η,α-1(t)ξ2}.
Now, we can set (8) in the operator form. Ifu={ξ,η} is the solution of (8) and ψ={f,g} is a test function in V, we multiply (8) by f and (9) by g, integrate over D, and add the resulting equation. The pressure term disappears and after simplification we find
(22)ddt(u,ψ)+α-1(u,u,ψ)+μ(u,ψ)+(R(t)u,ψ)=0,∀ψ∈V,
which can be reinterpreted as
(23)dudt+Au+α-1(t)B(u,u)+R(t)u=0.
Note that this equation differs from the determined case, and in determined case, the family R(t) of operator is independent of the time t. Initial condition (12) can be reinterpreted as
(24)u(0)=u0:={ξ0,η0}.
To solve (23)-(24), we also need some Sobolev norm estimates on the bilinear B and the operators R and A.Lemma 1.
The bilinear operatorsB:V×V→V′ and D(A)×D(A)→H are continuous and satisfy(i)
(B(u,v),v)=0,  for all  u,v∈V,(ii)
|(B(u,v),w)|≤c1|u|θ1∥u∥1-θ1∥v∥∥w∥θ1|w|1-θ1,  for all  u,v,w∈V,(iii)
|B(u,v)|+|B(v,u)|≤c2∥u∥∥v∥1-θ2|Av|θ2,  for all  u∈V, v∈D(A),(iv)
|B(u,v)|≤c3|u|θ3∥u∥1-θ3|Av|θ3∥v∥1-θ3,  for all  u∈V, v∈D(A),where  c1, c2, c3  are appropriate constants and  θi∈[0,1),  i=1,2,3.Proof.
The proof is the same as the deterministic case (see [10]).Lemma 2.
The linear continuous operatorsR:V→V′ and D(A)→H satisfy
(25)|R(t)u|≤c4(α(t))+α-1(t)∥u∥,∀u∈V,∀t≥0,(26)|(R(t)u,u)|≤c4(α(t)+α-1(t))∥u∥|u|,∀u∈V,∀t≥0.Proof.
By (21), we have
(27)|Ru|=|αe2η|+|α-1ξ2|≤α|η|+α-1|ξ|≤(α+α-1)(|ξ|+|η|)
which implies by the Poincare inequality
(28)|u|≤c4∥u∥,foru∈V
that (25) holds true. Since |(Ru,u)|≤|Ru||u|, it follows from (25) that (26) holds true.Lemma 3.
The bilinear formμ on V×V satisfies
(29)c5∥u∥2≤μ(u,u)≤c6∥u∥2,foru∈V.Proof.
By (15), we have
(30)μ(u,u)=β∥ξ∥2+λ∥η∥2≤(β+λ)(∥ξ∥+∥η∥)2=(β+λ)∥u∥2,μ(u,u)≥min{β,λ}(∥ξ∥2+∥η∥2)≥12min{β,λ}(∥ξ∥2+∥η∥2),
which imply (28).


## 3. Existence and Uniqueness

In this section, we will prove the existence and uniqueness of the global solution of (23)-(24), equivalently (8)–(12) or (1)–(3). We are working almost surely for ω∈Ω.Theorem 4.
Assume thatu0∈H, then there exists a unique solution of (23)-(24), such that
(31)u∈C([0,∞),H)∩Lloc2(0,∞;V),
and the mapping u0↦u(t) is continuous from H into D(A), for all t>0.Proof.
SinceA-1:H→D(A) is a self-adjoint compact operator in H, it follows from a classical spectral theorem that there exists a sequenceλj:0<λ1≤λ2≤⋯,λj→∞ and a family of elements wj∈D(A) which is completely orthogonal in H such that
(32)Awj=λjwj,∀j.
For each m, we look for an approximate solution um of the following form:
(33)um(t)=∑i=1mgim(t)wi
satisfying
(34)ddt(um,wj)+μ(um,wj)+α-1γ(um,um,wj)+(Rum,wj)=0,j=1,2,…,m
and initial condition
(35)um(0)=Pmu0,
where Pm is the projector in H (or V) on the space spanned byw1,w2,…,wm. Since A and Pm commute, the above equation is also equivalent to
(36)dumdt+Aum+α-1PmB(um,um)+Pm(Rum)=0,
where
(37)Pm(Rum)=∑j=1mgjm(t)PmRwj,
in view of the linearity of Pm,R.
The existence ofum on any finite interval [0,Tm) follows from standard results of the existence of solutions of ordinary differential equations that Tm=+∞ is a consequence of these results and of the following priori estimates:
(38)umremainsboundedinL∞(0,T;H)∩L2(0,T;V),∀T>0.
Indeed, multiplying (34) by gjm, summing these relations for j=1,2,…,m, and noting that α-1μ(um,um,um)=0 (by Lemma 1), we find
(39)12ddt|um|2+μ(um,um)+(Rum,um)=0,
which implies by Lemma 2, (29), and the Young inequality that
(40)12ddt|um|2+c5∥um∥2≤|(Rum,um)|≤c4sup0≤t≤T(α(t)+α-1(t))∥um∥·|um|≤c52∥um∥2+c4′|um|2,
that is,
(41)ddt|um|2+c5∥um∥2≤c′|um|2,
where c5 is defined in (29) and c′ is a appropriate constant. Using the classical Gronwall lemma we find
(42)|um(t)|2≤|um(0)|2ec′T≤|u0|2ec′T=M1,∀0<t≤T.
Integrating (41) for t from 0 to T and using above estimates we have
(43)∫0T∥um∥2dt≤1c5(|um(0)|2-|um(T)|2)+1c5∫0T|um(t)|2dt≤M2,
where M1and M2 are independent of m. Thus, we have proved (38).
We also claim that(44)dumdtremainsboundedinL2(0,T;V′).
Indeed, it follows from Lemma 1 that |B(um,um)|V′≤c|um|∥um∥ with appropriate constant c, which, together with (38), implies that B(um,um) and thus PmB(um,um) remain bounded in L2(0,T;V′). Since both operators R:V→V′ and A:V→V′ are continuous (Lemmas 2 and 3), it follows from (38) that Aum,Rum and thus PmRum remain bounded in L2(0,T;V′). Therefore, by (36),
(45)dumdt=-Aum-α-1PmB(um,um)-Pm(Rum)
remains bounded in L2(0,T;V′), which proved (44).
By weak compactness, it follows from (38) and (44) that there exists a u∈L∞(0,T;H)∩L2(0,T;V), for all T>0 subsequence still denoted by um, such that
(46)um⟶uinL2(0,T;V)weakly,um⟶uinL∞(0,T;H)weakly star,dumdt⟶dudtinL2(0,T;V′)weakly.
We pass to the limit in (34) and find that
(47)ddt(u,φ)+μ(u,φ)+α-1γ(u,u,φ)+(Ru,φ),∀φ∈V,
which implies that u satisfies (23). In particular,  u′=(du/dt)∈L2(0,T;V′)∈L1(0,T;V′). This implies by [10, Lemma II.3.1] that u is almost everywhere equal to a continuous function from [0,T] into V′. Therefore initial condition (24) follows by a passage to the limit in (35). u∈C([0,T],H) follows from [10, Lemma II.3.2] and the facts that H⊂V⊂V′ and u′∈L2(0,T;V′). Furthermore, if we show that uniqueness, then the fact that u∈C([0,T],H), for all T>0, implies that u∈C([0,∞),H).
To prove the uniqueness and continuous dependence ofu(t) on u0 (in H), we let u be a solution of (23)-(24) such that u∈C([0,∞),H)∩Lloc2(0,∞;V). Similar to (39), u  must satisfy the energy equality
(48)12ddt|u|2+μ(u,u)+(Ru,u)=0.
By using Lemmas 1–3 and Gronwall lemma, we get the following similar estimates:
(49)|u(t)|2≤|u(0)|2ect,
which has proved the continuous dependence. For the uniqueness, we let u1,u2 be two solutions of (23)-(24) and u=u1-u2. Then u is also a solution with u(0)=0. Thus, (49) implies that |u(t)|2≤0, that is, u1=u2.


## 4. Regularity Results

In this section, we will consider further regularity results for the unique solution. The main result is thatu∈D(A), and thus u∈H2(D)2×H2(D) provided the initial function u0∈V. More precisely, we have the following.Theorem 5.
Assume thatu0∈V, and let u be the unique solution of (23)-(24). Then,
(50)u∈C([0,∞),V)∩Lloc2(0,∞;D(A)).Proof.
Letum be the approximate solution (33) in the proof of Theorem 4. We first claim that
(51)umremains  bounded  inL∞(0,T;V)∩L2(0,T;D(A)),∀T>0.
Indeed, multiplying (34) by λjgjm, summing these relations for j=1,2,…,m, and using (32), we find
(52)(dumdt,Aum)+μ(um,Aum)+α-1γ(um,um,Aum)+(Rum,Aum)=0.
By Lemma 1(iv) and the Young inequality, we find
(53)|α-1γ(um,um,Aum)|=|α-1(B(um,um),Aum)|≤sup0≤s≤Tα-1(s)·c3|um|θ3∥um∥2(1-θ3)|Aum|(1+θ3)≤14|Aum|2+c3′(T)∥um∥4|u|2θ3/(1-θ3).
For 0≤t≤T, by Lemma 2, (25), and the Young inequality, we have
(54)|(Rum,Aum)|≤sup0≤s≤T(α(s)+α-1(s))c4∥um∥|Aum|≤14|Aum|2+c4′∥um∥2.
Noting also that
(55)(dumdt,Aum)=(Aum,um′)=μ(um,um′)=12ddtμ(um,um)
and μ(um,Aum)=|Aum|2, we find from (52) and all the above estimates that
(56)ddtμ(um,um)+|Aum|2≤2c4′∥um∥2+2c3′∥um∥4|um|2θ3/(1-θ3).
By (38), um is bounded in L∞(0,T;H). This, together with Lemma 3, implies that (56) can be rewritten as
(57)ddtμ(um,um)+|Aum|2≤2c(1+∥um∥2)μ(um,um),0≤t≤T
for some appropriate constant c>0. By Gronwall lemma, it follows from (57) and (38) that
(58)μ(um(t),um(t))≤∥um(0)∥2exp(∫0tc(1+∥um(s)∥2)ds)≤M1(T),0≤t≤T,
which implies by Lemma 3 again that um remains bounded in L∞(0,T;V). Integrating in (57) from t=0 to  t=T, we have
(59)∫0T|Aum(t)|2ds≤2M1+∫0TM1(1+∥um(t)∥2)ds≤M2.
which proved the second argument of (51), and thus (51) holds.
Taking the limit in (51) (by weak compactness), we then find that u is in L∞(0,T;V)∩L2(0,T;D(A)). We need also to prove that u is continuous from [0,T] into V. This is proved as follows.
Sinceu∈C([0,T],H)∩L∞(0,T;V) and V⊂H with densely continuous injection, it follows from [10, Lemma II.3.3] that u:[0,T]→V is weakly continuous; that is, t↦((u(t),v))  is continuous for every v∈V. Similarly t↦μ((u(t),v))  is continuous for every v∈V. Thus, by taking the limit in (52) and applying [10, Lemma II.3.2], we obtain an equality similar to (52) for u:
(60)ddtμ(u,u)+2|Au|2+2α-1λ(u,u,Au)+2(Ru,u)=0,
which holds in the distribution sense on (0,T). Since u∈L∞(0,T;V)∩L2(0,T;D(A)),  it follows from Lemma 1 and Lemma 2 that
(61)|Au|2+α-1γ(u,u,Au)+(Ru,u)∈L1(0,T;ℝ)
and thus (d/dt)μ(u(t),u(t))∈L1(0,T;ℝ), which implies by [10, Lemma II.3.1] that the function t↦μ(u(t),u(t)) is continuous. Therefore, since μ(φ,φ)1/2 is a norm on V equivalent to ∥φ∥ (by Lemma 3), it follows that u:[0,T]→V is continuous for the norm topology.



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