Stochastic quasi-geostrophic equation
aa r X i v : . [ m a t h . P R ] A ug Stochastic quasi-geostrophic equation ∗ Michael R¨ockner a, Rongchan Zhu a,b , , Xiangchan Zhu a,c , † aDepartment of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany,bInstitute of Applied Mathematics, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, China,cSchool of Mathematical Sciences, Peking University, Beijing 100871, China Abstract
In this note we study the 2d stochastic quasi-geostrophic equation in T for general parameter α ∈ (0 ,
1) and multiplicative noise. We prove the existence of martingale solutions and pathwiseuniqueness under some condition in the general case , i.e. for all α ∈ (0 ,
1) . In the subcritical case α > /
2, we prove existence and uniqueness of (probabilistically) strong solutions and construct aMarkov family of solutions. In particular, it is uniquely ergodic for α > provided the noise isnon-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast.In the general case, we prove the existence of Markov selections.
1. Introduction and notation ——Consider the following two dimensional (2D) stochastic quasi-geostrophic equation in the periodic domain T = R / (2 π Z ) : ∂θ ( t, x ) ∂t = − u ( t, x ) · ∇ θ ( t, x ) − κ ( −△ ) α θ ( t, x ) + G ( θ, ξ )( t, x ) , (1 . θ (0 , x ) = θ ( x ) , where θ ( t, x ) is a real-valued function of x and t , 0 < α < , κ > u is determined by θ through a stream function ψ via the following relations: u = ( u , u ) = ( − R θ, R θ ) . (1 . R j is the j -th periodic Riesz transform and ξ ( t, x ) is a Gaussian random field, white noise in time,subject to the restrictions imposed below. The case α = is called the critical case, the case α > sub-critical and the case α < super-critical. The existence of weak solutions in the deterministiccase has been obtained in [7]. In the following, we will restrict ourselves to flows which have zeroaverage on the torus, i.e. R T θdx = 0 . Set H = L ( T ) and let | · | and h ., . i denote the norm and innerproduct in H , respectively. We recall that on T , sin( k · x ) , cos( k · x ) form an eigenbasis of −△ . Here k ∈ Z \{ } , x ∈ T and the corresponding eigenvalues are | k | . Define k f k H s := P k | k | s h f, e k i andlet H s denote the Sobolev space of all f for which k f k H s is finite. Set Λ = ( −△ ) / . Define the linearoperator A : D ( A ) ⊂ H → H as Au = κ ( −△ ) α u. The operator A is positive definite and selfadjoint.Denote the eigenvalues of A by 0 < λ ≤ λ ≤ · · · , and by e , e , ... the corresponding completeorthonormal system in H of eigenvectors of A . We also denote k u k = | A / u | , then k θ k ≥ λ | θ | . ∗ Research supported by 973 project, NSFC, key Lab of CAS, the DFG through IRTG 1132 and CRC 701 † E-mail address: [email protected](M. R¨ockner), [email protected](R. C. Zhu), [email protected](X. C. Zhu) . Existence and uniqueness of solutions —–By the above definitions Eqs (1.1)-(1.2) turn intothe abstract stochastic evolution equation (cid:26) dθ ( t ) + Aθ ( t ) dt + u ( t ) · ∇ θ ( t ) dt = G ( θ ( t )) dW ( t ) ,θ (0) = x, (2 . u satisfies (1.2) and W ( t ) is a cylindrical Wiener process in a separable Hibert space K definedon a probability space (Ω , F , P ). Here G is a mapping from H α to L ( K, H ). Definition 2.1
We say that there exists a martingale solution to (2.1) if there exists a stochastic basis(Ω , F , {F t } t ∈ [0 ,T ] , P ), a cylindrical Wiener process W on the space K and a progressively measurableprocess θ : [0 , T ] × Ω → H , such that for P -a.e ω ∈ Ω θ ( · , ω ) ∈ L ∞ (0 , T ; H ) ∩ L (0 , T ; H α ) ∩ C ([0 , T ]; H w ) (2 . P -a.s. h θ ( t ) , ψ i + Z t h A / θ ( s ) , A / ψ i ds − Z t h u ( s ) · ∇ ψ, θ ( s ) i ds = h x, ψ i + h Z t G ( θ ( s )) dW ( s ) , ψ i , (2 . t ∈ [0 , T ] and all ψ ∈ C ( T ). Here C ([0 , T ]; H w ) denotes the space of H -valued weaklycontinuous functions on [0 , T ]. Remark 2.2
Note that for regular functions θ and v , we have h u ( s ) · ∇ ( θ ( s ) + ψ ) , θ ( s ) + ψ i = 0 , so h u ( s ) · ∇ θ ( s ) , ψ i = −h u ( s ) · ∇ ψ, θ ( s ) i . Thus the integral equation (2.3) corresponds to equation (2.1).
Definition 2.3
We say that there exists a (probabilistically strong) solution to (2.1) over the timeinterval [0 , T ] if for every probability space (Ω , F , {F t } t ∈ [0 ,T ] , P ) with an F t -Wiener process W , thereexists a a progressively measurable process θ : [0 , T ] × Ω → H such that (2.2) and (2.3) hold. —–Consider the following condition:( G. G : H → L ( K, H ) is continuous and | G ( θ ) | L ( K,H ) ≤ λ | θ | + ρ, θ ∈ H for some positive realnumbers λ and ρ .By the compactness method based on fractional Sobolev spaces in [3], we obtain the existence ofmartingale solutions. Theorem 2.1.1
Under condition (G.1), there is a martingale solution (Ω , F , {F t } , P, W, θ ) to (2.1).Moreover, if x ∈ H and G satisfies Z ( X j | G ( θ )( e j ) | ) p/ dx ≤ C (1 + Z | θ | p dx ) , ∀ t > . < p < ∞ , then we have sup t ∈ [0 ,T ] k θ ( t ) k L p < ∞ , P − a.s. Theorem 2.1.2 If G ∈ L ( K, H ) does not depend on θ , then there is a martingale solution (Ω , F , {F t } , P, W, θ )to (2.1). Moreover, if x ∈ L p ( T ), then sup t ∈ [0 ,T ] k θ ( t ) k L p < ∞ , P − a.s. Theorem 2.1.3
Let G satisfy the Lipschitz condition k G ( u ) − G ( v ) k L ( K,H ) ≤ β | u − v | for all u, v ∈ dom ( G ), for some β ∈ R independent of u, v . Then (2.1) admits at most one (probabilisticallystrong) solution such that sup t ∈ [0 ,T ] k Λ − α + ε θ ( t ) k L p < ∞ , P − a.s. with p ≤ α + ε and ε ∈ (0 , α ]. —–In this section, we will consider the subcritical case. Theorem 2.2.1
Assume α > / G does not depend on θ with Tr(Λ − α + ε GG ∗ ) < ∞ forsome ε >
0. Then for each initial condition x ∈ H , there exists a (probabilistically strong) solution θ to (2.1) over [0 , T ] with initial condition θ (0) = x . Theorem 2.2.2
Assume α > . If G satisfies the following condition k Λ − / ( G ( u ) − G ( v )) k L ( K,H ) ≤ β | Λ − / ( u − v ) | (2 . u, v ∈ dom ( G ), for some β ∈ R independent of u, v , then (2.1) admits at most one (probabilis-tically strong) solution θ such that sup t ∈ [0 ,T ] k θ ( t ) k L q < ∞ , P − a.s. for some q with 0 ≤ /q < α − Corollary 2.2.3
Assume α > . If there exists a (probabilistically strong) solution θ such thatsup t ∈ [0 ,T ] k θ ( t ) k L q < ∞ , P − a.s. for some q with 0 ≤ /q < α − and G satisfies (2.5), then (2.1)admits only one such solution. Theorem 2.2.4
Assume α > and G satisfies (2.4), (2.5) and (G.1). Then for each initial condition x ∈ H , there exists a pathwise unique (probabilistic strong) solution θ of equation (2.1) over [0 , T ]with initial condition θ (0) = x . Moreover, the solution satisfies sup t ∈ [0 ,T ] k θ ( t ) k L p < ∞ , P − a.s. Theorem 2.2.5
Assume α > and that G ∈ L ( K, H ) does not depend on θ . Then for each initialcondition x ∈ L p for some p with 0 ≤ /p < α − , there exists a pathwise unique (probabilisticallystrong) solution θ of equation (2.1) over [0 , T ] with initial condition θ (0) = x . Moreover, this solutionsatisfies sup t ∈ [0 ,T ] k θ ( t ) k L p < ∞ , P − a.s. Theorem 2.2.6 (Markov property) Assume α > and that G ∈ L ( K, H ) does not depend on θ . If x ∈ L p for some p with 0 ≤ /p < α − , for every bounded, B ( H )-measurable F : H → R , and all s, t ∈ [0 , T ], s ≤ t, E ( F ( θ ( t )) |F s )( ω ) = E ( F ( θ ( t, s, θ ( s )( ω )))) for P − a.e.ω ∈ Ω , where θ ( t, s, θ ( s )( ω ))denotes the solution starting from θ ( s ) at time s .Set p t ( x, dy ) := P ◦ ( θ ( t, x )) − ( dy ) , ≤ t ≤ T, x ∈ H, and for B ( H )-measurable F : H → R , and t ∈ [0 , T ] , x ∈ H, P t F ( x ) := R F ( y ) p t ( x, dy ) , provided F is p t ( x, dy )-integrable. Then by Theorem2.2.6, we have for F : H → R , bounded and B ( H )-measurable, s, t ≥ , P s ( P t F )( x ) = P s + t F ( x ) , x ∈ L p for some p with 0 ≤ /p < α − .
3. Ergodicity and Exponential convergence for α > Assumption 3.1
There are an isomophism Q of H and a number s ≥ G = A − s − α α Q / .Set W = D (Λ s ) and | x | W = | Λ s x | . Then by a similar method as in [4] and using the abstractresults in [5] for exponential convergence, we obtain the following results. Theorem 3.2
Assume α > and Assumption 3.1. Then there exists a unique invariant measure ν on W for the transition semigroup ( P t ) t ≥ . Moreover:(i) The invariant measure ν is ergodic in the sense of [2].(ii) The transition semigroup ( P t ) t ≥ is W -strong Feller, irreducible, and therefore strongly mixingin the sense of [2].(iii) There is C exp > a > k P ∗ t δ x − µ k T V ≤ C exp (1 + | x | ) e − at , for all t > x ∈ H , where k · k T V is the total variation distance for measures.
4. Markov Selections in the general case —–By using the abstract results for Markov selectionsin [6], we obtain the following results.
Theorem 4.1
Assume G satisfies (G.1). Then there exists an almost sure Markov family ( P x ) x ∈ H for Eq. (2.1). Acknowledgement
We thank Wei Liu for very helpful discussions.
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