\mathbb{L}^p-solutions of deterministic and stochastic convective Brinkman-Forchheimer equations
aa r X i v : . [ m a t h . P R ] F e b L p -SOLUTIONS OF DETERMINISTIC AND STOCHASTIC CONVECTIVEBRINKMAN-FORCHHEIMER EQUATIONS MANIL T. MOHAN Abstract.
In the first part of this work, we establish the existence and uniqueness of alocal mild solution to the deterministic convective Brinkman-Forchheimer (CBF) equationsdefined on the whole space, by using properties of the heat semigroup and fixed pointarguments based on an iterative technique. The second part is devoted for establishing theexistence and uniqueness of a pathwise mild solution upto a random time to the stochasticCBF equations perturbed by L´evy noise by exploiting the contraction mapping principle.We also discuss the local solvability of the stochastic CBF equations subjected to fractionalBrownian noise. Introduction
Deterministic convective Brinkman-Forchheimer equations.
The Cauchy prob-lem for the convective Brinkman-Forchheimer equations (CBF) in R d , d ≥ ∂ u ( t, x ) ∂t − µ ∆ u ( t, x ) + ( u ( t, x ) · ∇ ) u ( t, x ) + α u ( t, x ) + β | u ( t, x ) | r − u ( t, x )+ ∇ p ( t, x ) = f ( t, x ) , in (0 , T ) × R d , (1.1)with the conditions ∇ · u ( t, x ) = 0 , in (0 , T ) × R d , u (0 , x ) = u ( x ) in { } × R d , | u ( t, x ) | → | x | → ∞ , t ∈ (0 , T ) . (1.2)In (1.1), u ( t, x ) ∈ R d stands for the velocity field at time t and position x , p ( t, x ) ∈ R represents the pressure field, f ( t, x ) ∈ R d is an external forcing. The constant µ denotes thepositive Brinkman coefficient (effective viscosity), the positive constants α and β representthe Darcy (permeability of porous medium) and Forchheimer (proportional to the porosity ofthe material) coefficients, respectively. For α = β = 0, we obtain the classical Navier-Stokesequations (NSE). The absorption exponent r ∈ [1 , ∞ ) and the case r = 3 is known as thecritical exponent. The critical homogeneous CBF equations (1.1) have the same scaling asNSE only when α = 0 (see Proposition 1.1, [14] and no scale invariance property for other Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar High-way, Roorkee, Uttarakhand 247667, INDIA. e-mail: [email protected], [email protected]. * Corresponding author.
Key words: convective Brinkman-Forchheimer equations, L´evy noise, fraction Brownian motion, mildsolution.Mathematics Subject Classification (2010): Primary 76D06; Secondary: 35Q30, 76D03, 47D03. values of α and r ). Since α does not play a major role in our analysis, we fix α = 0 and wescale µ and β to unity in the rest of the paper. The existence and uniqueness of weak aswell as strong solutions of the system (1.1)-(1.2) in the whole space and periodic domains isdiscussed in the works [4, 21, 14, 23], etc and the references therein.1.2. Abstract formulation and mild solution.
The
Helmholtz-Hodge projection denotedby P is a bounded linear operator from L p ( R d ) to J p := P L p ( R d ), 1 < p < ∞ . Note thatthe space J p is a separable Banach space with L p ( R d )-norm denoted by k·k p and the operator P is an orthogonal projection of L ( R d ) onto the subspace H := J . Remember that P can be expressed in terms of the Riesz transform (cf. [22] for more details). We use thenotation L ( H , J p ) for the space of all bounded linear operators from H to J p . Let us applythe projection operator P to the system (1.1) to obtain d u ( t )d t + A u ( t ) + B( u ( t )) + C ( u ( t )) = P f ( t ) , t ∈ (0 , T ) , u (0) = x , (1.3)where A u = − P ∆ u , with domain D p (A) = D p (∆) ∩ J p , B( u ) = B( u , u ) , with B( u , v ) = P [( u · ∇ ) v ] = P [ ∇ · ( u ⊗ v )] , C ( u ) = P [ | u | r − u ] , and x ∈ J p . For r ≥
1, the operator C ( · ) is Gateaux differentiable with the Gateauxderivative C ′ ( u ) v = P ( v ) , for r = 1 , ( P ( | u | v ) + ( r − P (cid:16) u | u | − r ( u · v ) (cid:17) , if u = , , if u = , for 1 < r < , P ( | u | r − v ) + ( r − P ( u | u | r − ( u · v )) , for r ≥ , (1.4)for all u , v ∈ L p ( R d ), for p ∈ [2 , ∞ ). It should be recalled that P ∆ = ∆ P (cf. [22]), andhence A is essentially equal to − ∆ and e − t A is substantially the heat semigroup (Gauss-Weierstrass semigroup, [16]) and is given by( e − t A u )( x ) = Z R d Ψ( t, x − y ) u ( y )d y, where Ψ( t, x ) = 1(4 πt ) d e − | x | t , t > , x ∈ R d , and u ∈ L q ( R d ), q ∈ [1 , ∞ ). Thus, the operator system (1.3) can be transformed into anonlinear integral equation as follows: u ( t ) = e − t A x − Z t e − ( t − s )A [B( u ( s )) + C ( u ( s ))]d s + Z t e − ( t − s )A P f ( s )d s, (1.5)for all t ∈ [0 , T ]. For a given x ∈ J p and f ∈ L (0 , T ; J p ), a function u ∈ C([0 , T ]; J p ) , formax n d, d ( r − o < p < ∞ satisfying (1.5) is called a mild solution to the system (1.3).Since e − t A is an analytic semigroup, we infer that e − t A : L p → L q is a bounded mapwhenever 1 < p ≤ q < ∞ and t >
0, and there exists a constant C depending on p and q such that (see [16]) k e − t A g k q ≤ Ct − d ( p − q ) k g k p , (1.6) TOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS 3 k∇ e − t A g k q ≤ Ct − − d ( p − q ) k g k p , (1.7)for all t ∈ (0 , T ] and g ∈ L p ( R d ). Using the estimates (1.6)-(1.7), one can estimate k e − t A B( u , v ) k p as k e − t A B( u , v ) k p ≤ Ct − ( + d p ) k u k p k v k p , (1.8)for all t ∈ (0 , T ] and u , v ∈ J p . Furthermore, using the estimate (1.6), we calculate k e − t A C ( u ) k p and k e − t A C ′ ( u ) v k p as k e − t A C ( u ) k p ≤ Ct − d ( r − p k u k rp , (1.9) k e − t A C ′ ( u ) v k p ≤ Ct − d ( r − p k u k r − p k v k p , (1.10)for all t ∈ (0 , T ] and u , v ∈ J p . For the existence of local mild solution in L p to the 3D NSE inwhole space and bounded domains, the interested readers are referred to see [11, 31, 16, 17],etc.1.3. Stochastic CBF equations perturbed by L´evy noise.
Let (Ω , F , P ) be a completeprobability space equipped with an increasing family of sub-sigma fields { F t } ≤ t ≤ T of F satisfying the usual conditions. On taking the external forcing as L´evy noise, one can rewritethe stochastic counterpart of the problem (1.3) for t ∈ (0 , T ) as d u ( t ) + [A u ( t ) + B( u ( t )) + C ( u ( t ))]d t = ΦdW( t ) + Z Z γ ( s − , z ) e π (d s, d z ) , u (0) = x . (1.11)In (1.11), W = { W( t ) } ≤ t ≤ T is a cylindrical Wiener process and for an orthonormal basis { e j ( x ) } ∞ j =1 in H := J , W( · ) can be represented as W( t ) = ∞ X j =1 e j ( x ) β j ( t ), where { β j ( · ) } ∞ j =1 ’sare a sequence of one-dimensional mutually independent Brownian motions ([7]). Thebounded linear operator Φ : H → J p , p ∈ [2 , ∞ ) is a γ -radonifying operator in J p suchthat ΦdW( t ) = ∞ X j =1 Φ e j ( x )d β j ( t ) = ∞ X j =1 Z R d K ( x, y ) e j ( y )d y d β j ( t ) , where K ( · , · ) is the kernel of the operator Φ (Theorem 2.2, [1]). In particular, the operatorΦ ∈ γ ( H , J p ) satisfies k Φ k γ ( H , J p ) ≡ (Z R d (cid:20)Z R d | K ( x, y ) | d y (cid:21) p/ d x ) /p < + ∞ , where γ ( H , J p ) is the space of all γ -radonifying operators from H to J p . Let us denote by Z, a measurable subspace of some Hilbert space (for example measurablesubspaces of R d , L ( R d ), etc) and λ (d z ), a σ -finite L´evy measure on Z with an associated Let U be a real separable Hilbert space and X be a Banach space. A bounded linear operator R ∈ L ( U , X )is γ -radonifying provided that there exists a centered Gaussian probability ν on X such that R X ϕ ( x )d ν ( x ) = k R ∗ ϕ k U , ϕ ∈ X ∗ . Such a measure is at most one, and hence we set k R k γ ( U , X ) := R X k x k X d ν ( x ) . We denote γ ( U , X ) for the space of γ -radonifying operators, and γ ( U , X ) equipped with the norm k·k γ ( U , X ) is a separableBanach space. M. T. MOHAN
Poisson random measure π (d t, d z ). We define e π (d t, d z ) := π (d t, d z ) − λ (d z )d t as the com-pensated Poisson random measure. The jump noise coefficient γ ( t, z ) := γ ( t, z, x ) is suchthat γ : [0 , T ] × Z × J p → J p , p ∈ [2 , ∞ ) and in particular, γ satisfies Z T Z Z k γ ( t, z ) k p λ (d z )d t < + ∞ . The processes W( · ) and π ( · , · ) are mutually independent. The existence and uniqueness ofpathwise strong solutions to the stochastic CBF equations and related models perturbedby Gaussian as well as jump noises in the whole space or periodic domains are available inthe literature and the interested readers are referred to see [28, 2, 9, 24, 25], etc, and thereferences therein.We transform the operator system (1.3) into a stochastic nonlinear integral equation asfollows: u ( t ) = e − t A x − Z t e − ( t − s )A [B( u ( s )) + C ( u ( s ))]d s + Z t e − ( t − s )A ΦdW( s )+ Z t Z Z e − ( t − s )A γ ( s − , z ) e π (d s, d z ) , (1.12)for all t ∈ [0 , T ]. The existence of pathwise mild solutions for 2D and 3D NSE perturbed byGaussian as well as jump noise is available in [8, 13, 22, 33], etc and the references therein.1.4. Stochastic CBF equations perturbed by fractional Brownian noise.
Let usnow consider the stochastic CBF equations perturbed by fractional Brownian noise as ( d u ( t ) + [A u ( t ) + B( u ( t )) + C ( u ( t ))]d t = ΦdW H ( t ) , u (0) = x . (1.13)where Φ ∈ L ( H , J p ) and W H ( · ) is the cylindrical fractional Brownian motion with Hurstparameter H ∈ (cid:0) max (cid:8) , d (cid:9) , (cid:1) , where d = 2 , u ( t ) = e − t A x − Z t e − ( t − s )A [B( u ( s )) + C ( u ( s ))]d s + Z t e − ( t − s )A ΦdW H ( s ) , (1.14)for all t ∈ [0 , T ]. For the well-posedness and existence of density for 2D stochastic NSE per-turbed by fractional Brownian noise, we refer the interested readers to [12, 15], respectively.1.5. Major objectives.
The purpose of this work is two folded.(i) In the first part, we show the existence of a unique local mild solution to the de-terministic CBF equations (1.3) in L p -spaces, for max n d, d ( r − o < p < ∞ , d ≥ J p , for max n d, d ( r − o < p < ∞ , d ≥ TOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS 5 (b) By considering the noise as fractional Brownian motion, we show the existenceand uniqueness of local pathwise mild solutions (up to a random time) to thestochastic CBF equations in J p , for max n d, d ( r − o < p < ∞ and max (cid:8) , d (cid:9) In this section, we present the existence and uniqueness of local mild solution to theproblem (1.3). We use fixed point arguments (by using a simple iterative technique) toobtain the required result. Theorem 2.1. For max n d, d ( r − o < p < ∞ , let x ∈ J p and f ∈ L (0 , T ; J p ) be given. Then,there exists a time < T ∗ < T such that (1.3) has a unique mild solution given by (1.5) in C([0 , T ∗ ]; J p ) .Proof. As discussed in [11, 16], etc, in order to prove the theorem, we use an iterativetechnique. Let us set u ( t ) = e − t A x , (2.1) u n +1 ( t ) = u + G( u n )( t ) , n = 0 , , , . . . , (2.2)where G( u )( t ) = − Z t e − ( t − s )A [B( u ( s )) + C ( u ( s ))]d s + Z t e − ( t − s )A P f ( s )d s, which is continuous for all t ∈ [0 , T ]. Since e − t A is a contraction semigroup on L p ( R d ), firstwe note that k u ( t ) k p = k e − t A x k p ≤ k x k p . Using the estimates (1.6)-(1.10), we find k u n +1 ( t ) k p ≤ k x k p + Z t k e − ( t − s )A B( u n ( s )) k p d s + Z t k e − ( t − s )A C ( u n ( s )) k p d s M. T. MOHAN + Z t k e − ( t − s )A P f ( s ) k p d s ≤ k x k p + C Z t ( t − s ) − ( + d p ) k u n ( s ) k p d s + C Z t ( t − s ) − d ( r − p k u n ( s ) k rp d s + C Z t k f ( s ) k p d s ≤ (cid:26) k x k p + C Z t k f ( s ) k p d s (cid:27) + Ct − d p sup s ∈ [0 ,t ] k u n ( s ) k p + Ct − d ( r − p sup s ∈ [0 ,t ] k u n ( s ) k rp ≤ (cid:26) k x k p + C Z T k f ( s ) k p d s (cid:27) + CT − d p f n + CT − d ( r − p f rn , (2.3)for all t ∈ [0 , T ], where f n = sup t ∈ [0 ,T ] k u n ( t ) k p . For f = n k x k p + C R T k f ( s ) k p d s o , from the above relation, it is immediate that f n +1 ≤ f + CT − d p f n + CT − d ( r − p f rn , n = 0 , , , . . . , (2.4)which is a nonlinear recurrence relation. One can easily show by induction that if12 min ((cid:18) CT − d p (cid:19) , (cid:18) CT − d ( r − p (cid:19) r − ) > f , then f n ≤ min ((cid:18) CT − d p (cid:19) , (cid:18) CT − d ( r − p (cid:19) r − ) =: K, for all n = 1 , , , . . . , so that the sequence { f n } is uniformly bounded.Let us now consider v n +2 ( t ) = u n +2 ( t ) − u n +1 ( t )= − Z t e − ( t − s )A [B( u n +1 ( s )) − B( u n ( s ))]d s − Z t e − ( t − s )A [ C ( u n +1 ( s )) − C ( u n ( s ))]d s, (2.5)for all t ∈ [0 , T ]. Once again using the estimates (1.6)-(1.10), we obtain k v n +2 ( t ) k p = k u n +2 ( t ) − u n +1 ( t ) k p ≤ Z t k e − ( t − s )A B( u n +1 ( s ) − u n ( s ) , u n +1 ( s )) k p d s + Z t k e − ( t − s )A B( u n ( s ) , u n +1 ( s ) − u n ( s )) k p d s + Z t (cid:13)(cid:13)(cid:13)(cid:13) e − ( t − s )A Z C ′ ( θ u n +1 ( s ) + (1 − θ ) u n ( s ))( u n +1 ( s ) − u n ( s ))d θ (cid:13)(cid:13)(cid:13)(cid:13) p d s ≤ C Z t ( t − s ) − ( + d p )( k u n +1 ( s ) k p + k u n ( s ) k p ) k u n +1 ( s ) − u n ( s ) k p d s TOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS 7 + C Z t ( t − s ) − d ( r − p ( k u n +1 ( s ) k r − p + k u n ( s ) k r − p ) k u n +1 ( s ) − u n ( s ) k p d s ≤ Ct − d p sup s ∈ [0 ,t ] ( k u n +1 ( s ) k p + k u n ( s ) k p ) sup s ∈ [0 ,t ] k v n +1 ( s ) k p + Ct − d ( r − p sup s ∈ [0 ,t ] (cid:0) k u n +1 ( s ) k r − p + k u n ( s ) k r − p (cid:1) sup s ∈ [0 ,t ] k v n +1 ( s ) k p , (2.6)for all t ∈ [0 , T ]. Therefore, we deduce thatsup t ∈ [0 ,T ] k v n +2 ( t ) k p ≤ C (cid:16) KT − d p + K r − T − d ( r − p (cid:17) sup t ∈ [0 ,T ] k v n +1 ( t ) k p ≤ C n +1 (cid:16) KT − d p + K r − T − d ( r − p (cid:17) n +1 sup t ∈ [0 ,T ] k v ( t ) k p ≤ KC n +1 (cid:16) KT − d p + K r − T − d ( r − p (cid:17) n +1 , n = 0 , , , . . . . (2.7)Let us now consider the infinite series of the form u ( t ) + v ( t ) + v ( t ) + · · · + v n ( t ) + · · · . (2.8)The n th partial sum of the series is u n ( t ), that is, u n ( t ) = u ( t ) + n − X m =0 v m +1 ( t ) . (2.9)Therefore, the sequence { u n ( t ) } converges if and only if the series (2.8) converges. From theinequality (2.7), we havesup t ∈ [0 ,T ] k u ( t ) k p + ∞ X m =0 sup t ∈ [0 ,T ] k v m +1 ( t ) k p ≤ K ∞ X m =0 KC m (cid:16) KT − d p + K r − T − d ( r − p (cid:17) m = K K − C (cid:16) KT − d p + K r − T − d ( r − p (cid:17) < + ∞ , (2.10)provided C (cid:16) KT − d p + K r − T − d ( r − p (cid:17) < . Thus, we can choose a time 0 < T ∗ < T in such a way that the above condition is satisfied.Therefore the series (2.8) converges uniformly in [0 , T ∗ ] and we denote the sum of the seriesby u ( t ). Then, the relation (2.9) provides u ( t ) = lim n →∞ u n ( t ) . The uniform convergence of u n ( t ) to u ( t ) and the continuity of the operator B( · ) + C ( · ) givesus u ( t ) = u + G( u )( t ) , M. T. MOHAN which is a mild solution to the problem (1.3) in the interval [0 , T ∗ ]. The continuity ofthe function u ( · ) follows from the uniform convergence and the continuity of the sequence { u n ( · ) } ∞ n =0 .Let us now show the uniqueness. Let u ( · ) and u ( · ) be two local mild solutions of theproblem (1.3). Then u = u − u satisfies: u ( t ) = − Z t e − ( t − s )A [B( u ( s )) − B( u ( s ))]d s − Z t e − ( t − s )A [ C ( u ( s )) − C ( u ( s ))]d s. (2.11)A calculation similar to (2.6) yields k u ( t ) k p ≤ CT ∗ − d p sup t ∈ [0 ,T ∗ ] ( k u ( s ) k p + k u ( s ) k p ) sup t ∈ [0 ,T ∗ ] k u ( s ) k p + CT ∗ − d ( r − p sup t ∈ [0 ,T ∗ ] (cid:0) k u ( s ) k r − p + k u ( s ) k r − p (cid:1) sup t ∈ [0 ,T ∗ ] k u ( s ) k p ≤ C (cid:16) KT ∗ − d p + K r − T ∗ − d ( r − p (cid:17) sup t ∈ [0 ,T ∗ ] k u ( s ) k p , (2.12)for all t ∈ [0 , T ∗ ]. One can choose a T ∗ such that C (cid:16) KT ∗ − d p + K r − T ∗ − d ( r − p (cid:17) < u ∈ C([0 , T ∗ ]; J p ) follows. (cid:3) Existence and Uniqueness of Stochastic CBF equations This section is devoted for establishing the existence and uniqueness of mild solution upto a random time to the system (1.11). We use the contraction mapping principle to obtainthe required result.3.1. The linear problem. For p ∈ [2 , ∞ ), we know that e − t A is a C -contraction semigroupon L p ( R d ), and L p ( R d ) is an martingale type 2 Banach space and also a 2-smooth Banachspace. Let us now consider the stochastic Stokes equation: d w ( t ) + A w ( t )d t = ΦdW( t ) + Z Z γ ( t − , z ) e π (d t, d z ) , w (0) = . (3.1)Making use of Theorem 3.6, [32], the unique solution of the problem (3.1) with paths inL ∞ (0 , T ; J p ), p ∈ [2 , ∞ ) , P -a.s., can be represented by the stochastic convolution w ( t ) = Z t e − ( t − s )A ΦdW( s ) + Z t Z Z e − ( t − s )A γ ( s − , z ) e π (d s, d z ) , (3.2)for all t ∈ [0 , T ], and (3.2) has a c`adl`ag modification such that E (cid:20) sup ≤ t ≤ T k w ( t ) k p (cid:21) ≤ C (cid:18) k Φ k γ ( H , J p ) T + Z T Z Z k γ ( t, z ) k p λ (d z )d t (cid:19) , (3.3)and sup ≤ t ≤ T k w ( t ) k p < ∞ , P -a.s. TOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS 9 The nonlinear problem. Let us now establish the existence of a local mild solutionto the stochastic CBF system (1.11). Definition 3.1. A J p -valued and F t -adapted stochastic process u : [0 , T ] × R d × Ω → R with P -a.s. c`adl`ag trajectories for t ∈ [0 , T ] , is a mild solution to the system (1.11), if for any T > , u ( t ) := u ( t, · , · ) satisfies the integral equation (1.12) P -a.s., for each t ∈ [0 , T ] . Let us set v = u − w . Then, v ( · ) satisfies the following system P -a.s.: d v ( t )d t + [A v ( t ) + B( v ( t ) + w ( t )) + C ( v ( t ) + w ( t ))] = , u (0) = x . (3.4)Note that for each fixed ω ∈ Ω, (3.4) is a deterministic system. The operator system (3.4)can be transformed into an nonlinear integral equation as v ( t ) = e − t A x − Z t e − ( t − s )A [B( v ( s ) + w ( s )) + C ( v ( s ) + w ( s ))]d s, (3.5)for all t ∈ [0 , T ]. As in the case of deterministic CBF equations, we obtain the existence ofa unique local mild solution to the system (3.4) by using the contraction mapping principlein the space C([0 , e T ]; J p ), P -a.s., for max n d, d ( r − o < p < ∞ , where 0 < e T < T is a randomtime. Let us setΣ( M, e T ) = n v ∈ C([0 , e T ]; J p ) : k v ( t ) k p ≤ M, P -a.s., for all t ∈ [0 , e T ] o . (3.6)Clearly the space Σ( M, e T ) equipped with supremum topology is a complete metric space. Theorem 3.2. For max n d, d ( r − o < p < ∞ , let the F -measurable initial data x ∈ J p , P -a.s. be given. For M > k x k p , there exists a random time e T such that (3.4) has a uniquemild solution in Σ( M, e T ) .Proof. Let us take any v ∈ Σ( M, e T ) and define y ( t ) = F( v )( t ) by y ( t ) = e − t A x − Z t e − ( t − s )A [B( v ( s ) + w ( s )) + C ( v ( s ) + w ( s ))]d s, (3.7)for all t ∈ [0 , e T ]. Let us first establish that G : Σ( M, e T ) → Σ( M, e T ) . Making use of theestimates (1.6)-(1.10), we find k y ( t ) k p ≤ k e − t A x k p + Z t k e − ( t − s )A [B( v ( s ) + w ( s )) + C ( v ( s ) + w ( s ))] k p d s ≤ k x k p + C Z t ( t − s ) − ( + d p ) k v ( s ) + w ( s ) k p d s + C Z t ( t − s ) − d ( r − p k v ( s ) + w ( s ) k rp d s ≤ k x k p + Ct − d p sup s ∈ [0 ,t ] k v ( s ) + w ( s ) k p + Ct − d ( r − p sup s ∈ [0 ,t ] k v ( s ) + w ( s ) k rp ≤ k x k p + C e T − d p sup t ∈ [0 , e T ] k v ( t ) k p + sup t ∈ [0 , e T ] k w ( t ) k p ! + C e T − d ( r − p sup t ∈ [0 , e T ] k v ( t ) k rp + sup t ∈ [0 , e T ] k w ( t ) k rp ! ≤ k x k p + C e T − d p ( M + µ p ) + C e T − d ( r − p ( M r + µ rp ) , (3.8) P -a.s., for all t ∈ [0 , e T ], where µ p = sup t ∈ [0 ,T ] k w ( t ) k p . Now, since M > k x k p , P -a.s., and max n d, d ( r − o < p < ∞ , one can choose 0 < e T < T insuch a way that k y ( t ) k p ≤ M , for all t ∈ [0 , e T ], provided k x k p + C e T − d p ( M + µ p ) + C e T − d ( r − p ( M r + µ rp ) ≤ M. Therefore y ∈ Σ( M, e T ).Our next aim is to show that F : Σ( M, e T ) → Σ( M, e T ) is a contraction. Let us consider v , v ∈ Σ( M, e T ) and set y i ( t ) = G( v i )( t ), for all t ∈ [0 , e T ] and i ∈ { , } and y = y − y .Then y ( · ) satisfies y ( t ) = − Z t e − ( t − s )A [B( v ( s ) + w ( s )) − B( v + w ( s )) + C ( v ( s ) + w ( s )) − C ( v ( s ) + w ( s ))]d s, P -a.s., for all t ∈ [0 , e T ]. Once again using the bilinearity of B( · ) and Taylor’s formula, wefind k y ( t ) k p ≤ Z t k e − ( t − s )A B( v ( s ) − v ( s ) , v ( s ) + w ( s )) k p d s + Z t k e − ( t − s )A B( v ( s ) + w ( s ) , v ( s ) − v ( s )) k p d s + Z t (cid:13)(cid:13)(cid:13)(cid:13) e − ( t − s )A Z C ′ ( θ v ( s ) + (1 − θ ) v ( s ) + w ( s ))( v ( s ) − v ( s ))d θ (cid:13)(cid:13)(cid:13)(cid:13) p d s ≤ C Z t ( t − s ) − ( + d p ) k v ( s ) + w ( s ) k p k v ( s ) − v ( s ) k p d s + C Z t ( t − s ) − ( + d p ) k v ( s ) + w ( s ) k p k v ( s ) − v ( s ) k p d s + C Z t ( t − s ) − d ( r − p ( k v ( s ) k p + k v ( s ) k p + k w ( s ) k p ) r − k v ( s ) − v ( s ) k p d s ≤ Ct − d p sup s ∈ [0 ,t ] ( k v ( s ) k p + k v ( s ) k p + k w ( s ) k p ) sup s ∈ [0 ,t ] k y ( s ) k p + Ct − d ( r − p sup s ∈ [0 ,t ] (cid:0) k v ( s ) k r − p + k v ( s ) k r − p + k w ( s ) k r − p (cid:1) sup s ∈ [0 ,t ] k y ( s ) k p ≤ C (cid:16) e T − d p ( M + µ p ) + e T − d ( r − p ( M r − + µ r − p ) (cid:17) sup t ∈ [0 , e T ] k y ( t ) k p , (3.9) TOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS 11 for all t ∈ [0 , e T ]. For max n d, d ( r − o < p < ∞ , one can choose 0 < e T < T in such a waythat C (cid:16) e T − d p ( M + µ p ) + e T − d ( r − p ( M r − + µ r − p ) (cid:17) < . Hence, F is a strict contraction in Σ( M, e T ) and an application of the contraction mappingprinciple provides the existence of mild solution to the problem (3.4) up to a random time0 < e T < T . Uniqueness follows form the representation (3.5). (cid:3) Since u = v + w , we immediately obtain the following Theorem on the existence of mildsolution to the system (1.11). Theorem 3.3. For max n d, d ( r − o < p < ∞ , let the F -measurable initial data x ∈ J p , P -a.s. be given. Then there exists a random time < e T < T such that (1.11) has a uniquemild solution u ∈ L ∞ (0 , e T ; J p ) , P -a.s. with a c`adl`ag modification. Stochastic CBF equations subjected to fraction Brownian motion In this section, we obtain the existence and uniqueness of a local mild solution up to arandom time for the stochastic CBF equations (1.13), for d = 2 , Fractional Brownian motion. The first study on fractional Brownian motion (fBm)within the Hilbertian framework is reported in [19]. Due to various practical applications,the stochastic analysis of fBm has been intensively developed starting from the nineties.For a comprehensive study, the interested readers are referred to see [27, 26], etc. In thissubsection, we provide a brief description of fBm and its stochastic integral representationin separable Hilbert spaces (cf. sections 4 and 5, [18] for separable Banach spaces). Let usconsider a time interval [0 , T ], where T is an arbitrary fixed time horizon. Definition 4.1. A fractional Brownian motion (fBm) with Hurst parameter H ∈ (0 , is acentered Gaussian process W H with covariance R H ( t, s ) := E (cid:2) W H ( t )W H ( s ) (cid:3) = 12 (cid:0) t H + s H − | t − s | H (cid:1) , where s, t ∈ [0 , T ] . Note that if H = , then W is the standard Brownian motion. It should be recalled thatfBm is not a Markov process except in the case H = . The fBm is the only H -self-similarGaussian process (that is, for any constant a > 0, the processes { a − H W H ( at ) } ≤ t ≤ T andW H = { W H ( t ) } ≤ t ≤ T have the same distribution) with stationary increments (Proposition1.1, [30]) E (cid:2) (W H ( t ) − W H ( s )) (cid:3) = | t − s | H . Furthermore, the process W H admits the Wiener integral representation of the formW H ( t ) = Z t K H ( t, s )dW( s ) , (4.1)where W = { W( t ) } ≤ t ≤ T is a Wiener process, and K H ( · , · ) is the kernel given by K H ( t, s ) = d H ( t − s ) H − + s H − F (cid:18) ts (cid:19) , where d H is a constant andF( z ) = d H (cid:18) − H (cid:19) Z z − θ H − (cid:16) − ( θ + 1) H − (cid:17) d θ. For H > , the kernel K H ( · , · ) has the simpler expression K H ( t, s ) = c H s − H Z ts ( u − s ) H − u H − d u, where t > s and c H = (cid:16) H ( H − β (2 − H,H − ) (cid:17) , β ( · , · ) being the beta function. The fact that theprocess defined by (4.1) is a fBm follows from the equality Z t ∧ s K H ( t, u ) K H ( s, u )d u = R H ( t, s ) . Moreover, the kernel K H ( · , · ) satisfies the condition ∂∂t K H ( t, s ) = d H (cid:18) H − (cid:19)(cid:16) st (cid:17) − H ( t − s ) H − . Note that the fBm is an α -regular Volterra process for α = H − , where H > (see Remark2.2, [6] for more details).Let U be a separable Hilbert space with scalar product ( · , · ). Let E H denote the linearspace of U -valued step functions on [0 , T ] of the form ϕ ( t ) = m − X i =0 x i [ t i ,t i +1 ) ( t ) , (4.2)where 0 = t , t , t , . . . , t m ∈ [0 , T ], m ∈ N , x i ∈ U . The space E H is equipped with the innerproduct m − X i =0 x i [0 ,t i ) , n − X j =0 y j [0 ,s j ) ! H = m − X i =0 n − X j =0 ( x i , y j ) R H ( t i , s j ) . Note that E H is a pre-Hilbert space and we denote the completion of E H with respect to( · , · ) H by H . For ϕ ∈ E H of the form (4.2), let us define its Wiener integral with respect tothe fBm as Z T ϕ ( s )dW H ( s ) = m − X i =0 x i (W H ( t i +1 ) − W H ( t i )) . It is clear that the mapping ϕ = P mi =1 x i ( t i ,t i +1 ] R T ϕ ( s )dW H ( s ) is an isometry between E H and the linear space span { W H ( t ) : t ∈ [0 , T ] } viewed as a subspace of L (Ω; U ), since E "(cid:13)(cid:13)(cid:13)(cid:13)Z T ϕ ( s )dW H ( s ) (cid:13)(cid:13)(cid:13)(cid:13) U = k ϕ k H . The image of an element ϕ ∈ H under this isometry is called the Wiener integral of ϕ withrespect to the fBm W H . For 0 < s < T , we consider the operator K ∗ : E H → L (0 , T ; U ) as( K ∗ T ϕ )( s ) = K ( T, s ) ϕ ( s ) + Z Ts ( ϕ ( r ) − ϕ ( s )) ∂K∂r ( r, s )d r. TOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS 13 For H > , the operator K ∗ has the simpler expression( K ∗ T ϕ )( s ) = Z Ts ϕ ( r ) ∂K∂r ( r, s )d r. The integrals appearing on the right-hand side are both Bochner integrals. Since the operator K ∗ satisfies ( K ∗ ϕ, K ∗ ψ ) L (0 ,T ; U ) = ( ϕ, ψ ) H , for all ϕ, ψ ∈ E H , K ∗ can be extended to anisometry between H and L (0 , T ; U ) in the sense that E "(cid:13)(cid:13)(cid:13)(cid:13)Z T ϕ ( s )dW H ( s ) (cid:13)(cid:13)(cid:13)(cid:13) U = k K ∗ ϕ k (0 ,T ; U ) = k ϕ k H , for all ϕ ∈ H . Hence we have the following connection with the Wiener process W Z t ϕ ( s )dW H ( s ) = Z t ( K ∗ t ϕ )( s )dW( s ) , (4.3)for every t ∈ [0 , T ], and ϕ [0 ,t ] ∈ H if and only if K ∗ ϕ ∈ L (0 , T ; U ). Furthermore, if ϕ, ψ ∈ H are such that R T R T | ϕ ( t ) || ψ ( t ) || t − s | H − d s d t < ∞ , then their scalar product in H is given by ( ϕ, ψ ) H = Z T Z T ϕ ( t ) ψ ( t ) | t − s | H − d s d t. In general, careful justification is needed for the existence of right hand side of (4.3) (cf.section 5.1, [26]). As we are discussing the case of Wiener integrals over the Hilbert space U , we point out that if ϕ ∈ L (0 , T ; U ) is a deterministic function, then the relation (4.3)holds, and the right hand is well defined in L (Ω; U ) if K ∗ H ϕ ∈ L (0 , T ; U ).4.2. Cylindrical Brownian motion. For a Hilbert space U , let us now define the standardcylindrical fractional Brownian motion in U as the formal series (cf. [10, 29])W H ( t ) = ∞ X n =0 e n W Hn ( t ) , (4.4)where { e n } ∞ n =1 is a complete orthonormal basis in U and W Hn is an one dimensional fBm.It should be noted that the series (4.4) does not converge in L (Ω; U ) , and thus W H ( t ) isnot a well-defined U -valued random variable. But, one can consider a Hilbert space U suchthat U ⊂ U , the linear embedding is a Hilbert-Schmidt operator, therefore, the series (4.4)defines a U -valued Gaussian random variable and { W H ( t ) } t ∈ [0 ,T ] is a U -valued cylindricalfBm.Let Y be an another real and separable Hilbert space and L ( U , Y ) denote the spaceof Hilbert-Schmidt operators from U to Y . As discussed in [7], it is possible to define astochastic integral of the form: Z T ϕ ( t )dW H ( t ) , (4.5)where ϕ : [0 , T ] L ( U , Y ), and the integral (4.5) is a Y -valued random variable, whichis independent of choice of U . Let ϕ be a deterministic function with values in L ( U , Y )satisfying:(i) for each x ∈ U , ϕ ( · ) x ∈ L p (0 , T ; Y ), for p > H ,(ii) R T R T k ϕ ( s ) k L ( U , Y ) k ϕ ( t ) k L ( U , Y ) | s − t | H − d s d t < ∞ . Then the stochastic integral (4.5) can be expressed as Z T ϕ ( t )dW H ( t ) := ∞ X n =1 Z t ϕ ( s ) e n dW Hn ( s ) = ∞ X n =1 Z t ( K ∗ H ϕe n )dW n ( s ) , (4.6)where W n is the standard Brownian motion connected to fBm W Hn by the representationformula (4.1). Since H ∈ ( , 1) implies ϕe n ∈ L (0 , T ; Y ), for each n ∈ N , so that theterms of the series (4.6) are well-defined. Moreover, the sequence of random variables nR t ϕ ( s ) e n dW Hn ( s ) o ∞ n =1 are mutually independent Gaussian random variables (cf. [10]).For cylindrical Brownian motions in a separable Banach space Y , the interested readersare referred to see sections 4 and 5, [18]. For stochastic integrals in Y , a series expansionsimilar to (4.6) is available, where the Hilbert-Schmidt operators from U to Y are replacedby γ -radonifying operators from U to Y (see [18] for more details). One can refer [5, 20],etc for the local solvability in L p -spaces for some mathematical models like semilinear heatequation, Hardy-H´enon parabolic equations, etc perturbed by fBm.4.3. SCBF equations perturbed by fractional Brownian motion. We consider U = H = J , { e j } ∞ j =1 as the complete orthonormal basis of J , and we take d = 2 , 3. Next, weconsider the following stochastic Stokes equation perturbed by fractional Brownian noise asd w ( t ) + A w ( t )d t = ΦdW H ( t ) , w (0) = , ) (4.7)where Φ ∈ L ( H , J p ) and W H = { W H ( t ) } t ∈ [0 ,T ] is a cylindrical fractional Brownian process.Since the operator A generates an analytic semigroup on J p , by standard estimates on Green’sfunction, we obtain (cf. [6]) k S( t )Φ k γ ( H , J p ) ≤ Ct − d , for t > . (4.8)Using Corollary 4.1, [6] (see Remark 4.2 and Section 5.2 also), under the assumptionmax (cid:26) , d (cid:27) < H < , the unique solution of the problem (3.1) with paths in C([0 , T ]; J p ), p ∈ [2 , ∞ ) , P -a.s., canbe represented by the stochastic convolution w ( t ) = Z t e − ( t − s )A ΦdW H ( s ) , (4.9)for all t ∈ [0 , T ] has a modification such thatsup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)(cid:13)Z t e − ( t − s )A dW H ( s ) (cid:13)(cid:13)(cid:13)(cid:13) p < ∞ , P -a.s. (4.10)Then the following theorem can be established in a similar way as that of Theorems 3.2 and3.3. Theorem 4.2. For max n d, d ( r − o < p < ∞ and max (cid:8) , d (cid:9) < H < , d = 2 , , let the F -measurable initial data x ∈ J p , P -a.s. be given. Then there exists a random time b T suchthat (1.11) has a unique mild solution u ∈ C([0 , b T ]; J p ) , P -a.s. satisfying (1.14) . TOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS 15 Remark 4.3. One can also consider the stochastic CBF equations perturbed by α -regularVolterra processes as (cid:26) d u ( t ) + [A u ( t ) + B( u ( t )) + C ( u ( t ))]d t = Φd B ( t ) , u (0) = x , (4.11) where Φ ∈ L ( H , J p ) satisfies (4.8) and B is an infinite-dimensional α -regular cylindricalVolterra process with α ∈ (0 , ) , which belongs to a finite Wiener chaos (see [6] for moredetails on α -regular Volterra processes). Then for α > d − , d = 2 , , the process w ( t ) = Z t e − ( t − s )A Φd B ( s ) , has a modification in C([0 , T ]; J p ) , p ∈ [ α , ∞ ) , P -a.s. Thus a result similar to Theorem 4.2can be obtained in this case also for the system (4.11) , that is, the existence and uniquenessof a mild solution u ( t ) = e − t A x − Z t e − ( t − s )A [B( u ( s )) + C ( u ( s ))]d s + Z t e − ( t − s )A Φd B ( s ) , for t ∈ [0 , T ] , where < T < T is a random time, to the system (4.11) with P -a.s. continuousmodification with trajectories in J p , for max n d, d ( r − o < p < ∞ . Conclusions and future plans: The existence and uniqueness of a local mild solution in L p ( R d ) with max n d, d ( r − o < p < ∞ for deterministic and stochastic CBF equations in R d (for various kinds of noises) is established in this work. The case of p = max n d, d ( r − o isan interesting problem and it will be addressed in a future work (for similar works, see [16]for the deterministic NSE and [22] for stochastic NSE). Acknowledgments: M. T. Mohan would like to thank the Department of Science andTechnology (DST), India for Innovation in Science Pursuit for Inspired Research (INSPIRE)Faculty Award (IFA17-MA110). Conflict of interest: The author has no conflicts of interest to declare that are relevant tothe content of this article. References [1] Z. Brze´zniak and H. Long, A note on γ -radonifying and summing operators, Stochastic Analysis , BanachCenter Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa, (2015), 43–57.[2] Z. Brze´zniak and Gaurav Dhariwal, Stochastic tamed Navier-Stokes equations on R : the existence andthe uniqueness of solutions and the existence of an invariant measure, Journal of Mathematical FluidMechanics , , Article number: 23 (2020).[3] Z. Brze´zniak, E. Hausenblas and J. Zhu, Maximal inequality for stochastic convolutions driven bycompensated Poisson random measures in Banach spaces, Ann. Inst. Henri Poincar´e Probab. Stat., (2017), 937–956. [4] Z. Cai and Q. Jiu, Weak and Strong solutions for the incompressible Navier-Stokes equations withdamping, Journal of Mathematical Analysis and Applications , (2008), 799–809.[5] J. Clarke and C. Olivera, Local L p -solution for semilinear heat equation with fractional noise, https://arxiv.org/abs/1902.06084 .[6] P. Coupek, B. Maslowski and M. Ondrejat, L p -valued stochastic convolution integral driven by Volterranoise, Stoch. Dyn. , (6) (2018), 1850048.[7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions , Cambridge University Press,1992.[8] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems , London Mathematical SocietyLecture Notes, , Cambridge University Press, 1996.[9] Z. Dong and R. Zhang, 3D tamed Navier-Stokes equations driven by multiplicative L´evy noise: Exis-tence, uniqueness and large deviations, https://arxiv.org/pdf/1810.08868.pdf .[10] T.E. Duncan, B. Pasik-Duncan,and B. Maslowski, Fractional Brownian motion and stochastic equationsin Hilbert spaces, Stoch. Dyn. (2002), 225–250.[11] E. B. Fabes, B. F. Jones and N. M. Riviere, The Initial value problem for the Navier-Stokes equationswith data in L p , Archive for Rational Mechanics and Analysis , (1972), 222–240.[12] L. Fang, and P. Sundar and F. G. Viens, Two-dimensional stochastic Navier-Stokes equations withfractional Brownian noise, Random Oper. Stoch. Equ. , (2) (2013), 135–158.[13] B. P. W. Fernando, B. R¨udiger and S. S. Sritharan, Mild solutions of stochastic Navier-Stokes equationwith jump noise in L p -spaces, Mathematische Nachrichten , (2015), 1615–1621.[14] K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimerequations, Journal of Differential Equations , (2017), 7141–7161.[15] E. Hausenblas and P. A. Razafimandimby, Existence of a density of the 2-dimensional stochastic NavierStokes equation driven by L´evy processes or fractional Brownian motion, Stochastic Process. Appl. , (7) (2020), 4174–4205.[16] T. Kato, Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions, Mathematische Zeitschrift , (1984), 471–480.[17] Y. Giga and T. Miyakawa, Solutions in L r of the Navier-Stokes initial value problem, Arch. Ration.Mech. Anal. , (3) (1985), 267–281.[18] E. Issoglio, and M. Riedle, Cylindrical fractional Brownian motion in Banach spaces, Stochastic Process.Appl. , (11) (2014), 3507–3534.[19] A.N. Kolmogorov, Wienerische Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady). Acad. URSS (N.S.) , (1940), 115–118.[20] M. Majdoub and E. Mliki, Well-posedness for Hardy-H´enon parabolic equations with fractional Brow-nian noise, Analysis and Mathematical Physics Nonlinearity , (4), 2016, 1292-1328.[22] M. T. Mohan and S. S. Sritharan, L p -solutions of the stochastic Navier-Stokes equations subject toL´evy noise with L m ( R m )-initial data, Evol. Equ. Control Theory , (3) (2017), 409–425.[23] M. T. Mohan, On convective Brinkman-Forchheimer equations, Submitted .[24] M. T. Mohan, Stochastic convective Brinkman-Forchheimer equations, Submitted , https://arxiv.org/abs/2007.09376 .[25] M. T. Mohan, Well-posedness and asymptotic behavior of the stochastic convec-tive Brinkman-Forchheimer equations perturbed by pure jump noise, Submitted , https://arxiv.org/abs/2008.08577 .[26] D. Nualart, The Malliavin calculus and related topics , 2nd Ed. Probability and Its Application (NewYork), Springer, Berlin (2006).[27] V.Pipiras and M.Taqqu, Integration questions related to the fractional Brownian motion, Probab. TheoryRelat. Fields , (2) (2001), 251–281.[28] M. R¨ockner and X. Zhang, Stochastic tamed 3D Navier-Stokes equation: existence, uniqueness andergodicity, Probability Theory and Related Fields , (2009) 211–267.[29] S. Tindel, C. A. Tudor, and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields , (2) (2003), 186–204. TOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS 17 [30] C. A. Tudor, Analysis of Variations for Self-similar Processes, A Stochastic Calculus Approach , SpringerInternational Publishing Switzerland 2013.[31] F. B. Weissler, The Navier-Stokes initial value problem in L p , Arch. Ration. Mech. Anal. . , 219–230(1980).[32] J. Zhu, Z. Brze´zniak, and W. Liu, Maximal inequalities and exponential estimates for stochastic convo-lutions driven by L´evy-type processes in Banach spaces with application to stochastic quasi-geostrophicequations , SIAM J. Math. Anal. , (3) (2019), pp. 2121–2167.[33] J. Zhu, Z. Brze´zniak, and W. Liu, L p -solutions for stochastic Navier-Stokes equations with jump noise, Statist. Probab. Lett. ,155