Semi-linearized compressible Navier-Stokes equations perturbed by noise

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1 Electronic Journal of ifferential Equations, Vol. 2323), No. 2, pp ISSN: URL: or ftp ejde.math.swt.edu login: ftp) Semi-linearized compressible Navier-Stokes equations perturbed by noise Hakima Bessaih Abstract In this paper, we study semi-linearized compressible barotropic Navier- Stokes equations perturbed by noise in a 2-dimensional domain. We prove the existence and uniqueness of solutions in a class of potential flows. 1 Introduction We consider the following system of equations with a stochastic perturbation ρu t + pρ) = µ u + µ + λ) div u + G t in Q T, ρ t + divρu) = in Q T, 1.1) where Q T =, T ), =, 1) 2 ), ρ, λ, µ are constants such that ρ >, µ >, µ + λ ; while G is a stochastic process in a function space, which we will precise below, and u t and G t denote the derivative with respect to t in the distribution sense. and div are the gradient and divergence operators with respect to the space variables, is the Laplace operator. The space variables are denoted by x = x 1, x 2 ) and the time by t. In absence of the random perturbation G t, 1.1) is reduced to the system ρu t + pρ) = µ u + λ + µ) div u, ρ t + divρu) =. 1.2) This system can be considered as a semi-linearized approximation of the compressible Navier-Stokes equations of a barotropic viscous fluid n ρ t u i + u )u i ) i pρ) = j µ j u i + i u j )) + i λ div u), j=1 ρ t + divρu) =, 1.3) where i = 1,..., n; u, ρ, pρ) represent respectively the velocity vector, the density, and the pressure; while µ, λ are viscosity coefficients which according to the thermodynamic principles should satisfy the inequalities µ > and 3λ+2µ. Mathematics Subject Classifications: 35Q3, 76N1, 6G99. Key words: Compressible Navier-Stokes equations, noise. c 23 Southwest Texas State University. Submitted October Published January 2, 23. 1

2 2 Semi-linearized compressible Navier-Stokes equations EJE 23/2 System 1.3) has been investigated mostly for one-dimensional flows n = 1). For many-dimensional flows, considerably less is known except for small initial data or in small time interval. A global existence theorem for the model 1.3) has been proved by P.L.Lions [9, 1] and Vaigant-Kazhikhov [16]. Notice that in the first µ and λ are considered constants while some particular requirements on the growth of the viscosity coefficient λ and the pressure as functions of the density ρ are imposed for the last result. The semi-linearized system 1.2) is studied in [15], which proves the existence and uniqueness of the strong solution. As far as the stochastic equations for incompressible viscous fluids are concerned, some existence theorems and some results on various aspects are known see [2, 5, 6] etc... But in the compressible case, the variation of the fluid density gives some difficulties. For this reason, only the two dimensional space is considered here with some other restrictions. In the one dimensional case, the full equation 1.3) subject to a perturbation is studied in [13] and [14]. We use the standard notation W l,p for the Sobolev spaces consisting in the functions which are integrable in power p as well as their derivatives up to the order l and H l = W l,2 ; C[, T ]; X) denotes the space of the continuous functions with values in a Banach space X. In this paper, we use the Orlicz space L φ ) associated to the convex function φr) = 1+r) log1+r) r, r. We denote by.,. the inner product in L 2 and by. the corresponding norm. We use the abbreviated notation j = x j, j 2 = 2 x 2. j The propose of the present paper is to prove the existence and uniqueness) of a global solution to 1.1). The solution will be constructed in the class of periodic and potential flows as in [15], i.e., in the case where u has the form u = ϕ, with some function ϕ, which is periodic in x 1 and x 2. More precisely we suppose that every function appearing in 1.1) is periodic of period 1 in x 1 and x 2 and take the equation of state pρ) = cρ, c = const >. We also suppose that the perturbation G is the gradient of a potential i.e. G = W. For simplicity, we assume that the constants c and ρ are equal to 1 and the constants λ and µ are respectively equal to 1/4 and 1/2 and impose ϕt, x)dx =. When W t, x)dx = which will follows from the assumptions of section 2, integrating the momentum equation 1.1) 1, the system acquires the form dϕ = ϕ + 1 ρ)dt + dw in Q T, ρ t + divρ ϕ) =. 1.4)

3 EJE 23/2 Hakima Bessaih 3 Below, W will be a Wiener process taking values in a particular Hilbert space. The unknown functions are assumed to take prescribed values at the initial time, ρ t= = ρ x), ρ x)dx = 1, ϕ t= = ϕ x), ϕ x)dx =. In addition, we impose the following natural requirement on the solution, 2 Main result ρx, t) in Q T. Before stating the existence results, we have to precise some conditions on the noise term appearing in 1.1). We set A) = { u H 2 ) : u is periodic of period 1 in x 1 and x 2, udx = }. and define a linear operator A : A) { u L 2 ) : u is periodic of period 1 in x 1 and x 2 }, as Au = u. The operator A is self-adjoint with compact resolvent. We denote by < λ 1 λ 2... lim λ j = ) the eigenvalues of A and by e 1, e 2... the corresponding complete orthonormal system of eigenvectors. As well known, for the space of periodic functions the eigenvectors are trigonometric functions and we see easily that e jx)dx =, j = 1, Let W t) = σ j β j t)e j x). 2.1) j=1 where {σ j } j=1 is a sequence of constants satisfying the condition λ δ+2 j σj 2 <, 2.2) j=1 with some δ > while β 1, β 2,... are independent standard 1-dimensional Brownian motions defined on a complete probability space Ω, F, P) adapted to a filtration {F t } t. We denote by E the expectation relative to Ω, F, P). Now, we state the main theorem of this paper. Theorem 2.1 Let Ω, F, P) be a probability space and T a positive number. Suppose that W is a Wiener process satisfying 2.1) and the condition 2.2), and that ρ and ϕ are two Random variables with values respectively in L ) and W 1,q ) H 2 ) q 1) satisfying respectively the conditions 1.5) and 1.6) P-a.s. and inf x ρ x) > and sup x ρ x) < P-a.s. Then there exists a unique solution to 1.4) up to a modification. Besides ρ satisfies inf QT ρx, t) > and sup QT ρx, t) < P-a.s.

4 4 Semi-linearized compressible Navier-Stokes equations EJE 23/2 3 Reduction of the problem via the Ornstein- Uhlenbeck equation Let us consider an auxiliary problem, the Ornstein-Uhlenbeck equation, dzt) + Azt)dt = dw t), z) =. 3.1) This equation has a solution given by the process see [4]) zt) = t e t s)a dw s), 3.2) where e ta denotes a C -semigroup generated by A. The regularity of zt) depends on the regularity of W t). Indeed, we have for an arbitrary k > A k zt) = t j=1 λ k j e t s)λj σ j dβ j s)e j. E zt) 2 A k ) = E A k zt) 2 = E = = j=1 j=1 j=1 t t ) 2 λ k j e t s)λj σ j dβ j s) λ k j e t s)λj σ j 2 ds λ 2k j σ2 j 1 e 2tλj ). 2λ j According to 2.2), W t) belongs in A δ+2)/2 ) for some δ > which yields, for k = δ + 3)/2 in the above equality, that zt) has continuous trajectories taking values in A 3+δ)/2 ) as we will need in the next sections), i.e. zt) C[, T ]; A 3+δ/2 )) P-a.s. for some δ >. Following the idea of Bensoussan-Temam [2], we set yt) = ϕt) zt). 3.3) Using this change of variable in 1.4) and equation 3.1), one obtains the system y t y = 1 ρ in Q T, ρ t + divρ y + z)) = in Q T. 3.4)

5 EJE 23/2 Hakima Bessaih 5 4 Reduced deterministic problem In this section, we study the following reduced deterministic problem y t y = 1 ρ in Q T, ρ t + divρ y + z)) = in Q T, 4.1) where zt) is a continuous function taking values in H 3+δ ), δ >. For this problem, we state the following existence and uniqueness theorem. Theorem 4.1 Let T be positive number and suppose that y W 2,s ) and ρ L s ), s 2. We suppose also that z C [, T ]; H 3+δ )) δ > )). Then there exists at least one solution y, ρ) to Problem 4.1) which satisfies y L, T ; W 1.q )) L 2, T ; H 2 )), y t L, T ; L s )) L 2, T ; H 1 )), ρ L, T ; L φ )) L s Q T ), where q 2. Moreover, if inf x ρ x) > and sup x ρ x) <, then inf QT ρx, t) >, sup QT ρx, t) <, and 4.1) is uniquely solvable. The proof follows the lines of Vaigant-Kazhikhov [15], of which we will use the ideas without quote them explicitly. 4.1 A priori estimates and existence of solutions In this section, we obtain a priori estimates that permit us to prove the existence of a solution. The first energy estimate is obtained by multiplying the first equation in 4.1) by y and the second equation by log ρ, followed by integrating over. More precisely, we obtain d 1 dt 2 y 2 + ρ log ρ ρ + 1) dx + y 2 z L. 4.2) This relation implies that the solution is bounded in the norms of the spaces y L, T ; H 1 )), y L 2, T ; L 2 )), ρ L, T ; L φ )). The following lemma may be derived from the second equation in system 4.1). Lemma 4.2 If ρ L p 1 ) then there exists a constant C depending on p such that the inequality t ρt) p 1 L p 1 ) + ρτ) p L p ) dτ t C ρ p 1 L p 1 ) + t ) 4.3) y τ τ) p L p ) dτ + zτ) p L p ) dτ holds for any exponent p, 2 < p < and any t [, T ].

6 6 Semi-linearized compressible Navier-Stokes equations EJE 23/2 Proof: form For r > 1 using 4.1) 1, the expression 4.1) 2 can be rewritten in the ρ r t + ρr y + z)) + r 1)ρ r+1 = r 1)ρ r y t 1 z). 4.4) Integrating over and estimating ρ r y t and ρ r z by the Young inequality, ρ r y t ɛ 1 ρ r+1 + C 1 y t r+1, ρ r z ɛ 2 ρ r+1 + C 2 z r+1, with convenient small numbers ɛ 1, ɛ 2, we obtain 4.3) for p = r + 1. Lemma 4.3 If ρ L p 1 ), then there exists a constant C depending on p such that the inequality y W 2,p Q T ) C ρ L p 1 ) + y t Lp Q T ) + z Lp Q T )) 4.5) holds for 2 < p <. The proof of this lemma is is a consequence of 4.3) and the first equation of 4.1). Now, to obtain additional a priori estimates, we differentiate 4.1) 1 with respect to x 1, x 2, t so that we obtain y t y = ρ, 4.6) y tt y t = ρ t = divρ y + z)). 4.7) Lemma 4.4 If ρ L 3 ), y W 1,q ) H 2 ), where q 4 then there exists a constant C depending on q such that the inequality sup y q + y t 2) t + y t 2 C 4.8) <τ<t holds for all t [, T ]. Proof: For arbitrary q 2 and s 2, we multiply 4.6) by q y q 2 y and 4.7) by s y t s 2 y t, sum these equations and integrate over to obtain d y q + y t s ) + q j k y) 2 y q 2 dt + qq 2) j k y) j l y) k y) l y) y q 4 + ss 1),l = q ρ y y q 2 + qq 2) ρ j y) k y) j k y) y q 4 ss 1) ρ y + z) y t y t s 2. y t 2 y t s 2 4.9)

7 EJE 23/2 Hakima Bessaih 7 By taking q = 4 and s = 2 in 4.9) and substituting ρ = 1 + y y t, we have d y 4 + y t 2) + 4 dt = 4 y y = I I 1. j k y) 2 y y t 2 y)y t y 2 y) 2 y 2 4 j y) k y) j k y) + 8 y t j y) k y) j k y) 2 y + z). y t y t 8,l y j y) k y) j k y) y + z). y t 2 y y + z). y t j k y) j l y) k y) l y) The assumption that is a two-dimensional region is essential for this estimate. I 2 = 4 y 2 y 2 y y L 4 ) y 2 L 4 ). On the other hand y 2 L4 ) y 2 1/2 y 2 1/2. Then using Young s inequality twice, we obtain for arbitrary positive small numbers ɛ 1 and ɛ 2 such that Since and one obtains I 2 ɛ 1 y y 2 L 4 ) + ɛ 2 y C y 2 y 2 2. y 2 2 = y 2 2 = I 2 ɛ 1 y y 2 L 4 ) + ɛ 2 y 2 2 = 2 y 2 ) 2 = y 2 j k y) 2 y 4, y 2 j k y) 2 + C y 4 L 4 ) y 2. By Young s inequality and for arbitrary small positive constant ɛ, ɛ 1 and ɛ 2 we can estimate I 1, I 3, I 4, I 5 and I 6 as follows: I 1 = 4 y 2 y ɛi 2 + C y 2,

8 8 Semi-linearized compressible Navier-Stokes equations EJE 23/2 I 4 = 8 I 5 = 8 I 3 = 4 y y 2 y t ɛ 1 y 2 y 2 + ɛ 2 y t 2 + C y t 2 y 2 y 2, I 6 = 8 ɛ 1 j y) k y) j k y) ɛ I 7 = 2 y j y) k y) j k y) ɛ y t j y) k y) j k y) k y) 2 j k y) 2 + C y 2, k y) 2 j k y) 2 + ɛ 2 y t 2 + C k y) 2 j k y) 2 + CI 2. y t 2 y 2 j k y 2. y + z) y t ɛ y t 2 + C y + z) 2. I 8 = 2 y y + z) y t ɛ y t 2 + CI 2 + z 2 L ) y 2. I 9 = 2 y + z) y t y t ɛ y t 2 + C y t 2 y + z) 2 y + z) 2. I 1 ɛ 1 j k y) 2) 1/2 j l y) 4) 1/4 k y) 8) 1/8 l y) 8) 1/8,l j k y j l y 2 L 4 ) + ɛ 2 y 2 2,l + C ɛ1,ɛ 2 y 2 2 j k y 2. We set αt) = sup y 4 + y t 2, <t<t βt) = j k y) 2 y 2 + y t 2. On the other hand from 4.5) in particular for p = 4 and ρ L 3 ), we have y 2 W 2,4 ) C ρ L 3 ) + y t 2 L 4 ) + z 2 L 4 )).

9 EJE 23/2 Hakima Bessaih 9 Consequently, by using Cauchy s inequality and the energy estimate 4.2) we obtain t t 1/2. ɛ y y 2 L 4 ) Cɛ y 4 L )) 4 But in the two-dimensional space we have so that ɛ t y t 4 L 4 ) C y t 2 y t 2, 4.1) y y 2 L 4 ) Cɛ αt) 1/2 t 1/2 ) βτ)dτ) + z 2 L4 Q T ). It is the same for ɛ 1 2,l t j k y j l y 2 L 4 ). Set Λt) = yt) 2. Since this is integrable on [, T ], we have αt)+ t βτ)dτ α)+ɛαt) 1/2 t βτ)dτ ) 1/2 t +C 1+ Λτ)+1)ατ)dτ ). Using Gronwall s lemma we obtain, sup y 4 + y τ 2) t + <τ<t) According to this equation, T y t 4 T y τ 2 C. 4.11) y t 2 y t 2 C. 4.12) Hence, using this inequality in 4.3) and 4.5) for p = 4, it follows that t ρ 4 + t y 4 C. 4.13) Now, let us consider the equation 4.9) for q 4 and s = 2. We obtain the equality d y q + y t 2) + q dt = 2 ρ y + z) y t + q + qq 2) qq 2),l j k y) 2 y q y t 2 ρ y y q 2 ρ j y) k y) k j y) y q 4 j k y) j l y) k y) l y) y q )

10 1 Semi-linearized compressible Navier-Stokes equations EJE 23/2 The first term on the right hand side of 4.14) by Young s inequality is bounded by T T T ρ y + z) y t ɛ y t 2 + C ρ 2 L 4 ) y + z) 2 L 4 ). In view of 4.11) and 4.13), ρ and y are bounded respectively in L 4 Q T ) and in L, T ; L 4 )). Hence T T T ρ y + z) y t ɛ y t 2 + C z 4 L 4 ). Using the Young and the Hölder s inequality for the second term on the right hand side of 4.14), one obtains ρ y y q 2 ɛ ɛ y 2 y q 2 + C ρ 2 y q 2 y 2 y q 2 + C ρ 2 L 4 ) y 2q 2)) 1/2. We write the last integral in the above inequality as y 2q 2)) 1/2 = y q/2 2q 2)/q. L 4q 2)/q Using the embedding inequality see [8], pp 62) f L 4q 2)/q C f a f 1 a, 4.15) for the function f = u q/2, where a = q 4)/2q 2)), yields ρ y y q 2 C ρ 2 L 4 y q/2 ) q 4)/q y q/2 + ɛ y 2 y q 2. Since y q/2 = y q ) 1/2, and y q/2 ) = q 2 j k ) 2 y q 2) 1/2, and in virtue of the Young s inequality with p = 2q/q 4) and p = 2q/q + 4) we obtain ρ y y q 2 ɛ j k ) 2 y q 2 + C ρ 4q/q+4) L 4 ) y q) q/q+4).

11 EJE 23/2 Hakima Bessaih 11 By using the same argument for the third term on the right hand side of 4.14) we obtain, for an arbitrary ɛ, ɛ ρ j y) k y) k j y) y q 4 y q 2 j k y) + ρ 4q/q+4) L 4 ) y q) q/q+4). For the last term on the right hand side of 4.14), we use the same arguments as before, we have,l ɛ j k y) j l y) k y) l y) y q 4,l y q 2 j k y) 2 + C ɛ j,l j l y 4q/q+4) L 4 ) y q) q/q+4) Consequently, using 4.5) for p = 4 and *), we have y q + y t 2) T + y t 2 C y q L q ) + ρ 2 + y 2 + z 2 ) L 4 Q T ) T + ρ 4q/q+4) L 4 ) + y t 2q/q+4) y t 2q/q+4) ) + z 4q/q+4) L 4 Q T ) + ρ 4q/q+4) L 3 ) y q) q/q+4) Now using Gronwall s lemma, 4.12) and 4.13), we obtain 4.8). Lemma 4.5 If ρ L s ), y W 2,s ) then the inequality y τ s dx C 4.16) holds for s > 2 and t [, T ]. sup <τ<t Proof: We multiply 4.7) by s y t s 2 y t, s > 2 and then we integrate over, to obtain d y t s + ss 1) y t s 2 y t 2 = ss 1) ρ y + z) y t y t s 2. dt 4.17) Cauchy s inequality applied to the right hand side of the above quality gives ρ y + z) y t y t s 2 ɛ y t s 2 y t 2 + C ρ 2 y + z) 2 y t s 2.

12 12 Semi-linearized compressible Navier-Stokes equations EJE 23/2 According to 4.1) 1, with ρ = 1 + y y t, 4.17) yields d dt C y t s dx + y t s 2 y t 2 y + z) 2 y t s 2 + y + z) 2 C) yt s + y 2 y t s 2) ). 4.18) The first term on the right hand side of 4.18), using Young s inequality with p = s/2 and p = s/s 2), is estimated as According to 4.5) y + z) 2 y t s 2 y 2 L s ) + ) z 2 L s ) yt s 2 L s ). y + z) 2 y t s 2 C y t s L s ) + y t s 2 L s ) z 2 L s )). Using Hölder s inequality, the third term on the right hand side of 4.18) yields y 2 y t s 2 y t s 2 L s ) y 2 L s ). On the other hand, from the Gagliardo-Nirenberg s inequality it follows y C) y b L 4 ) y 1 b L q ). For b = 4/q + 4) and by 4.8) and 4.13), we obtain t t y q+4 C) y 4 L 4 ) y q L q ) C. 4.19) We set β s t) = sup <τ<t y τ s Ls. Then the expression on the right-hand side of 4.18), after integrating on,t), can be estimated as t y τ s L s ) + C y τ ) s L s ) + t + + t t y τ s 2 y τ 2 y τ s 2 L s ) z 2 L s ) ) 1 + y + z) 2 C) y τ s L s ) y + z) 2 C) y τ s 2 L s ) y s L s ) ). 4.2)

13 EJE 23/2 Hakima Bessaih 13 By Young s inequality p = s/s 2), p = s/2), the last term on the right-hand side of the above inequality can be estimated as t y + z) 2 C) y τ s 2 L s ) y 2 L s ) sup y τ s 2 L <τ<t s ) t β s t)) s 2)/s t From 4.5), we have t y + z) 2 C) y 2 L s ) y + z) 2s/s 2) C) y s L s ) C 1 + ) s 2)/s t t ) β s τ)dτ. In view of 4.19) and Young s inequality it follows that t y + z) 2 C) y τ s 2 L s ) y 2 L s ) dτ ɛβ st) On the other hand the second term of 4.2) can be estimated as t t y τ s 2 L s ) z 2 L s ) β st) s 2)/2 z 2 L s ) ɛβ s t) + C ɛ t y s L s )) 2/s. t z 2 L s )) s/2. ) β s τ)dτ. We conclude 4.16) by using Gronwall s lemma. These estimates are sufficient for proving the existence of solutions. One can use the scheme of constructing solutions given in [1]. According to this scheme, approximate solutions y k, ρ k ) are found by the Galerkin method; y k is sought as a finite sum of basis and ρ k is determined from the transport equation. In particular y k is compact in L 2 Q T ). Thus, the passage to the limit in the nonlinear terms in equations 4.1) is justified. 4.2 Upper and lower bounds for the density, and uniqueness of the solution The estimates obtained in the preceding section permit us to establish that the density ρ is bounded provided that the initial density ρ is bounded. To this end, we write out a special equation for logρ). This idea has been used in [8]. Lemma 4.6 If y W 2,s ), s 2 and ρ L ) then ρt) L ) M, t [, T ]. 4.21)

14 14 Semi-linearized compressible Navier-Stokes equations EJE 23/2 Proof: Assuming that ρx, t) >, let us rewrite the second equation in 4.1) in the form log ρ + y + z) log ρ + y + z) =. t By adding the above equation to 4.1) 1, we obtain log ρ + y) + y + z) log ρ + y) + ρ = 1 z + y y + z). 4.22) t Set γ = log ρ + y, γ + = max{, γx, t)}. Considering 4.22) as the transport equation for γ and taking into account the fact that ρ is nonnegative, we conclude that γ + x, t) γ + t= L ) + t According to 4.8), y L Q T ) is bounded; indeed y L Q T ) C sup <t<t ) 1 + z 2 C) + y 2 C). 4.23) y L q ) C, q > ) Hence 4.21) follows by the hypotheses of the lemma and from 4.24) and 4.19) with the constant M = exp y L Q T ) + γ + t= L ) + t 1 + z 2 C) + y 2 ) ) C). Lemma 4.7 If the initial density ρ x) is strictly positive under the hypotheses of the lemma 4.6, then ρx, t) remains a strictly positive function in Q T i.e. ρx, t) m > a.e. in Q T. 4.25) Proof: Let us change the sign in equation 4.22) and rewrite it for γ. We wish to find an upper bound for the function γ = max {, γ}. By analogy, we obtain γ x, t) γ t= L ) + t ) z 2 L ) + y 2 C) ρ L ). 4.26) Hence 4.25) follows with the constant m = exp y L ) + γ t= L ) t ) + z 2 + y 2 C)) + MT ).

15 EJE 23/2 Hakima Bessaih 15 If the density ρ is bounded, then the solution of 4.1) is unique. Indeed, if ρ, y ) and ρ, y ) are two solutions, then their difference ρ = ρ ρ, y = y y is a solution to the linear problem y t y = ρ, ρ t + divρ y + z)) + divρ y) =, 4.27) with the zero initial conditions. Let us introduce an auxiliary function ψ, which is a solution to periodic Neumann problem ψ = ρ, ψdx =. 4.28) Since ρ is bounded, ψ is also bounded i.e. ψ M a.e. in Q T. 4.29) We multiply 4.27) 1 by y and 4.27) 2 by ψ, we sum these equations and integrate over, we have 1 d y 2 + ψ 2) + y 2 2 dt = y, ψ ρ y, ψ ψ y + z), ψ. The first two terms on the right-hand side of this equation are bounded by 2 1 y C ψ 2 2 in view of the Cauchy s inequality, since ρ is bounded. The last term can be trasformed by integrating by parts as follows ψ y + z), ψ = ψ ) y + z), ψ 1 2 y + z), ψ 2. For the other part, by Hölder s inequality ψ 2 2 y + z) 2 y + z) p) 1/p ψ 2p ) 1/p, with p = ɛ 1 > 1 and p = 1/1 ɛ) where < ɛ < 1 is arbitrary. ψ 2 2 y + z) M 2ɛ ψ 21 ɛ) 2 y + z) L 1/ɛ. Thus, for the nonnegative function expression 4.29) yields the inequality dy t) dt Y t) = y 2 + ψ 2, Y ) =, C 1 Y t) + M 2ɛ 2 y + z) L 1/ɛY t) 1 ɛ.

16 16 Semi-linearized compressible Navier-Stokes equations EJE 23/2 Consequently, t Y ɛ t) ɛm 2ɛ e C1+C)ɛt e Cɛτ 2 y + z) L 1/ɛdτ t ɛcm 2ɛ e C1+C)ɛt e Cɛτ 2 y L 1/ɛ + 1 ) dτ We apply Hölder inequality to the integral with the exponent p = 1/ɛ and p = 1/1 ɛ); we obtain Y ɛ t) ɛcm 2ɛ e C1+C)ɛt t e Cɛτ ) p ) 1/p t 2 y L 1/ɛ + 1) p dτ ɛcm 2ɛ ) 1 ɛ 2 y L 1/ɛ Q T ) + 1 e Cɛt/1 ɛ) 1 )) 1 ɛ. Cɛ 4.3) Considering y as a solution to the parabolic equation y t y = 1 ρ with bounded right-hand side and using estimates for the higher derivatives in the L 1/ɛ -norm see [9]), we obtain 2 y L 1/ɛ Q T ) Cɛ 1 1 ρ L 1/ɛ Q T ) Cɛ 1. Therefore, from 4.3) follows that Y t) C 1/ɛ M 2 1 ɛ e Cɛt/1 ɛ) 1 )) 1 ɛ)/ɛ. Cɛ It is easy to check that if t [, τ], where Cτ < 1 on [, τ], then the righthand side of the last inequality vanishes as ɛ. Hence Y t) = on [, τ]. Repeating the argument for the interval [τ, 2τ] and so on, we obtain Y t) =, which proves the uniqueness of the solution. Conclusion The solution of deterministic system of equations 4.1) will allow us constructing the solution of the stochastic system 1.4). Proof of Theorem 2.1 We can apply Theorem 4.1 to obtain existence and uniqueness of solution for problem 3.4) for fixed ω. The measurability is an obvious fact using the uniqueness of the solution see [17]). As a consequence, using 3.3) and the properties measurability) of z, this theorem is proved. References [1] S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids, 199. [2] A. Bensoussan, R. Temam, Equations stochastiques du type Navier-Stokes, J. Func. Anal, 13, , ) p

17 EJE 23/2 Hakima Bessaih 17 [3] H. Brezis, Analyse fonctionnelle. Théorie et applications, Masson, [4] G. da Prato, J.Zabczyk, Stochastic equations in infinite dimensions, Cambridge, [5] B. Ferrario, The Benard problem with Random perturbations: dissipativity and invariant measures, NoEA, 4, no. 1, , [6] F. Flandoli, issipativity and invariant measures for stochastic Navier- Stokes equations, NoEA, 1, , [7] H. Fujita-Yashima,Equations de Navier-Stokes Stochastiques non Homogènes et applications, Scuola Normale Superiore Pisa, thesis, [8] A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural tseva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23, [9] P. L. Lions, Existence globale de solutions pour les équations de Navier- Stokes compressibles isentropiques, C. R. Acad. Sci. Paris, t. 316, Série I, p , [1] P. L. Lions, Compacité des solutions des équations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris, t. 317, Série I, p , [11] P. L. Lions, Limites incompressibles et acoustique pour des fluides visqueux compressibles et isentropiques, C. R. Acad. Sci. Paris, t. 317, Série I, p , [12] R. H. Nochetto, C. Verdi, Convergence past singularities for a fully discrete approximation of curvature driven interfaces, to appear in SIAM J. Numer. Anal. [13] E. Tornatore H. Fujita-Yashima, Equazioni monodimensionale di un gas viscoso barotropico con una perturbazione poco regolare, Annli Univ. Ferrara Sez. Mat, [14] E. Tornatore H. Fujita-Yashima, Equazione stocastica monodimensionale di un gas viscoso barotropico, Preprint N. 19, Università degli studi di Palermo, [15] V. A. Vaigant, A. V. Kazhikhov, Global solutions to the potential flow equations for a compressible viscous fluid at small Reynolds numbers, ifferential Equations, Vol 3, No 6, [16] V. A. Vaigant, A. V.K azhikhov, On existence of global solutions to the two-dimenional Navier-Stokes equations for a compressible viscous fluid, Siberian Mathematical Journal, Vol 36, No. 6, 1995.

18 18 Semi-linearized compressible Navier-Stokes equations EJE 23/2 [17] M. I. Visik, A. V. Fursikov, Mathematical problems of statistical hydromechanics, in russian) Nauka, Mosca, 198; English translation) Kluver, ordrecht, 198; German translation) Akad, Verlag, Leipzig, 198. Current address: ipartimento di matematica applicata, Univ. ini Via Bonanno 25/B, Pisa, Italy Permanent address: Faculty of Mathematics U.S.T.H.B Algiers, Algeria