We consider a viscous, incompressible fluid whose motion is subject to a random force. The container of the fluid is supposed to be the torus T

Stochastic Power Law Fluid Equations

寺澤祐高

(Yutaka Terasawa),

PD

吉田伸生氏（京大・理）との共同研究

We consider a viscous, incompressible fluid whose motion is subject to a random force. The container of the fluid is supposed to be the torus T

= ( R / Z )

∼ = [0, 1)

as a part of idealization. For a differentiable vector field v : T

→ R

, which is interpreted as the velocity field of the fluid, we denote the rate of strain tensor by:

e(v) =

( ∂

v

+ ∂

v

2

)

: T

→ R

⊗ R

. (0.1) We assume that the extra stress tensor

τ (v) : T

→ R

⊗ R

depends on e(v) polynomially. More precisely, for ν > 0 (the kinematic viscosity) and p > 1,

τ (v) = 2ν(1 + | e(v) |

)

e(v). (0.2) Given an initial velocity u

: T

→ R

, the dynamics of the fluid is described by the following SPDE:

div u = 0, (0.3)

∂

u + (u · ∇ )u = −∇ Π + div τ (u) + ∂

W, (0.4) where

u · ∇ =

∑

j=1

u

∂

and div τ(u) = (

∑

j=1

∂

τ

(u) )

i=1

. (0.5)

The unknown process in the SPDE are the velocity field u = u(t, x) = (u

(t, x))

and the pressure Π = Π(t, x). The Brownian motion W = W (t, x) = (W

(t, x))

with values in L

( T

→ R

) (the set of vector fields on T

with L

components) is added as the random force. Physical interpretation of (0.3) and (0.4) are the mass conservation, and the motion equation, respectively. We note that the SPDE (0.3)–(0.4) for the case p = 2 is the stochastic Navier-Stokes equation [1, 2].

Our motivation comes from works by J. M´ alek, J. Ne˘ cas, M. Rokyta, and M.

R u˘

zi˘ cka [3], where the deterministic equation (where the colored noise ∂

W in (0.3)–(0.4) is dropped) is investigated.

We need the following definition.

Definition 1 Let H be a Hilbert space, and Γ : H → H be a self-adjoint, non- negative definite operator of trace class. A random variable (W

)

with values in C([0, ∞ ) → H) is called a H-valued Brownian motion with the covariance operator Γ (abbreviated by BM(H, Γ) below) if, for each ϕ ∈ H and 0 ≤ s < t,

E [exp (i h ϕ, W

− W

i ) | (W

)

] = exp (

− t − s

2 h ϕ, Γϕ i )

, a.s.

We also need some function spaces. Let V be the set of R

-valued divergence free, mean-zero trigonometric polynomials. We equip the torus T

with the Lebesgue measure. For p ∈ [1, ∞ ) and α ∈ R , we introduce:

V

= the completion of V with respect to the norm k · k

, (0.6) where

k v k

=

∫

T^d

| (1 − ∆)

v |

. (0.7) Definition 2 Suppose that

I Γ : V

→ V

is a bounded self-adjoint, non-negative definite operator of trace class;

I µ

is a Borel probability measure on V

.

I (X, Y ) = ((X

, Y

))

is a process defined on a probability space (Ω, F , P ) such that:

X ∈ L

([0, ∞ ) → V

) ∩ L

([0, ∞ ) → V

) ∩ C([0, ∞ ) → V

), (0.8) for some β > 0, and (Y

)

is a BM(V

, Γ) (cf. Definition 1).

Then, the process (X, Y ) is said to be a weak solution to the SDE (stochastic differential equation)

X

= X

+

∫

b(X

)ds + Y

(0.9)

with the initial law µ

if the following conditions are satisfied;

P (X

∈ · ) = µ

;

Y

− Y

and {h ϕ, X

i ; s ≤ t, ϕ ∈ V} are independent for any t ≥ 0;

h ϕ, X

i = h ϕ, X

i +

∫

h ϕ, b(X

) i ds + h ϕ, Y

i , for all ϕ ∈ V and t ≥ 0.

We can now state our existence result.

Theorem 3 Let Γ and µ

be as in Definition 2 and suppose additionally that p ∈ (

, ∞ ) (d = 2), p ∈ (

, ∞ ) (d = 3) · · · ,

I ∆Γ = Γ∆ and both Γ, ∆Γ are of trace class;

I µ

is a probability measure on V

and m

=

∫

k ξ k

µ

(dξ) < ∞ for α = 0, 1. (0.10)

hold. Then, there exists a weak solution to the SDE (0.9) with the initial law µ

(cf. Definition 2) such that (0.8) holds with β = β(p) > 0. Moreover, for any T > 0,

E [

sup

t≤T

k X

k

+

∫

k X

k

dt ]

≤ (1 + T )C < ∞ , (0.11)

where C = C(d, p, Γ, m

) < ∞ .

The proof is done by using Galerkin approximation. We obtain Stochastic ODEs by this. By combining the deterministic method from [3] and Ito’s formula, we get their uniform apriori estimates needed for passing to the limit.

References

[1] Flandoli, Franco : An introduction to 3D stochastic fluid dynamics. SPDE in hy- drodynamic: recent progress and prospects, 51–150, Lecture Notes in Math., 1942, Springer, Berlin, 2008.

[2] Flandoli, Franco ; Gatarek, Dariusz: Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Related Fields 102 (1995), no.

3, 367–391.

[3] M´ alek, J.; Ne˘ cas, J.; Rokyta, M.; R u˘

zi˘ cka, M.: Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computation, 13.

Chapman & Hall, London, 1996. xii+317 pp. ISBN: 0-412-57750-X

[4] Yutaka Terasawa and Nobuo Yoshida, Stochastic Power Law Fluids: Existence and

Uniqueness of Weak Solutions, Annals of Applied Probability 21 (2011), 1827–1859.