Existence and Uniqueness Results for Two-Dimensional Stochastic Linearised Boussinesq Equation

1. Introduction

The water flow for an unconfined aquifer of the soil is described by Boussinesq equation (1904). The equation results from the application of the mass conservation principle, Darcy’s law, and the Dupuit-Forchheimer hypothesis (Bear, 1972). The two-dimensional Boussinesq equation is:

∈ D× (0, T], where x=(x₁, x₂) ∈ D ⊂R² denote the space coordinates (variables), t is the time variable, D denotes a bounded spatial domain with smooth boundary ∂D, T > 0 is a constant, x= (x₁, x₂), |x|²= x²₁ + x²₂ , dx = dx₁* dx₂, u = u(x, t) represents the elevation of the free surface (or hydraulic head) above the impervious layer, d denotes aquifer’s depth (measured from the impervious layer), denotes the saturated hydraulic conductivity of the soils along i direction, ET denotes the evapotranspiration, r denotes the recharge, s denotes the drainable porosity.

The Boussinesq equation associated with initial condition:

u (x, 0) = u ₀(x) , x∈ D, (2)

and with Dirichlet boundary condition

(3)

where u₀ (x) and are given function. Similarly, we can consider the Neumann boundary condition.

The linearised Boussinesq equation is

(4)

where (constant) represent the average depth of the aquifer.

In the present study we investigate the linearised stochastic Boussinesq equation

(x, t, ω)∈D×(0, T]×Ω (5)

Where Ω={ω} denotes the sample space and denotes the probabilitary variable.

The equation (5) associated with initial condition

(6)

and with Dirichlet boundary condition

(7)

Where and are given random function and random field, respectively.

Similarly, we can consider von Neumann boundary condition.

The Modern Soil science considers the soil as a complex dynamical system which evolves under total impact of the interactions between natural and biological factors as well as the human activity. Because these interactions are random processes, it is reasonable to assume that the water flow in soil is described by a random field. Measurement errors of hydraulic conductivity, drainable porosity, recharge and evapotranspiration are the other reason for the presence of randomness in the soil’s water flow process. These arguments have a lot of supporters among the soil scientists and physicists, who are showing a greater interest for stochastic models of water flow in soil. They use the stochastic models effectively in their work (research), see Freeze (1975), Cordova and Bras (1981), Chung and Austin (1987), etc.

The experimental data from many countries, including Albania, support the hypothesis that hydraulic conductivity of the saturated soil, drainable porosity, recharge and evapotranspiration are random variables, see Averjanov (1972), Van Schilfgaarde (1974-1979), Freeze (1975), Hubert (1976), Skaggs and Tang (1976) , Sagar and Preller (1980), Cordova and Bras (1981), Kolaneci, Xinxo and Bica (1983), Chung and Austin (1987), etc.

Depending on how the randomness is present (introduced) in unsteady water flow in soil, there are four mathematical problems, with increasing level of complexity:

1. The problem with random initial condition.

2. The problem with random boundary condition.

3. The problem with random recharge or evapotranspiration.

4. The problem with random hydraulic conductivity or drainable porosity.

The important achievements in the study of stochastic linearised Boussinesq problem are:

The paper is organized as follows. In section 2 we formulate the problem (5), (6), (7) in an appropriate functional setting. In section 3 we give the existence and uniqueness theorem for problem (5), (6), (7). Section 4 contains conclusion.

2. Functional Setting and Formulation of the Problem.

Let (Ω, F, µ) or (Ω, F, P) be a complete probability space, where Ω ={ ω} denotes the space of elementary events (or the space of basic outcomes), F is the σ-algebra associated with Ω, and µ (or P) is the probability measure defined on F. The σ -algebra F can be interpreted as a collection of all random events ω ∈ Ω and that have a well-defined probability with respect to F. In the present study use the Theory of Sobolev Spaces, see Adams (1975), Tribel (1986).

A real-valued random variable X=X(ω) is a mapping X:Ω->R.

Assume that the probability measure µ has a numerable basis D denotes a bounded domain with smooth boundary and

The separable Hilbert space L²(Ω) is well-known

M= {f (x, ω): D → L² (Ω)} denotes the set of second order random functions f(x,) over the domain.

Define the space

H= L² (D; L² (Ω))= {f= f (x, ) ∈ M,

L² (D)},

Equipped with scalar product

=, ∀ f, g∈H.

The induced norm from the scalar product is:

=,

Proved that H is separable Hilbert.

Definition of the functional spaces (D),

(D), 0, (D; L²(Ω)) are well known.

For f= f(x, ω) with respect to x:

f (φ)=φ, φ∈D(D), α= (α₁, α₂) , |α|=α₁+ α₂).

Denotes the outward normal in an arbitrary point x∈∂D.

if f=(D; L²(Ω)), m- trace of f on is defined:

, …, }, m=2,3,4,…

It is obvious that , where the right-hand side denotes Cartesian product.

Consider the space.

= {f, .

The Banach space (Ω), and ( are well known.

Consider the separable Hilbert space V such that V⊂H, V dense in H, V⊂ H (V continuously embedded in H).

That is, (, H, V') represent a Gelfand triplet.

( ,

( ,

( ,

The induced norms are || |, || |, || |

Respectively.

Defined the space

Equipped with scalar product

The induced norm from the scalar product is

is separable Hilbert space.

Similarly defined the separable Hilbert space, , and , see Tribel (1986).

Defined the space

Equipped with scalar product

The induced norm in from the scalar product is

is separable Hilbert space ,

,

see Dautary and Lions (1985), Tribel (1986).

The symbol <.,.> denotes the duality between the Hilbert spaces V’ and V.

Given a family of the continous bilinear forms defined on V with the parameter

Assume that satisface the conditions:

positive constant real number with that

, (8)

∃ the constant real numbers μ and μ>0 with that

, (9)

The condition (9) is V-chercivate hypothesis

The family of random operators A(t) associated with is defined by

(10)

Is proved that A(t) and , see Lions (1972).

If then we can prove that

(11)

Consider the random evolutionary equation:

(12)

With initial condition

(13)

The family of random operators A(t) is defined by (10). Assume that

Problem 1

Given , defined by (10). Find the random field

Which satisfies (12) for almost all and (13).

By definition, u=u(x, t, ω) is a solution of the problem (12), (13) if the above mentioned conditions are satisfied?

3. Existence and Uniqueness Results

In this section we will prove:

Theorem 1

Suppose that the assumptions (8), (9) are satisfied. Prove that the problem 1 has a unique solution u=u(x, t, ω)∈W(0, T) which depends continuously on the data. The proof of Theorem 1 undergoes through several steps.

Step1. Preliminary reduction of the problem

We can assume that (9) is satisfied for λ=0. Substituting where k is an arbitrary real number, the problem 1 transformed in equivalent problem:

Find z=z(x, t, ω) which satisfies the identity

With Z(x, t, ω)≡0 for t<0. In this identity, a(t, u, v) substituded by . Choosing k=λ we obtain the desirable result.

Step II. Proving the existence of the solution

To prove the existence of the solution of stochastic problem (12), (13), we modify the Galerkin method, developed by Dautray and Lions (1985) for deterministic parabolic partial differential equations, see Dautray and Lions (1985), pp 619-627.

The Hilbert space V is separable. Therefore, ∃ is the basis …. Of V in the following sense: ∀m∈N the elements …. are the linearly independent and the set of the all finite linear combinations

is dense in V.

For each m=1,2,3,…. Define approximate solution of the problem (12) ,(13) by using the following method:

(14)

(15)

(16)

where is the orthogonal projection of over the subspace spanned on

More generally, denotes each element of the above mentioned subspace, which satisfies the condition:

The system (15) with initial condition (16) represents the Cauchy problem for the unknown deterministic functions

:.

The matrices are:

is a non degenerate matrix, because of are linearly independent. Therefore, the system (15), (16) has unique solution for t∈(0, T), see Arnold (1975).

Implies that are integrable functions.

Therefore, the

and (17)

The proof of Theorem 1 continues similarly to the arguments presented by Dautray and Lions (1985), pp 619—627.

We obtain the following results:

(18)

where α denotes the cohercivity constant in (9) ,

,

the sequence {is strogly bounded in the space , (19)

the sequence {is strogly bounded in the space . (20)

There exist the element and the subsequence of natural numbers m’, such that in * weak topology of the space as m’→+∞:

such that

(21)

There exist and the subsequence m” of the sequence m’, such that in weak topolgy of as m”→+∞

, (22)

and , , (23)

(24)

is solution of the problem (12),(13). That is, u satisfies the equation (12) and the initial condition (13)

Step III. Proving the uniqueness of the solution

Assume that u and are solutions of the problem (12), (13) with data {f, } and { , }, respectively. Using Vishik-Ladizhenskaja lemma, we prove that

Where α denotes the cohercivity constant, see (9). The proof of Theorem 1 is complete.

Remark 1

The appropriate boundary condition for the stochastic problem (12), (13) depends on the choice of the Hilbert space V and random field f=f(x, t, ω)

Choose and suppose that where

and

Under these conditions, the appropriate boundary condition for the problem (12), (13) is von Neumann condition

Remark 2

We distinguish between the deterministic and stochastic Boussinesq Problems:

Galerikin method is an interior approximation f the Hilbert space V, see Team (1981). Therefore, in the stochastic Boussinesq problem are discredited the spatial variables as well as the probabilitary variable ω.

Conclusion

In this paper we have studied the stochastic linearised two-dimensional Boussinesq problem in the Hilbert spaces. The randomness is the present in hydraulic conductivity, drainable porosity, (recharge-evapotranspiration), initial condition and boundary condition. We use Sobolev spaces and Galerkin method. Under suitable assumptions, we prove the existence and uniqueness theorem for the solution of the problem (12), (13).Numerical approximations of the solution of stochastic linearised Boussinesq equation is the next step of the present study. One important application of our study is in agricultural drainage.

References