Korteweg-de Vries Equation in Bounded Domains

where μ, ν are positive constants. This equation, in the case μ = 0, was derived independently by Sivashinsky [1] and Kuramoto [2] with the purpose to model amplitude and phase expansion of pattern formations in different physical situations, for example, in the theory of a flame propagation in turbulent flows of gaseous combustible mixtures, see Sivashinsky [1], and in the theory of turbulence of wave fronts in reaction-diffusion systems, Kuramoto [2]. The generalized KdV-KS equation (1.1) arises in modeling of long waves in a viscous fluid flowing down on an inclined plane. When ν = 0, we have the KdV equation studied by various authors [6-12]. From the mathematical point of a view, the history of the KdV equation is much longer than the one of the KS equation. Well-posedness of the Cauchy problem for the KdV equation in various classes of solutions was studied in [6-9]. Solvability of mixed problems for the KdV equation and for the KdV equation with dissipation in bounded domains studied Bubnov [11], Hublov [12], see also


Introduction
The goal of this paper is to prove the existence, uniqueness and the energy decay of global regular solutions of the KdV equation in a bounded domain approximating it by the Kuramoto-Sivashinsky equations.
From the mathematical point of a view, the history of the KdV equation is much longer than the one of the KS equation.Well-posedness of the Cauchy problem for the KdV equation in various classes of solutions was studied in [6][7][8][9].Solvability of mixed problems for the KdV equation and for the KdV equation with dissipation in bounded domains studied Bubnov [11], Hublov [12], see also [19].In [10], Bui An Ton proved well-posedness of the mixed problem for the KdV equation in (0, ∞) × (0, T ) approximating the KdV equation by the KS type equations.Mixed problems for some classes of third order equations studied Kozhanov [13] and Larkin [18].The Cauchy problem for (1.1) was considered by Biagioni et al [6].They proved the existence of a unique strong global solution and studied asymptotic behaviour of solutions as ν tends to zero.This gave a solution to the Cauchy problem for the KdV equation as a limit of a sequence of solutions to the Cauchy problem for the KdV-KS equations.The Cauchy problem for the KS equation considered Tadmor [3] and Guo [5].In [5], Guo studied also solvability of the mixed problem for the KS equation in bounded domains in onedimensional and multi-dimensional cases.Cousin and Larkin [4] proved global well-posedness of the mixed problem for the KS equation in classes of regular solutions in bounded domains with moving boundaries.The exponential decay of L 2 − norms of solutions as t → ∞ was proved.
In the present paper we study asymptotics of solutions to a mixed problem for (1.1) when ν tends to zero in order to prove therewith that solutions to a mixed problem for the KdV equation may be obtained as singular limits of solutions to a corresponding mixed problem for the KS equation.The passage to the limit as ν tends to zero is singular because we loose one boundary condition in x = 0.
We consider in the rectangle Q the mixed problem for (1.1) which is different from the one considered in [4,5,10].In Section 2, we state our main results.In Section 3, exploiting the Faedo-Galerkin method with a special basis, we prove solvability of the mixed problem for (1.1) when ν > 0. In Section 4, we prove the existence and uniqueness of a strong solution to the mixed problem for the KdV equation letting ν tend to zero.It must be noted that the Fourier transform, commonly used to solve the Cauchy problem, see [6][7][8][9], is not suitable in the case of the mixed problem.Instead, we use the Faedo-Galerkin method to solve the mixed problem for (1.1) and weighted estimates to pass to the limit as ν tends to zero.In Section 5, we show that if u 0 L 2 (0,1) is sufficiently small, then u(t) L 2 (0,1) decreases exponentially in time and no dissipativity on the boundaries of the domain is needed for this.

Notations and results
We use standard notations, see Lions-Magenes [16], some special cases will be given below.We denote Our result on solvability of (2.1)-(2.3) is the following.
Then there exists a unique solution to (2.1)-( 2.3) from the class, When ν tends to zero, we obtain the following result.
Proof: It is easy to see that if u, v ∈ H 4 (0, 1) and satisfy boundary conditions of Lemma 1, then This means that the operator corresponding to the problem above is selfadjoint and positive.Hence, assertions of Lemma 1 follow from the well-known facts, see Coddington and Levinson [15], Mikhailov [14].
We construct approximate solutions to (2.1)-(2.3) in the form, where w j (x) are defined in Lemma 1 and g N j (t) are to be found as solutions to the Cauchy problem for the system of N ordinary differential equations, g N j (0) = (u 0 , w j ), j = 1, ..., N.
2) is a normal nonlinear ODE system, hence, there exist on some interval 0, T N ) functions g N 1 (t), ..., g N N (t).To extend them to any T < ∞ and to pass to the limit as N → ∞ , we prove the following estimates: where C 1 does not depend on N, t ∈ (0, T ), ν > 0.
where C 2 , C 3 do not depend on N, t ∈ (0, T ).Estimates (3.4), (3.5), (3.6) imply that u N (x, t) can be extended to all T ∈ (0, ∞) and that approximations (u N ) converge as N → ∞.Passing to the limit in (3.2), we prove the existence part of Theorem 1. Uniqueness can be proved by the standard methods, see [ 4 ].Thus Theorem 1 is proved.

Solvability of the KdV equation
Theorem 1 guarantees well-posedness of the problem (2.1)-(2.3)for all ν > 0. Our aim now is to pass to the limit as ν tends to zero.For this purpose we need a priori estimates of solutions to (2.1)-(2.3)independent of ν > 0. First we observe that estimate (3.4) does not depend on ν, but (3.5), (3.6) do depend.
Due to Theorem 1, for all ν > 0 we have the integral identity, which is true for any v ∈ L 2 (0, 1).It can be shown that u ν satisfy uniformly in ν > 0 the following inclusions:

Proof of Theorem 2
Proof: Letting ν → 0, we have a sequence of functions u ν satisfying (4.1).The last inclusions imply that there exists a subsequence of u ν , which we denote also by u ν , and a function U such that
Proof: Due to Theorem 1, for all ν ∈ (0, 1/2) the following identity is valid where v is an arbitrary function from L 2 (0, T ; L 2 (0, 1)), in particularly, we can take v an arbitrary function from W. Then, taking into account boundary conditions (2.3), we can rewrite the last identity in the form, Passing to the limit as ν → 0, we obtain for a.e.t ∈ (0, T ) and for all v ∈ W. The boundary conditions U (0, t) = U (1, t) = 0 obviously are fulfilled and the boundary condition U x (1, t) = 0 is fulfilled in a weak sense.It is clear that functions U and v have conjugate boundary conditions. 2 Taking into account properties of U, we can write where It means that U is a weak solution to the following boundary value problem, ) Now we must prove that a weak solution is regular.To prove this fact, we use the following Lemma 2 A weak solution to (4.2)-(4.4) is uniquely defined.
On the other hand, it is easy to verify that the function belongs to H 3 (0, 1), U 0 (0) = 0 for any F ∈ L 2 (0, 1), and satisfies the equation, Given F (x), the constants K 1 , K 2 can be found to satisfy the boundary conditions, By Lemma 2, U − U 0 = 0, hence, U = U 0 a.e. in (0, 1), It implies that U ∈ H 3 (0, 1).Returning to (4.2), we rewrite it as This proves the existence part of Theorem 2.

Stability
We have the following result.
This implies the assertion of Theorem 5.