GAIN OF REGULARITY FOR AN NONLINEAR DISPERSIVE EQUATION KORTEWEG-DE VRIES-BURGERS TYPE

In this papers we study smoothness properties of solutions. We consider the equation of Korteweg de Vries Burgers type (1) { ut + ∂xf(u) =  ∂2 xu− δ ∂3 xu u(x, 0) = φ(x) with −∞ < x < +∞ and t > 0. The flux f = f(u) is a given smooth function satisfying certain assumptions to be listed shortly. It is shown under certain additional conditions on f that C∞ solutions u(x, t) are obtained for all t > 0 if the initial data u(x, 0) = φ(x) decays faster than polinomially on IR = {x ∈ IR ; x > 0 } and has certain initial Sobolev regularity.


Introduction
In 1976, J. C. Saut and R. Temam [ 22 ] have remarked that a solution u of an equation of Korteweg-de Vries type cannot gain or lose regularity: They show that if u(x, 0) = ϕ(x) ∈ H s (IR) for s ≥ 2, then u( • , t) ∈ H s (IR) for all t > 0. The same results were obtained independently by J. Bona and R. Scott [ 2 ] by different methods.For the Korteweg -de Vries ( KdV ) equation on the line, T. Kato [ 16 ], motivated by work of A. Cohen [ 6 ], showed that if u(x, 0) = ϕ(x) ∈ L 2  b ≡ H 2 (IR) L 2 (e b x dx) ( b > 0 ) then the solution u(x, t) of the KdV equation becomes C ∞ for all t > 0. A main ingredient in the proof was the fact that formally the semi-group S(t) = e −t ∂ 3 x in L 2 b is equivalent to S b (t) = e −t ( ∂x−b ) 3 in L 2 when t > 0. One would be inclined to believe this was a special property of the KdV equation.This is not, however, the case.The effect is due to the dispersive nature of the linear part of the equation.S. N. Kruzkov and A. V. Faminskii [ 20 ] for u(x, 0) = ϕ(x) ∈ L 2 such that x α ϕ(x) ∈ L 2 (( 0, +∞ )) is was proven that the weak solution of the KdV equation constructed there has l-continuous space derivatives for all t > 0 if l < 2α.The proof of this result is based on the asymptotic behavior of the Airy function and its derivatives, and on the smoothing effect of the KdV equation found in [16,20].Corresponding work for some special nonlinear Schrödinger equations was done by Hayashi et al. [12,13] and G. Ponce [21].While the proof of T. Kato appears to depend on special a priori estimates, some of its mystery has been resolved by results of local gain of finite regularity for various others linear and nonlinear dispersive equations due to P. Constantin and J. C. Saut [10], P. Sjolin [23], J. Ginibre and G. Velo [11] and others.However, all of them require growth conditions on the nonlinear term.
All the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property.To mention only a few, the KdV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrödinger equations are included.Continuing with the idea of W. Craig, T. Kappeler and W. Strauss [9] we study a equation of Korteweg -de Vries -Burgers Type with −∞ < x < +∞ and t > 0. The flux f = f (u) is a given smooth function satisfying certain assumptions to be listed shortly.It is shown under certain additional condition on f that C ∞ -solutions u(x, t) are obtained for all t > 0 if the initial data u(x, 0) decays faster than polinomially on IR + = { x ∈ IR ; x > 0 } and has certain initial Sobolev regularity.In section three we prove an important a priori estimate.In section four we prove basic local-in-time existence and uniqueness results for (1) used in the gain of regularity result in section 7. Specifically, we show that for initial ϕ(x) ∈ H N (IR), for N ≥ 3, there exists a unique u ∈ L ∞ ([ 0, T ]; H N (IR)) where the time of existence depends of the norm of ϕ(x) ∈ H 3 (IR).In section five we developed a serie of estimates for solutions of equation ( 1) in weighted Sobolev norms.We show that solution u in (1) also satisfies a persistence property.Indeed, we prove that if the initial data ϕ lies in a certain weighted Sobolev space, then the unique solution u of the nonlinear equation ( 1) lies in the same Sobolev space.At the conclusion of sections, we give a formal proof of our gain in regularity theorem for nonlinear equation (1).In section six we state our main results on the gain of regularity for the nonlinear equation ( 1) and prove the a priori estimate used in the main Theorem 7.2.In the section seven, we state and prove our main results concerning the gain of regularity for solutions to the nonlinear equation ( 1), including the main estimates for the remainder terms.Specifically, we prove the following principal theorem.
Theorem. ( Main Theorem ) Let T > 0 and u(x, t) be a solution of (1.1) for some L ≥ 2 and all σ > 0. Then

Preliminaries
We consider equation of Korteweg -de Vries -Burgers type with − ∞ < x < + ∞ and t > 0 is an arbitrary positive time, , δ > 0. The flux f = f (u) ≡ f (u(x, t)) is given smooth function satisfying certain assumptions.
Notation 1.We write Let T > 0, The assumptions on f are as follow: A.2 All the derivatives of f (u, x, t) are bounded for x ∈ IR, t ∈ [ 0, T ] and u in a bounded set.
A.3 x N ∂ j x f (0, x, t) is bounded for all N ≥ 0, j ≥ 0 and x ∈ IR, t ∈ ( 0, T ].Lemma 1.These assumptions imply that f has the form x, t) and h = h(x, t).f 0 and h are C ∞ and each of their derivatives is bounded for u bounded, x ∈ IR and t ∈ [0, T ]. Proof.Indeed for y 0 = 0 and h(x, t) = f (0, x, t).
Definition 2.1.An evolution equation enjoys a gain of regularity if its solutions are smoother for all t > 0 than its initial data.

Definition 2.2. A function ξ(x, t) belong to the weight class
Remark 1.We shall always take σ ≥ 0, k ≥ 0 and i ≥ 1.
Definition 2.3.Fixed ξ ∈ W σ i k define the space ( s is an integer positive ) Proof.This result has already been proved in [ 9 ].
From now on we consider the following equation The equation is considered for −∞ < x < +∞, t ∈ [ 0, T ] and T is an arbitrary positive time.

An Important a Priori Estimate
In this section we show an fundamental a priori estimate to demonstrate basic local-in-time existence theorem.Differentiating the equation (2.5) two times leads to The equation (3.2) is linearized by substituting a new variable w in each coefficient This linear (3.3) equation which is to be solved at each iteration has the form We consider the following lemma that to help us to set up the iteration scheme.
The next step is to estimate the corresponding solutions v = v(x, t) of equation (3.3) via the coefficients of that equation.
For each integer α there exist positive nondecreasing functions G, E and F such that for all t ≥ 0 where • α is the norm in H α (IR) and λ = max {1, α}.

Uniqueness and Local Existence Theorem
In this section, we study uniqueness and local existence of strong solutions for the problem (2.5).Specifically, we show that for initial ϕ(x) ∈ H N (IR), for N ≥ 3, there exists a unique u ∈ L ∞ ([ 0, T ]; H N (IR)) where the time of existence depends of the norm of ϕ(x) ∈ H 3 (IR).These results are used in the proof of Theorem 7.2.First we address the question of uniqueness.
) and with the same initial data.Then with (u − v)(x, 0) = 0.By the mean value theorem we have then there is smooth function d depending smoothly on u, x, t and v, x, t such that (4.1) takes the form In the firs term we have In the second term integrating by parts The others terms are treated the similar form.Replacing over (4.3) we have Using the assumptions on f and for a suitably chosen constant c , we have By Gronwall's inequality and the fact (u − v) vanishes at t = 0 it follows that u = v.This proves uniqueness.
To stablish the local existence of strong solutions to (2.5) we use of the a priori estimate together with a approximation procedure.
We construct the mapping T : where u (n−1) is in the position of w in equation (3.3) and u (n) is in the position of v which is the solution of equation (3.3).By to lemma 3.1.,u (n) exists and is unique in C(( 0, +∞ ); H N (IR)).A choice of c 0 and the use of the a priori estimate in §3 show that T : We know state our existence theorem for equation (2.5).
We prove that for ϕ ∈ H ∞ (IR) = k≥0 H k (IR) there exists a solution u ∈ L ∞ ([ 0, T ]; H N (IR)) with initial data u(x, 0) = ϕ(x) and which a time of existence T > 0 which only depends on the norm of ϕ.
We define a sequence of approximations to equation (3.3) as The first approximation is given by v (0) (x, 0) = ϕ(x) − ∂ 2 ϕ(x).Equation (4.4) is a linear equation at each iteration which can be solved in any interval of time in which the coefficients are defined.This is shown in lemma 3.1.By lemma 3.2 follows that and we obtain for j = 0, 1.
Choosing T = T (c 0 ) sufficiently small, but T not depending on n , one concludes that Claim.u = ∧v is the solution we are looking for.
Proof.We first need to show that v is a solution of (3.2).We do so by showing that each term in (4.4) converges to its corret limit.By equation (4.4), ∂ t v (n) is a sum of terms each of which is the product of a coefficient, bounded uniformly in n , so that the sequence Therefore, for a subsequence, v (n j ) def = v (n) we have v (n) −→ v a. e. in x and t.It follows that the fourth term on the right hand side of (4.4) . Similarly all other terms in (4.4) converge to their corrects limits, implying v . This way we have (2.5) for u = ∧v.We prove that there exists a solution of the equation (2.5) with u ∈ L ∞ ([ 0, T ]; H N (IR)) and N ≥ 4, where T depends only on the norm of ϕ.We already know that there is a solution ( previously ) . Take α = N − 2 and consider (4.5) for α ≥ 2. By the same arguments as for α = 1 we conclude that there exists T (α) > 0 depending on the norm of ϕ but independent n such that v (n) α ≤ c for all 0 ≤ t ≤ T (α) .Thus v ∈ L ∞ ([ 0, T (α) ]; H α (IR)).Now denote by 0 ≤ T * (α) ≤ +∞ the maximal number such that for all 0 < T ≤ T * (α) We claim that T (1) ≤ T * (α) for all α ≥ 2. Thus T can be chosen depending only on norm of ϕ.Approximating ϕ by Let u j be a solution of (2.5) with u j (x, 0) = ϕ j (x).According to the above argument, there exists T which is independent of n but depending only sup j ϕ j such that u j exists on [ 0, T ] and a subsequence . As a consequence of Theorem 4.1 and Theorem 4.2 and its proof one gets.
Let u and u (γ) be the corresponding unique solutions given by Theorems 4.

Main Inequality
Lemma 5.1.Let u be a solution of the initial value problem (2.5).
Then we have the following inequality.
Proof.Taking α-derivatives of the equation (2.5) (for α ≥ 3 ) over x ∈ IR Multiply (5.1) by 2 ξ u α , integrate over x ∈ IR we have integrating by parts we have where we obtain main inequality.

Persistence Theorem
As a starting point for the a priori gain of regularity results that will be discussed in next section, we need to develop some estimates for solutions of the equation (2.5) in weighted Sobolev norms.The existence of these weighted estimated is often called the persistence of a property of the initial data ϕ.We show that if The time interval of such persistence is at least as long as the interval guaranteed by the existence theorem 4.2.Theorem 6.1.( Persistence ) Let i ≥ 1 and L ≥ 3 be nonnegative integers, 0 < T < +∞.Assume that u is the solution to where σ is arbitrary, η ∈ W σ, i−1, 0 for i ≥ 1.

The Main Theorem
In this section we state and prove our main theorem, which tells us that if the initial data u(x, 0) decays faster than polinomially on IR + = { x ∈ IR ; x > 0 } and possesses certain initial Sobolev regularity, then the solution u(x, t) ∈ C ∞ for all t > 0. In the case of main theorem, we take 4 ≤ α ≤ L + 2. For α ≤ L + 2, we take any where c depends only on the norms of u in Theorem 7.2.( Main Theorem ) Let T > 0 and u(x, t) be a solution of (2.5) for some L ≥ 2 and all σ > 0. Then Remark 7.1.If the assumption (7.9) holds for all L ≥ 2, the solution is infinitely differentiable in the x -variable.From the equation (2.5) itself the solution C ∞ in both of its variables.

Lemma 5 . 3 .
an arbitrary weight function, then there exist ξ ∈ W σ, i+1, k which satisfies η = 3 δ + 2 The expression R in the main inequality is a sum of terms of the form