Directory UMM :Journals:Journal_of_mathematics:DM:
Divulgaciones Matem´
aticas Vol. 10 No. 2(2002), pp. 131–148
Existence and Uniqueness for a
Nonlinear Dispersive Equation
Existencia y Unicidad para una Ecuaci´on Dispersiva No Lineal
Octavio Paulo Vera Villagr´an (overa@ucsc.cl)
Facultad de Ingenier´ıa
Universidad Cat´olica de la Sant´ısima Concepci´on
Alonso de Rivera 2850, Concepci´on, Chile.
A Lorena Conde Baquedano, con admiraci´
on, mucha admiraci´
on.
Abstract
In this paper we study the existence and uniqueness properties of solutions of some nonlinear dispersive equations of evolution. We consider
the equation
½ ∂u
3
∂
∂
+ ∂x
[f (u)] = ǫ ∂x
[g( ∂u
)] − δ ∂∂xu3
∂t
∂x
(1) =
u(x, 0) = ϕ(x)
with x ∈ R, T an arbitrary positive time and t ∈ [0, T ]. The flux
f = f (u) and the (degenerate) viscosity g = g(λ) are given smooth
functions satisfying certain assumptions. This work presents a result a
priori that permits to obtain gain of regularity for equation (1), motivated by the results obtained by Craig, Kappeler and Strauss [3].
Key words and phrases: Evolution equations, Lions-Aubin Theorem, Weighted Sobolev Space.
Resumen
En este art´ıculo estudiamos las propiedades de existencia y unicidad de las soluciones de algunas ecuaciones de evoluci´
on dispersivas no
lineales. Consideramos la ecuaci´
on
½ ∂u
3
∂
∂
+ ∂x
[f (u)] = ǫ ∂x
[g( ∂u
)] − δ ∂∂xu3
∂t
∂x
(1) =
u(x, 0) = ϕ(x)
Recibido 2000/05/06. Revisado 2001/10/19. Aceptado 2002/03/15.
MSC (2000): Primary 35Q53; Secondary 47J35.
This research was partially supported by DIN 04/2001 Universidad Cat´
olica de la Sant´ısima
Concepci´
on, Concepci´
on, Chile.
132
Octavio Paulo Vera Villagr´an
con x ∈ R, t ∈ [0, T ] y T un tiempo positivo arbitrario. El flujo f = f (u)
y la viscosidad (degenerada) g = g(λ) son funciones suaves dadas que
satisfacen ciertas condiciones. En este trabajo se presenta un resultado a
priori que permite obtener una ganancia de regularidad para la ecuaci´
on
(1), motivado en los resultados obtenidos por Craig, Kappeler y Strauss
[3].
Palabras y frases clave: Ecuaciones de evoluci´
on, teorema de LionsAubin, Espacios pesados de Sobolev.
1
Introduction
In 1976, J. C. Saut and R. Temam [23] remarked that a solution u of an
equation of Korteweg-de Vries type cannot gain or lose regularity: they showed
that if u(x, 0) = ϕ(x) ∈ H s (R) for s ≥ 2, then u(·, t) ∈ H s (R) 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) ∈
T
L2b ≡ H 2 (R) L2 (ebx 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
3
3
formally the semigroup S(t) = e−t∂x in L2b is equivalent to Sb (t) = e−t(∂x −b)
2
in L 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) ∈ L2 such that xα ϕ(x) ∈ L2 ((0, +∞)),
proved that the weak solution of the KdV equation constructed there has lcontinuous 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 Schr¨odinger equations was done by Hayashi et
al. [12, 13] and G. Ponce [22]. While the proof of T. Kato seems to depend on
special a priori estimates, some of its mystery has been resolved by results of
local gain of finite regularity for various other linear and nonlinear dispersive
equations due to P. Constantin and J. C. Saut [10], P. Sjolin [24], 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 Schr¨odinger equations are included.
Divulgaciones Matem´
aticas Vol. 10 No. 2(2002), pp. 131–148
Existence and Uniqueness for a Nonlinear Dispersive Equation
133
Continuing with the idea of W. Craig, T. Kappeler and W. Strauss [9] we
study existence and uniqueness properties of solutions of some nonlinear dispersive equations of evolution. We consider the nonlinear dispersive equation
(
3
∂
∂
∂u
+ ∂x
[f (u)] = ǫ ∂x
[g( ∂u
)] − δ ∂∂xu3
∂t
∂x
(1) =
u(x, 0) = ϕ(x)
with x ∈ R, T an arbitrary positive time and t ∈ [0, T ]. The flux f = f (u)
and the (degenerate) viscosity g = g(λ) are given smooth functions satisfying
certain assumptions to be listed shortly.
In section 3 we prove an important a priori estimate.
In section 4 we prove basic local-in-time existence and uniqueness results
for (1). Specifically, we show that for initial ϕ(x) ∈ H N (R), for N ≥ 3, there
exists a unique u ∈ L∞ ([0, T ]; H N (R)) where the time of existence depends
of the norm of ϕ(x) ∈ H 3 (R).
In section 5 we develop a series of estimates for solutions of equation (1)
in weighted Sobolev norms. We show that a solution u of (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.
2
Preliminaries
We consider the nonlinear dispersive equation
· µ ¶¸
∂3u
∂u
∂
∂
∂u
g
−δ 3
+
[f (u)] = ǫ
∂t
∂x
∂x
∂x
∂x
(2.1)
with x ∈ R, t ∈ [0, T ] and T is an arbitrary positive time. The flux f = f (u)
and the (degenerate) viscosity g = g(λ) are given smooth functions satisfying
certain assumptions. ǫ, δ > 0.
Notation 1. We write ∂ =
∂
∂j u
∂xj ; ∂j = ∂uj .
∂
∂x ,
∂t u =
∂u
∂t
= ut and we abbreviate uj = ∂ j u =
Example. If ∂u/∂x = u1 then
· µ ¶¸
∂
∂
∂
∂
∂
∂u
[g(u1 )]
[g(u1 )] u2 = (∂1 g)u2
[g(u1 )] =
[u1 ] =
g
=
∂x
∂x
∂x
∂u1
∂x
∂u1
The assumptions on f are the following:
Divulgaciones Matem´
aticas Vol. 10 No. 2(2002), pp. 131–148
134
Octavio Paulo Vera Villagr´an
A.1 f : R2 × [0, T ] 7→ R is C ∞ in all its variables.
A.2 All the derivatives of f = f (u, x, t) are bounded for x ∈ R for t ∈ [0, T ]
and u ∈ R in a bounded set.
A.3 xN ∂xj f (0, x, t) is bounded for all N ≥ 0, j ≥ 0, and x ∈ R, t ∈ (0, T ].
Indeed, ∀N ≥ 0, ∀j ≥ 0, x ∈ R, t ∈ (0, T ], there exists c > 0 such that
|xN ∂xj f (0, x, t)| ≤ c.
The assumptions on g are the following:
B.1 g : R2 × [0, T ] 7→ R is C ∞ in all its variables.
B.2 All the derivatives of g(y, x, t) are bounded for x ∈ R, t ∈ [0, T ] and y in
a bounded set.
B.3 xN ∂xj g(0, x, t) is bounded for all N ≥ 0, j ≥ 0 and x ∈ R, t ∈ (0, T ].
B.4 There exists c > 0 such that ∂1 g(u1 , x, t) ≥ c > 0, for all u1 ∈ R, x ∈ R
and t ∈ [0, T ].
Lemma 1. These assumptions imply that f has the form f = u0 f0 + h ≡
uf0 + h where f0 = f0 (u0 , x, t) ≡ f0 (u, x, t) and h = h(x, t). f0 and h are C ∞
and each of their derivatives is bounded for u bounded, x ∈ R and t ∈ [0, T ].
Proof. Indeed, we define
f0 =
½
f (u0 ,x,t)−f (0,x,t)
u0
∂0 f (0, x, t)
for u0 6= 0
for u0 = 0
and h(x, t) = f (0, x, t).
Remark 1. The same for g.
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 Wσik if it is
a positive C ∞ function on R × [0, T ], ξx > 0 and there exists a constant
cj , 0 ≤ j ≤ 5 such that
0 < c1 ≤ t−k e−σx ξ(x, t) ≤ c2
0 < c3 ≤ t−k x−i ξ(x, t) ≤ c4
¡
t | ξt | + | ∂ j ξ
in R × [0, T ], for all j ∈ N.
for x < −1, 0 < t < T.
for x > 1, 0 < t < T.
¢
| /ξ ≤ c5
Remark 2. We shall always take σ ≥ 0, i ≥ 1 and k ≥ 0.
Divulgaciones Matem´
aticas Vol. 10 No. 2(2002), pp. 131–148
(2.2)
(2.3)
(2.4)
Existence and Uniqueness for a Nonlinear Dispersive Equation
135
Example 1. Let
ξ(x) =
½
1 + e−1/x
1
for x > 0
for x ≤ 0
then ξ ∈ W0i0 .
Definition 2.3. Fixed ξ ∈ Wσik define the space (for s a positive integer)
©
H s (Wσik ) = v : R → R; such that the distributional derivatives
s
X
∂j v
for 0 ≤ j ≤ s satisfy kvk2 =
j
∂x
j=0
Z
+∞
|∂ j v(x)|2 ξ(x, t)dx < +∞
−∞
ª
Remark 3. H s (Wσik ) depend t ( because ξ = ξ(x, t)).
Lemma 2. For ξ ∈ Wσi0 and σ ≥ 0, i ≥ 0 there exists a constant c such that,
for u ∈ H 1 (Wσi0 )
Z +∞
¡
¢
2
sup | ξu |≤ c
| u |2 + | ∂u |2 ξdx.
x∈R
−∞
Proof. See Lemma 7.3 in [9].
Definition 2.4. Fixed ξ ∈ Wσik define the space
L2 ([0, T ]; H s (Wσik )) = {v = v(x, t), v(·, t) ∈ H s (Wσik ) such that
Z T
2
||| v ||| =
kv(·, t)k2 dt < +∞}
0
L∞ ([0, T ]; H s (Wσik )) = {v = v(x, t), v(·, t) ∈ H s (Wσik ) such that
||| v |||∞ = ess sup kv(·, t)k < +∞}
t∈[0,T ]
Remark 4. The usual Sobolev space is H s (R) = H s (W000 ) without a weight.
Remark 5. We shall derive the a priori estimates assuming that the solution
is C ∞ , bounded as x → −∞, and rapidly decreasing as x → +∞, together
with all of its derivatives.
According to notation 1, for equation (1) we obtain
ut + δu3 − ǫ(∂1 g)u2 + (∂0 f )u1 = 0.
(2.5)
The equation is considered for −∞ < x < +∞, t ∈ [0, T ] and T is an arbitrary
positive time.
Divulgaciones Matem´
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136
3
Octavio Paulo Vera Villagr´an
An important a priori estimate
In this section we show a fundamental a priori estimate to demonstrate basic local-in-time existence theorem. We need to construct a mapping T :
L∞ ([0, T ]; H s (R)) → L∞ ([0, T ]; H s (R)) with the following property: Given
u(n) = T (u(n−1) ) and ku(n−1) ks ≤ c0 then ku(n) ks ≤ c0 , where s and c0 > 0
are constants. This property tells us, in fact, that T : Bc0 (0) → Bc0 (0) where
Bc0 (0) = {v(x, t); kv(x, t)ks ≤ c0 } is a ball in space L∞ ([0, T ]; H s (R)). To
guarantee this property, we will appeal to an a priori estimate which is the
main object of this section.
Differentiating the equation (2.5) two times leads to
∂t u2 + δu5 − ǫ(∂1 g)u4 + (∂0 f )u3 − 2ǫ∂(∂1 g)u3
+ 2∂(∂0 f )u2 − ǫ∂ 2 (∂1 g)u2 + ∂ 2 (∂0 f )u1 = 0 (3.1)
Let u = ∧v where ∧ = (I − ∂ 2 )−1 . Then ∂t u2 = −vt + ut . Replacing in (3.1)
we have
− vt + δ∧v 5 − ǫ(∂1 g)∧v 4 + (∂0 f )∧v 3 − 2ǫ∂(∂1 g)∧v 3 + 2∂(∂0 f )∧v 2
− ǫ∂ 2 (∂1 g)∧v 2 + ∂ 2 (∂0 f )∧v 1 − [δ∧v 3 − ǫ(∂1 g)∧v 2 + (∂0 f )∧v 1 ] = 0 (3.2)
where g = g(∧v 1 ) and f = f (∧v).
The equation (3.2) is linearized by substituting a new variable w in each
coefficient;
−vt + δ∧v 5 − ǫ∂1 g(∧w1 )∧v 4 + ∂0 f (∧w)∧v 3 − 2ǫ∂(∂1 g(∧w1 ))∧v 3
+2∂(∂0 f (∧w))∧v 2 − ǫ∂ 2 (∂1 g(∧w1 ))∧v 2 + ∂ 2 (∂0 f (∧w))∧v 1
−[δ∧v 3 − ǫ∂1 g(∧w1 )∧v 2 − ∂0 f (∧w)∧v 1 ] = 0.
(3.3)
Lemma 3.1. Let v, w ∈ C k ([0, +∞); H N (R)) for all k, N which satisfy (3.3).
Let
ξ ≥ c1 > 0. For each integer α there exist positive nondecreasing functions
E,F and G such that for all t ≥ 0
Z
∂t ξvα2 dx ≤ G(kwkλ )kvk2α + E(kwkλ )kwk2α + F (kwkα ) (3.4)
R
where k · kα is the norm in H α (R) and λ = max {1, α}.
Proof. Differentiating α-times the equation (3.3) for some α ≥ 0
−∂t vα + δ∧v α+5 − ǫ(∂1 g)∧v α+4 + ((∂0 f ) − (α + 2)ǫ∂(∂1 g))∧v α+3
+
α+2
X
h(j) ∧v j + q(∧w)∧wα+2 + p(∧wα+1 , . . .) = 0
j=2
Divulgaciones Matem´
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(3.5)
Existence and Uniqueness for a Nonlinear Dispersive Equation
137
where h(j) is a smooth function depending on ∧wi+3 , ∧wi+2 , . . . with i =
2 + α − j.
We multiply equation (3.5) by 2ξvα , integrate over x ∈ R
Z
Z
Z
−2 ξvα ∂t vα dx + 2δ ξvα ∧v α+5 dx − 2ǫ ξ(∂1 g)vα ∧v α+4 dx
R
R
Z
ZR
+2 ξ(∂0 f )vα ∧v α+3 dx − 2(α + 2)ǫ ξ∂(∂1 g)vα ∧v α+3 dx
R
α+2
XZ
+2
j=2
Z
+2
R
ξh(j) vα ∧v j dx + 2
Z
ξq(∧w)vα ∧wα+2 dx
R
R
ξvα p(∧wα+1 , . . .)dx = 0.
(3.6)
R
Each term in (3.6) is treated separately. The first two terms yield
Z
Z
Z
ξt vα2 dx
−2 ξvα ∂t vα dx = −∂t ξvα2 dx +
R
R
R
and
Z
Z
ξvα ∧v α+5 dx = 2δ ξ∧(I − ∂ 2 )vα ∧v α+5 dx
R
Z
Z
= 2δ ξ∧v α ∧v α+5 dx − 2δ ξ∧v α+2 ∧v α+5 dx
ZR
ZR
5
2
= −δ ∂ ξ(∧v α ) dx + 5δ ∂ 3 ξ(∧v α+1 )2 dx
Z
ZR
ZR
3
2
2
− 5δ ∂ξ(∧v α+2 ) dx + δ ∂ ξ(∧v α+2 ) dx − 3δ ∂ξ(∧v α+3 )2 dx.
2δ
R
R
R
R
The other terms are treated similarly, integrating by parts once again. Replacing over (3.6) we have
Z
Z
Z
Z
5
2
2
2
ξt vα dx − δ ∂ ξ(∧v α ) dx + 5δ ∂ 3 ξ(∧v α+1 )2 dx
∂t ξvα dx =
R
R
ZR
Z R
Z
3
2
2
− 5δ ∂ξ(∧v α+2 ) dx + δ ∂ ξ(∧v α+2 ) dx − 3δ ∂ξ(∧v α+3 )2 dx
R
R
Z
Z
Z R
4
2
3
2
∂ (ξ∂0 f )(∧v α ) dx − ǫ ∂ (ξ∂1 g)(∧v α ) dx + 4ǫ ∂ 2 (ξ∂1 g)(∧v α+1 )2 dx
−
R
R
R
Z
Z
− 2ǫ ξ(∂1 g)(∧v α+2 )2 dx + ǫ ∂ 2 (ξ∂1 g)(∧v α+2 )2 dx
Z
ZR
ZR
2
∂(ξ∂0 f )(∧v α+2 )2 dx
− 2ǫ ξ(∂1 g)(∧v α+3 ) dx + 3 ∂(ξ∂0 f )(∧v α+1 )2 dx +
R
R
R
Divulgaciones Matem´
aticas Vol. 10 No. 2(2002), pp. 131–148
138
Octavio Paulo Vera Villagr´an
Z
∂ 3 (ξ∂(∂1 g))(∧v α )2 dx − 3(α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+1 )2 dx
R
Z
ZR
2
(α+2)
2
)(∧v α )2 dx
− (α + 2)ǫ ∂(ξ∂(∂1 g))(∧vα+2 ) dx + ∂ (ξh
+ (α + 2)ǫ
Z
R
R
−2
+2
Z
Z
(α+2)
ξh
2
(∧v α+1 ) dx − 2
Z
ξh
(α+2)
2
(∧v α+2 ) dx + 2
R
R
ξq(∧w)vα ∧wα+2 dx + 2
α+1
XZ
j=2
Z
ξh(j) vα ∧v j dx
R
ξvα p(∧wα+1 , . . .)dx,
R
R
then we have
Z
Z
ξt vα2 dx
ξvα2 dx = − (3δ∂ξ + 2ǫξ(∂1 g))(∧v α+3 )2 dx +
R
R
Z R
Z
Z
2
3
2
+ δ ∂ ξ(∧v α+2 ) dx − 5δ ∂ξ(∧v α+2 ) dx + ǫ ∂ 2 (ξ(∂1 g))(∧v α+2 )2 dx
R
R
Z
ZR
∂(ξ(∂0 f ))(∧v α+2 )2 dx
− 2ǫ ξ(∂1 g)(∧v α+2 )2 dx +
R
R
Z
Z
− (α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+2 )2 dx − 2 ξh(α+2) (∧v α+2 )2 dx
R
R
Z
Z
2
3
2
+ 5δ ∂ ξ(∧v α+1 ) dx + 4ǫ ∂ (ξ∂1 g)(∧v α+1 )2 dx
R
Z
Z R
2
+ 3 ∂(ξ∂0 f )(∧v α+1 ) dx − 3(α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+1 )2 dx
R
Z
Z
ZR
5
(α+2)
2
(∧v α+1 ) dx − δ ∂ ξ(∧v α )2 dx − ǫ ∂ 4 (ξ∂1 g)(∧v α )2 dx
− 2 ξh
R
R
Z
Z R
3
3
2
∂ (ξ∂0 f )(∧v α ) dx + (α + 2)ǫ ∂ (ξ∂(∂1 g))(∧v α )2 dx
−
∂t
Z
+
Z
R
R
∂ 2 (ξh(α+2) )(∧v α )2 dx + 2
R
+2
α+1
XZ
j=2
Z
R
ξq(∧w)vα ∧wα+2 dx + 2
Z
ξh(j) vα ∧v j dx
R
ξvα p(∧wα+1 , . . .)dx.
R
The first term in the righthand side is nonpositive, hence
Divulgaciones Matem´
aticas Vol. 10 No. 2(2002), pp. 131–148
Existence and Uniqueness for a Nonlinear Dispersive Equation
139
Z
Z
Z
ξt vα2 dx + δ ∂ 3 ξ(∧v α+2 )2 dx − 5δ ∂ξ(∧v α+2 )2 dx
ξvα2 dx ≤
R
R
R
R
Z
Z
+ ǫ ∂ 2 (ξ∂1 g)(∧v α+2 )2 dx − 2ǫ ξ(∂1 g)(∧v α+2 )2 dx
R
Z
Z R
2
+ ∂(ξ∂0 f )(∧v α+2 ) dx − (α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+2 )2 dx
R
R
Z
Z
3
(α+2)
2
(∧v α+2 ) dx + 5δ ∂ ξ(∧v α+1 )2 dx
− 2 ξh
R
ZR
Z
+ 4ǫ ∂ 2 (ξ∂1 g)(∧v α+1 )2 dx + 3 ∂(ξ∂0 f )(∧v α+1 )2 dx
R
R
Z
Z
− 3(α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+1 )2 dx − 2 ξh(α+2) (∧v α+1 )2 dx
R
R
Z
Z
4
2
5
2
− δ ∂ ξ(∧v α ) dx − ǫ ∂ (ξ∂1 g)(∧v α ) dx
R
Z
Z R
3
2
− ∂ (ξ∂0 f )(∧v α ) dx + (α + 2)ǫ ∂ 3 (ξ∂(∂1 g))(∧v α )2 dx
∂t
Z
+
Z
R
R
∂ 2 (ξh(α+2) )(∧v α )2 dx + 2
R
+2
α+1
XZ
j=2
Z
ξq(∧w)vα ∧wα+2 dx + 2
Z
ξh(j) vα ∧v j dx
R
ξvα p(∧wα+1 , . . .)dx.
R
R
In the last term we have
¯ Z
¯
¯Z
¯
¯Z
¯
¯
¯
¯
¯
¯
¯
¯2 ξ(q∧wα+2 + p)vα dx¯ ≤ 2 ¯ ξq∧wα+2 vα dx¯ + 2 ¯ ξpvα dx¯ ,
¯
¯
¯
¯
¯
¯
R
R
R
but ∧wα+2 = ∧wα − wα then
¯
¯Z
¯
¯Z
¯
¯ Z
¯
¯
¯
¯
¯
¯
¯2 ξ(q∧wα+2 + p)vα dx¯ ≤ 2 ¯ ξqvα (∧wα − wα )dx¯ + 2 ¯ ξpvα dx¯
¯
¯
¯
¯
¯
¯
R
R
R
¯
¯
¯Z
¯Z
µZ
¶1/2 µZ
¶1/2
¯
¯
¯
¯
ξp2 dx
ξvα2 dx
≤ 2 ¯¯ ξq∧wα vα dx¯¯ + 2 ¯¯ ξqwα vα dx¯¯ + 2
R
¯ZR
¯
¯ZR
¯ Z R
Z
¯
¯
¯
¯
2
2
≤ 2 ¯¯ ξq∧wα vα dx¯¯ + 2 ¯¯ ξqwα vα dx¯¯ +
ξvα dx
ξp dx +
R
R
R
R
and since p = p(∧wα+1 , . . .) then
¯ Z
¯
¯
¯
¯2 ξ (q∧wα+2 + p) vα dx¯ ≤ r(kwkα )[kwk2α + kvk2α ] + kvkα + s(kwkα )
¯
¯
R
Divulgaciones Matem´
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140
Octavio Paulo Vera Villagr´an
in the same form, using that ∧v n = ∧v n−2 − vn−2 . By standard estimates,
the Lemma now follows.
We define a sequence of approximations to equation (3.3) as
(n)
−vt
(n)
(n)
(n)
(n)
+ δ∧v 5 − ǫ(∂1 g)∧v 4 + (∂0 f )∧v 3 − 2ǫ∂(∂1 g)∧v 3
(n)
(n−1)
−δ∧v 3 + O(∧v 2
(n−1)
, ∧v 1
, . . .) = 0
(3.7)
(n−1)
where g = g(∧v 1
), f = f (∧v (n−1) ) and where the initial condition is
(n)
given by v (x, 0) = ϕ(x) − ∂ 2 ϕ(x). The first approximation is given by
v (0) (x, 0) = ϕ(x)−∂ 2 ϕ(x). Equation (3.7) is a linear equation at each iteration
which can be solved in any interval of time in which the coefficients are defined.
This equation has the form
∂t v = δ∧v 5 − ǫ∧v 4 + b(1) ∧v 3 + b(0)
(3.8)
T
∞
N
Lemma 3.2. Given initial data in ϕ ∈ H (R) = N ≥0 H (R) there exists a
unique solution of (3.8). The solution is defined in any time interval in which
the coefficients are defined.
Proof. See [26].
4
Uniqueness and existence theorem
In this section, we study uniqueness and local existence of strong solutions for
problem (2.5). Specifically, we show that for initial ϕ(x) ∈ H N (R), for N ≥ 3,
there exists a unique u ∈ L∞ ([0, T ]; H N (R)) where the time of existence
depends of the norm of ϕ(x) ∈ H 3 (R). First we address the question of
uniqueness.
Theorem 4.1. (Uniqueness). Let ϕ ∈ H 3 (R) and 0 < T < +∞. Assume f
satisfies A.1-A.3. and g satisfies B.1-B.4, then there is at most one solution
u ∈ L∞ ([0, T ]; H 3 (R)) of (2.5) with initial data u(x, 0) = ϕ(x).
Proof. Assume u, v ∈ L∞ ([0, T ]; H 3 (R)) are two solutions of (2.5) with
ut , vt ∈ L∞ ([0, T ]; L2 (R)) and with the same initial data. Then
(u − v)t + δ(u − v)3 − ǫ[g(u1 ) − g(v1 )]1 + [f (u) − f (v)]1 = 0
(4.1)
with (u − v)(x, 0) = 0. Using the Mean Value Theorem there are smooth
functions d(1) and d(2) depending smoothly on u1 , x, t; v1 , x, t and u, x, t; v, x, t
respectively such that (4.1) has the form
(u − v)t + δ(u − v)3 − ǫ[d(1) ]1 (u − v)1 − ǫd(1) (u − v)2
+ [d(2) ]1 (u − v) + d(2) (u − v)1 = 0
Divulgaciones Matem´
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(4.2)
Existence and Uniqueness for a Nonlinear Dispersive Equation
We multiply equation (4.2) by 2ξ(u − v), integrate over x ∈ R
Z
Z
2 ξ(u − v)(u − v)t dx + 2δ ξ(u − v)(u − v)3 dx
R
R
Z
Z
(1)
− 2ǫ ξ[d ]1 (u − v)(u − v)1 dx − 2ǫ ξd(1) (u − v)(u − v)2 dx
R
Z
Z R
(2)
(2)
2
+ 2 ξ[d ]1 (u − v) dx + 2 ξd (u − v)(u − v)1 dx = 0.
141
(4.3)
R
R
Each term is treated separately. In the first term we have
Z
Z
Z
ξt (u − v)2 dx.
2 ξ(u − v)(u − v)t dx = ∂t ξ(u − v)2 dx −
R
R
R
The second term, integrating by parts, yields
Z
2δ ξ(u − v)(u − v)3 dx
R
Z
Z
= −2δ ∂ξ(u − v)(u − v)2 dx − 2δ ξ(u − v)1 (u − v)2 dx
Z R
Z
ZR
2
= 2δ ∂ ξ(u − v)(u − v)1 dx + 2δ ∂ξ(u − v)21 dx + δ ∂ξ(u − v)21 dx
R
R
R
Z
Z
2
3
2
(4.4)
= − δ ∂ ξ(u − v) dx + 3δ ∂ξ(u − v)1 dx.
R
R
Replacing over (4.3) we have
Z
Z
Z
∂t ξ(u − v)2 dx − ξt (u − v)2 dx − δ ∂ 3 ξ(u − v)2 dx
R
R
R
Z
Z
(1)
2
+ 3δ ∂ξ(u − v)1 dx + ǫ ∂(ξ[d ]1 )(u − v)2 dx
R
Z
Z R
2
(1)
2
− ǫ ∂ (ξd )(u − v) dx + 2ǫ ξd(1) (u − v)21 dx
R
Z
ZR
(2)
2
∂(ξd(2) )(u − v)2 dx = 0.
+ 2 ξ[d ]1 (u − v) dx −
R
R
Then
Z
Z
∂t ξ(u − v)2 dx + (3δ∂ξ + 2ǫξd(1) )(u − v)21 dx
R
Z
Z
ZR
3
2
2
ξt (u − v) dx + δ ∂ ξ(u − v) dx − ǫ ∂(ξ[d(1) ]1 )(u − v)2 dx
=
R
R
R
Z
Z
Z
(2)
2
(1)
2
∂(ξd(2) )(u − v)2 dx.
+ ǫ ∂ (ξd )(u − v) dx − 2 ξ[d ]1 (u − v)2 dx +
R
R
R
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142
Octavio Paulo Vera Villagr´an
Using B-4, the assumptions on f and g and for a suitably chosen constant c,
we have
Z
Z
Z
∂t ξ(u − v)2 dx + (3δ∂ξ + 2ǫξd(1) )(u − v)21 dx ≤ c ξ(u − v)2 dx.
R
R
R
By Gronwall’s inequality and the fact that (u − v) vanishes at t = 0 it follows
that u = v. This proves uniqueness.
We construct the mapping T : L∞ ([0, T ]; H s (R)) → L∞ ([0, T ]; H s (R)) by
defining that
u(0)
u(n)
=
=
ϕ(x)
T (u(n−1) )
n≥1
(n−1)
where u
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 3.2., u(n) exists and
is unique in C((0, +∞); H N (R)). A choice of c0 and the use of the a priori
estimate in §3 show that T : Bc0 (0) → Bc0 (0) with Bc0 (0) a bounded ball in
L∞ ([0, T ]; H s (R)).
Theorem 4.2. (Local existence). Assume f satisfies A.1 - A.4, and g satisfies
B.1 - B.4. Let N be an integers ≥ 3. If ϕ ∈ H N (R), then there is T > 0 and
u such that u is a strong solution of (2.5). u ∈ L∞ ([0, T ]; H N (R)) with initial
data u(x, 0) = ϕ(x).
T
Proof. We prove that for ϕ ∈ H ∞ (R) = k≥0 H k (R) there exists a solution
u ∈ L∞ ([0, T ]; H N (R)) with initial data u(x, 0) = ϕ(x) and which a time of
existence T > 0 which only depends ϕ.
We define a sequence of approximations to equation (3.2) as
− vt (n) + δ∧v 5 (n) − ǫ(∂1 g)∧v 4 (n) + (∂0 f )∧v 3 (n) − 2ǫ∂(∂1 g)∧v 3 (n)
+ 2∂(∂0 f )∧v 2 (n) − ǫ∂ 2 (∂1 g)∧v 2 (n) + ∂ 2 (∂0 f )∧v 1 (n)
− [δ∧v 3 −
(n)
ǫ(∂1 g)∧v 2
+
(n)
(∂0 f )∧v 1 ]
(4.5)
=0
(n−1)
where g = g(∧v 1
) and f = f (∧v (n−1) ) and whith initial data v (n) (x, 0) =
2
ϕ(x) − ∂ ϕ(x).
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.
By Lemma 3.1. it follows that
Z
∂t ξ[vα(n) ]2 dx ≤ G(kv (n−1) kλ )kv (n) k2α + E(kv (n−1) kλ )kv (n−1) k2α
R
+ F (kv (n−1) kα ).
Divulgaciones Matem´
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(4.6)
Existence and Uniqueness for a Nonlinear Dispersive Equation
143
Choose α = 1. Let c0 ≥ kϕ − ∂ 2 ϕk1 ≥ kϕk3 . For each iterate n, kv (n) (·, t)k is
c2 2
c +1. Let T (n)
continuous in t ∈ [0, T ], and kv (n) (·, 0)k1 ≤ c0 . Define c3 = 2c
1 0
be the maximum time such that kv (k) (·, t)k2 ≤ c3 for 0 ≤ t ≤ T (n) , 0 ≤ k ≤ n.
Integrating (4.5) over [0, t], we have for 0 ≤ t ≤ T (n) and j = 0, 1, . . .
¶
Z t
Z tµ Z
(n)
∂s ξ[vj ]2 dx ds ≤
G(kv (n−1) k1 )kv (n) k2j ds
0
+
0
R
Z
t
E(kv
(n−1)
0
k1 )kv (n−1) k2j ds
+
Z
t
Z
t
F (kv (n−1) kj )ds
0
and it follows that
Z t
Z
Z
(n)
(n)
G(kv (n−1) k1 )kv (n) k2j ds
ξ(x, 0)[vj (x, 0)]2 dx+
ξ(x, t)[vj (x, t)]2 dx ≤
0
R
R
+
Z
t
0
E(kv (n−1) k1 )kv (n−1) k2j ds +
hence
Z
(n)
c1 [vj (x, t)]2 dx
≤
R
≤
Z
F (kv (n−1) kj )ds,
0
(n)
R
ξ(x, t)[vj (x, t)]2 dx
Z t
(n)
G(kv (n−1) k1 )kv (n) k2j ds
ξ(x, 0)[vj (x, 0)]2 dx +
0
R
Z t
Z t
F (kv (n−1) kj )ds.
E(kv (n−1) k1 )kv (n−1) k2j ds +
+
Z
0
0
In this way,
Z
Z
c2
G(c3 ) 2
E(c3 ) 2
F (c3 )
(n) 2
(n)
[vj ] dx ≤
[vj (x, 0)]2 dx +
c3 t +
c3 t +
t
c
c
c
c1
1 R
1
1
R
and we obtain for j = 0, 1.
kv (n) k1 ≤
E(c3 ) 2
F (c3 )
c2 2 G(c3 ) 2
c0 +
c3 t +
c3 t +
t.
c1
c1
c1
c1
Claim. T (n) 6→ 0.
Proof. We suppose T (n) → 0. Since kv (n) (·, t)k is continuous in t > 0, there
exists τ ∈ [0, T ] such that kv (k) (·, τ )k1 = c3 for 0 ≤ τ ≤ T (n) , 0 ≤ k ≤ n.
Then
c23 ≤
c2 2 G(c3 ) 2 (n) E(c3 ) 2 (n) F (c3 ) (n)
c +
c T
+
c T
+
T
c1 0
c1 3
c1 3
c1
Divulgaciones Matem´
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144
Octavio Paulo Vera Villagr´an
we do n → +∞ follows
µ
¶2
c2
c2 2
c0 + 1 ≤ c20
2c1
c1
thus
c22 2
c +1≤0
4c21 0
(contradiction)
this way T (n) →
6 0. Choosing T = T (c0 ) sufficiently small, but T not depending on n, one concludes that
kv (n) k1 ≤ c
(4.7)
for 0 ≤ t ≤ T . This shows that T (n) ≥ T .
def
Hence of (4.6) there exists a subsequence v (nj ) = v (n) such that
∗
v (n) ⇀ v
weakly in L∞ ([0, T ]; H 1 (R))
(4.8)
Claim. u = ∧v is the solution we are looking for.
Proof. In the linearized equation (4.4) we have
(n)
∧v 5
(n)
=
∧(I − (I − ∂ 2 ))v3
=
∧v 3 − v3
=
∂ 2 ( ∧v 1 ) − ∂ 2 (v1 )
| {z }
| {z }
(n)
(n)
(n)
∈L2 (R)
(n)
∈H −2 (R)
(n)
since ∧ = (I − ∂ 2 )−1 is bounded in H 1 (R) then ∧v 5 belong to H −2 (R),
so still v (n) is bounded in L∞ ([0, T ]; H 1 (R)) ֒→ L2 ([0, T ]; H 1 (R)) and since
(n)
(n)
∧ : L2 (R) −→ H 2 (R) is a bounded operator, k∧v 1 kH 2 (R) ≤ ckv1 kL2 (R) ≤
(n)
(n)
c, kv1 kH 1 (R) hence ∧v 1 is bounded in L2 ([0, T ]; H 2 (R)) ֒→ L2 ([0, T ]; L2 (R)),
(n)
follows ∂ 2 (∧v 1 ) is bounded in L2 ([0, T ]; H −2 (R)). This way
(n)
∧v 5
is bounded in L2 ([0, T ]; H −2 (R))
(4.9)
(n)
Similarly all other terms are bounded. By equation (4.4), vt is a sum of
terms each of which is the product of a coefficient, bounded uniformly in n
(n)
and a function in L2 ([0, T ]; H −2 (R)) bounded uniformly n such that vt is
Divulgaciones Matem´
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Existence and Uniqueness for a Nonlinear Dispersive Equation
145
c
1/2
1
bounded in L2 ([0, T ]; H −2 (R)) for on the other hand Hloc
(R) ֒→ Hloc (R) ֒→
−2
H (R). By Lions-Aubin’s compactness Theorem there is a subsequence
def
1/2
v (nj ) = v (n) such that v (n) −→ v strongly in L2 ([0, T ]; Hloc (R)). Hence for
def
1/2
a subsequence v (nj ) = v (n) we have v (n) −→ v a. e. in L2 ([0, T ]; Hloc (R)).
(n)
Moreover from (4.8) ∧v 5 ⇀ ∧v 5 weakly in L2 ([0, T ]; H −2 (R)). Similarly
(n)
∧v 4 ⇀ ∧v 4 weakly in L2 ([0, T ]; H −2 (R)). Since k∧v (n) kH 4 (R) ≤ ckv (n) kH 1 (R)
1/2
≤ c, kv (n) kH 1/2 (R) and v1 (n) −→ v1 strongly in L2 ([0, T ]; Hloc (R)), then ∧v (n)
4
(R)), thus ∂(∧v (n) ) −→ ∂(∧v) strongly
−→ ∧v strongly in L2 ([0, T ]; Hloc
(n)
3
2
in L2 ([0, T ]; Hloc
(R)) ֒→ L2 ([0, T ]; Hloc
(R)), then ∧v 1 −→ ∧v 1 strongly
(n)
2
3
2
2
in L ([0, T ]; Hloc (R)) ֒→ L ([0, T ]; Hloc (R)). In this way ∂1 g(∧v 1 ) −→
2
3
2
2
∂1 g(∧v 1 ) strongly in L ([0, T ]; Hloc (R)) ֒→ L ([0, T ]; Hloc (R)). Thus the
(n−1)
(n)
third term on the right hand side of (4.4), ∂g(∧v 1
)∧v 4 ⇀ ∂g(∧v 1 )∧v 4
(n)
weakly in L2 ([0, T ]; L1loc (R)) as ∧v 4 ⇀ ∧v 4 weakly in L2 ([0, T ]; H −2 (R)),
(n−1)
2
and ∂g(∧v 1
) −→ ∂g(∧v 1 ) strongly in L2 ([0, T ]; Hloc
(R)). Similarly all
(n)
other terms in (4.4) converge to their correct limits, implying vt ⇀ vt weakly
in L2 ([0, T ]; L1loc (R)). Passing to limits,
vt
=
δ∧v 5 − ǫ∂1 g(∧v 1 )∧v 4 + ∂0 f (∧v)∧v 3 − 2ǫ∂(∂1 g(∧v 1 ))∧v 3
+ O(∧v 2 , ∧v 1 , . . .) − [δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 ],
then
vt
=
=
∂ 2 (δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 )
− (δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 )
−(I − ∂ 2 )(δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 ).
Thus vt + (I − ∂ 2 )(δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 ) = 0. This way we
have (2.5) for u = ∧v.
We prove that there exists a solution of equation (2.5), u ∈ L∞ ([0, T ]; H N (R)),
with N ≥ 4, where T depends only on ϕ. We already know that there
is a solution (previously) u ∈ L∞ ([0, T ]; H 3 (R)). It suffices to prove that
the approximating sequence v (n) is bounded in L∞ ([0, T ]; H N −2 (R)). 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) depending on the norm of ϕ but independent
n such that kv (n) kα ≤ c for all 0 ≤ t ≤ T (α) . Thus v ∈ L∞ ([0, T (α) ]; H α (R)).
Now denote by 0 ≤ T ∗(α) ≤ +∞ the maximal number such that for all
0 < T ≤ T ∗(α) , u = ∧v ∈ L∞ ([0, T ]; H N (R)) with T (1) ≤ T ∗(α) for all α ≥ 2.
Thus T can be chosen depending only on norm of ϕ. Approximating ϕ by
Divulgaciones Matem´
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146
Octavio Paulo Vera Villagr´an
j−→+∞
{ϕj } ∈ C0∞ (R) such that kϕ − ϕj kH N (R) −→ 0. Let uj be a solution of
(2.5) with uj (x, 0) = ϕj (x). According to the above argument, there exists T
which is independent of n but depending only supj kϕj k such that uj exists on
j−→+∞
[0, T ] and a subsequence uj −→ u in L∞ ([0, T ]; H N (R)). As a consequence
of Theorem 4.1 and Theorem 4.2 and its proof one gets
Corollary 4.3. Let ϕ ∈ H N (R) with N ≥ 3 such that ϕ(γ) −→ ϕ in H N (R).
Let u and u(γ) be the corresponding unique solutions given by Theorems 4.1
and 4.2 in L∞ ([0, T ]; H N (R)) with T depending only on supγ kϕ(γ) kH 3 (R) then
∗
u(γ) ⇀ u
weakly in
L∞ ([0, T ]; H N (R))
and
u(γ) −→ u
strongly in L2 ([0, T ]; H N +1 (R)).
Theorem 4.4. (Persistence) Let i ≥ 1 and L ≥ 3 be non-negative integers,
0 < T < +∞. Assume that u is the solution to (2.5) in L∞ ([0, T ]; H 3 (R))
with initial data ϕ(x) = u(x, 0) ∈ H 3 (R). If ϕ(x) ∈ H L (W0i0 ) then
\
u ∈ L∞ ([0, T ]; H 3 (R) H L (W0i0 ))
(4.10)
where σ is arbitrary, η ∈ Wσ,i−1,0 for i ≥ 1.
Proof. Similar to Theorem 4.2.
Acknowledgments
The author is very grateful to professor Felipe Linares (IMPA) for his valuable
suggestions.
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(1969), 159–172.
[26] Vera, O. Gain of regularity for the nonlinear dispersive equation of
Korteweg-de Vries-Burger type. Revista de Matem´atica Proyecciones.
Vol. 19, Nro. 3, Dic. 2000. pp. 207–226.
Divulgaciones Matem´
aticas Vol. 10 No. 2(2002), pp. 131–148
aticas Vol. 10 No. 2(2002), pp. 131–148
Existence and Uniqueness for a
Nonlinear Dispersive Equation
Existencia y Unicidad para una Ecuaci´on Dispersiva No Lineal
Octavio Paulo Vera Villagr´an (overa@ucsc.cl)
Facultad de Ingenier´ıa
Universidad Cat´olica de la Sant´ısima Concepci´on
Alonso de Rivera 2850, Concepci´on, Chile.
A Lorena Conde Baquedano, con admiraci´
on, mucha admiraci´
on.
Abstract
In this paper we study the existence and uniqueness properties of solutions of some nonlinear dispersive equations of evolution. We consider
the equation
½ ∂u
3
∂
∂
+ ∂x
[f (u)] = ǫ ∂x
[g( ∂u
)] − δ ∂∂xu3
∂t
∂x
(1) =
u(x, 0) = ϕ(x)
with x ∈ R, T an arbitrary positive time and t ∈ [0, T ]. The flux
f = f (u) and the (degenerate) viscosity g = g(λ) are given smooth
functions satisfying certain assumptions. This work presents a result a
priori that permits to obtain gain of regularity for equation (1), motivated by the results obtained by Craig, Kappeler and Strauss [3].
Key words and phrases: Evolution equations, Lions-Aubin Theorem, Weighted Sobolev Space.
Resumen
En este art´ıculo estudiamos las propiedades de existencia y unicidad de las soluciones de algunas ecuaciones de evoluci´
on dispersivas no
lineales. Consideramos la ecuaci´
on
½ ∂u
3
∂
∂
+ ∂x
[f (u)] = ǫ ∂x
[g( ∂u
)] − δ ∂∂xu3
∂t
∂x
(1) =
u(x, 0) = ϕ(x)
Recibido 2000/05/06. Revisado 2001/10/19. Aceptado 2002/03/15.
MSC (2000): Primary 35Q53; Secondary 47J35.
This research was partially supported by DIN 04/2001 Universidad Cat´
olica de la Sant´ısima
Concepci´
on, Concepci´
on, Chile.
132
Octavio Paulo Vera Villagr´an
con x ∈ R, t ∈ [0, T ] y T un tiempo positivo arbitrario. El flujo f = f (u)
y la viscosidad (degenerada) g = g(λ) son funciones suaves dadas que
satisfacen ciertas condiciones. En este trabajo se presenta un resultado a
priori que permite obtener una ganancia de regularidad para la ecuaci´
on
(1), motivado en los resultados obtenidos por Craig, Kappeler y Strauss
[3].
Palabras y frases clave: Ecuaciones de evoluci´
on, teorema de LionsAubin, Espacios pesados de Sobolev.
1
Introduction
In 1976, J. C. Saut and R. Temam [23] remarked that a solution u of an
equation of Korteweg-de Vries type cannot gain or lose regularity: they showed
that if u(x, 0) = ϕ(x) ∈ H s (R) for s ≥ 2, then u(·, t) ∈ H s (R) 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) ∈
T
L2b ≡ H 2 (R) L2 (ebx 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
3
3
formally the semigroup S(t) = e−t∂x in L2b is equivalent to Sb (t) = e−t(∂x −b)
2
in L 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) ∈ L2 such that xα ϕ(x) ∈ L2 ((0, +∞)),
proved that the weak solution of the KdV equation constructed there has lcontinuous 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 Schr¨odinger equations was done by Hayashi et
al. [12, 13] and G. Ponce [22]. While the proof of T. Kato seems to depend on
special a priori estimates, some of its mystery has been resolved by results of
local gain of finite regularity for various other linear and nonlinear dispersive
equations due to P. Constantin and J. C. Saut [10], P. Sjolin [24], 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 Schr¨odinger equations are included.
Divulgaciones Matem´
aticas Vol. 10 No. 2(2002), pp. 131–148
Existence and Uniqueness for a Nonlinear Dispersive Equation
133
Continuing with the idea of W. Craig, T. Kappeler and W. Strauss [9] we
study existence and uniqueness properties of solutions of some nonlinear dispersive equations of evolution. We consider the nonlinear dispersive equation
(
3
∂
∂
∂u
+ ∂x
[f (u)] = ǫ ∂x
[g( ∂u
)] − δ ∂∂xu3
∂t
∂x
(1) =
u(x, 0) = ϕ(x)
with x ∈ R, T an arbitrary positive time and t ∈ [0, T ]. The flux f = f (u)
and the (degenerate) viscosity g = g(λ) are given smooth functions satisfying
certain assumptions to be listed shortly.
In section 3 we prove an important a priori estimate.
In section 4 we prove basic local-in-time existence and uniqueness results
for (1). Specifically, we show that for initial ϕ(x) ∈ H N (R), for N ≥ 3, there
exists a unique u ∈ L∞ ([0, T ]; H N (R)) where the time of existence depends
of the norm of ϕ(x) ∈ H 3 (R).
In section 5 we develop a series of estimates for solutions of equation (1)
in weighted Sobolev norms. We show that a solution u of (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.
2
Preliminaries
We consider the nonlinear dispersive equation
· µ ¶¸
∂3u
∂u
∂
∂
∂u
g
−δ 3
+
[f (u)] = ǫ
∂t
∂x
∂x
∂x
∂x
(2.1)
with x ∈ R, t ∈ [0, T ] and T is an arbitrary positive time. The flux f = f (u)
and the (degenerate) viscosity g = g(λ) are given smooth functions satisfying
certain assumptions. ǫ, δ > 0.
Notation 1. We write ∂ =
∂
∂j u
∂xj ; ∂j = ∂uj .
∂
∂x ,
∂t u =
∂u
∂t
= ut and we abbreviate uj = ∂ j u =
Example. If ∂u/∂x = u1 then
· µ ¶¸
∂
∂
∂
∂
∂
∂u
[g(u1 )]
[g(u1 )] u2 = (∂1 g)u2
[g(u1 )] =
[u1 ] =
g
=
∂x
∂x
∂x
∂u1
∂x
∂u1
The assumptions on f are the following:
Divulgaciones Matem´
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134
Octavio Paulo Vera Villagr´an
A.1 f : R2 × [0, T ] 7→ R is C ∞ in all its variables.
A.2 All the derivatives of f = f (u, x, t) are bounded for x ∈ R for t ∈ [0, T ]
and u ∈ R in a bounded set.
A.3 xN ∂xj f (0, x, t) is bounded for all N ≥ 0, j ≥ 0, and x ∈ R, t ∈ (0, T ].
Indeed, ∀N ≥ 0, ∀j ≥ 0, x ∈ R, t ∈ (0, T ], there exists c > 0 such that
|xN ∂xj f (0, x, t)| ≤ c.
The assumptions on g are the following:
B.1 g : R2 × [0, T ] 7→ R is C ∞ in all its variables.
B.2 All the derivatives of g(y, x, t) are bounded for x ∈ R, t ∈ [0, T ] and y in
a bounded set.
B.3 xN ∂xj g(0, x, t) is bounded for all N ≥ 0, j ≥ 0 and x ∈ R, t ∈ (0, T ].
B.4 There exists c > 0 such that ∂1 g(u1 , x, t) ≥ c > 0, for all u1 ∈ R, x ∈ R
and t ∈ [0, T ].
Lemma 1. These assumptions imply that f has the form f = u0 f0 + h ≡
uf0 + h where f0 = f0 (u0 , x, t) ≡ f0 (u, x, t) and h = h(x, t). f0 and h are C ∞
and each of their derivatives is bounded for u bounded, x ∈ R and t ∈ [0, T ].
Proof. Indeed, we define
f0 =
½
f (u0 ,x,t)−f (0,x,t)
u0
∂0 f (0, x, t)
for u0 6= 0
for u0 = 0
and h(x, t) = f (0, x, t).
Remark 1. The same for g.
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 Wσik if it is
a positive C ∞ function on R × [0, T ], ξx > 0 and there exists a constant
cj , 0 ≤ j ≤ 5 such that
0 < c1 ≤ t−k e−σx ξ(x, t) ≤ c2
0 < c3 ≤ t−k x−i ξ(x, t) ≤ c4
¡
t | ξt | + | ∂ j ξ
in R × [0, T ], for all j ∈ N.
for x < −1, 0 < t < T.
for x > 1, 0 < t < T.
¢
| /ξ ≤ c5
Remark 2. We shall always take σ ≥ 0, i ≥ 1 and k ≥ 0.
Divulgaciones Matem´
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(2.2)
(2.3)
(2.4)
Existence and Uniqueness for a Nonlinear Dispersive Equation
135
Example 1. Let
ξ(x) =
½
1 + e−1/x
1
for x > 0
for x ≤ 0
then ξ ∈ W0i0 .
Definition 2.3. Fixed ξ ∈ Wσik define the space (for s a positive integer)
©
H s (Wσik ) = v : R → R; such that the distributional derivatives
s
X
∂j v
for 0 ≤ j ≤ s satisfy kvk2 =
j
∂x
j=0
Z
+∞
|∂ j v(x)|2 ξ(x, t)dx < +∞
−∞
ª
Remark 3. H s (Wσik ) depend t ( because ξ = ξ(x, t)).
Lemma 2. For ξ ∈ Wσi0 and σ ≥ 0, i ≥ 0 there exists a constant c such that,
for u ∈ H 1 (Wσi0 )
Z +∞
¡
¢
2
sup | ξu |≤ c
| u |2 + | ∂u |2 ξdx.
x∈R
−∞
Proof. See Lemma 7.3 in [9].
Definition 2.4. Fixed ξ ∈ Wσik define the space
L2 ([0, T ]; H s (Wσik )) = {v = v(x, t), v(·, t) ∈ H s (Wσik ) such that
Z T
2
||| v ||| =
kv(·, t)k2 dt < +∞}
0
L∞ ([0, T ]; H s (Wσik )) = {v = v(x, t), v(·, t) ∈ H s (Wσik ) such that
||| v |||∞ = ess sup kv(·, t)k < +∞}
t∈[0,T ]
Remark 4. The usual Sobolev space is H s (R) = H s (W000 ) without a weight.
Remark 5. We shall derive the a priori estimates assuming that the solution
is C ∞ , bounded as x → −∞, and rapidly decreasing as x → +∞, together
with all of its derivatives.
According to notation 1, for equation (1) we obtain
ut + δu3 − ǫ(∂1 g)u2 + (∂0 f )u1 = 0.
(2.5)
The equation is considered for −∞ < x < +∞, t ∈ [0, T ] and T is an arbitrary
positive time.
Divulgaciones Matem´
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136
3
Octavio Paulo Vera Villagr´an
An important a priori estimate
In this section we show a fundamental a priori estimate to demonstrate basic local-in-time existence theorem. We need to construct a mapping T :
L∞ ([0, T ]; H s (R)) → L∞ ([0, T ]; H s (R)) with the following property: Given
u(n) = T (u(n−1) ) and ku(n−1) ks ≤ c0 then ku(n) ks ≤ c0 , where s and c0 > 0
are constants. This property tells us, in fact, that T : Bc0 (0) → Bc0 (0) where
Bc0 (0) = {v(x, t); kv(x, t)ks ≤ c0 } is a ball in space L∞ ([0, T ]; H s (R)). To
guarantee this property, we will appeal to an a priori estimate which is the
main object of this section.
Differentiating the equation (2.5) two times leads to
∂t u2 + δu5 − ǫ(∂1 g)u4 + (∂0 f )u3 − 2ǫ∂(∂1 g)u3
+ 2∂(∂0 f )u2 − ǫ∂ 2 (∂1 g)u2 + ∂ 2 (∂0 f )u1 = 0 (3.1)
Let u = ∧v where ∧ = (I − ∂ 2 )−1 . Then ∂t u2 = −vt + ut . Replacing in (3.1)
we have
− vt + δ∧v 5 − ǫ(∂1 g)∧v 4 + (∂0 f )∧v 3 − 2ǫ∂(∂1 g)∧v 3 + 2∂(∂0 f )∧v 2
− ǫ∂ 2 (∂1 g)∧v 2 + ∂ 2 (∂0 f )∧v 1 − [δ∧v 3 − ǫ(∂1 g)∧v 2 + (∂0 f )∧v 1 ] = 0 (3.2)
where g = g(∧v 1 ) and f = f (∧v).
The equation (3.2) is linearized by substituting a new variable w in each
coefficient;
−vt + δ∧v 5 − ǫ∂1 g(∧w1 )∧v 4 + ∂0 f (∧w)∧v 3 − 2ǫ∂(∂1 g(∧w1 ))∧v 3
+2∂(∂0 f (∧w))∧v 2 − ǫ∂ 2 (∂1 g(∧w1 ))∧v 2 + ∂ 2 (∂0 f (∧w))∧v 1
−[δ∧v 3 − ǫ∂1 g(∧w1 )∧v 2 − ∂0 f (∧w)∧v 1 ] = 0.
(3.3)
Lemma 3.1. Let v, w ∈ C k ([0, +∞); H N (R)) for all k, N which satisfy (3.3).
Let
ξ ≥ c1 > 0. For each integer α there exist positive nondecreasing functions
E,F and G such that for all t ≥ 0
Z
∂t ξvα2 dx ≤ G(kwkλ )kvk2α + E(kwkλ )kwk2α + F (kwkα ) (3.4)
R
where k · kα is the norm in H α (R) and λ = max {1, α}.
Proof. Differentiating α-times the equation (3.3) for some α ≥ 0
−∂t vα + δ∧v α+5 − ǫ(∂1 g)∧v α+4 + ((∂0 f ) − (α + 2)ǫ∂(∂1 g))∧v α+3
+
α+2
X
h(j) ∧v j + q(∧w)∧wα+2 + p(∧wα+1 , . . .) = 0
j=2
Divulgaciones Matem´
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(3.5)
Existence and Uniqueness for a Nonlinear Dispersive Equation
137
where h(j) is a smooth function depending on ∧wi+3 , ∧wi+2 , . . . with i =
2 + α − j.
We multiply equation (3.5) by 2ξvα , integrate over x ∈ R
Z
Z
Z
−2 ξvα ∂t vα dx + 2δ ξvα ∧v α+5 dx − 2ǫ ξ(∂1 g)vα ∧v α+4 dx
R
R
Z
ZR
+2 ξ(∂0 f )vα ∧v α+3 dx − 2(α + 2)ǫ ξ∂(∂1 g)vα ∧v α+3 dx
R
α+2
XZ
+2
j=2
Z
+2
R
ξh(j) vα ∧v j dx + 2
Z
ξq(∧w)vα ∧wα+2 dx
R
R
ξvα p(∧wα+1 , . . .)dx = 0.
(3.6)
R
Each term in (3.6) is treated separately. The first two terms yield
Z
Z
Z
ξt vα2 dx
−2 ξvα ∂t vα dx = −∂t ξvα2 dx +
R
R
R
and
Z
Z
ξvα ∧v α+5 dx = 2δ ξ∧(I − ∂ 2 )vα ∧v α+5 dx
R
Z
Z
= 2δ ξ∧v α ∧v α+5 dx − 2δ ξ∧v α+2 ∧v α+5 dx
ZR
ZR
5
2
= −δ ∂ ξ(∧v α ) dx + 5δ ∂ 3 ξ(∧v α+1 )2 dx
Z
ZR
ZR
3
2
2
− 5δ ∂ξ(∧v α+2 ) dx + δ ∂ ξ(∧v α+2 ) dx − 3δ ∂ξ(∧v α+3 )2 dx.
2δ
R
R
R
R
The other terms are treated similarly, integrating by parts once again. Replacing over (3.6) we have
Z
Z
Z
Z
5
2
2
2
ξt vα dx − δ ∂ ξ(∧v α ) dx + 5δ ∂ 3 ξ(∧v α+1 )2 dx
∂t ξvα dx =
R
R
ZR
Z R
Z
3
2
2
− 5δ ∂ξ(∧v α+2 ) dx + δ ∂ ξ(∧v α+2 ) dx − 3δ ∂ξ(∧v α+3 )2 dx
R
R
Z
Z
Z R
4
2
3
2
∂ (ξ∂0 f )(∧v α ) dx − ǫ ∂ (ξ∂1 g)(∧v α ) dx + 4ǫ ∂ 2 (ξ∂1 g)(∧v α+1 )2 dx
−
R
R
R
Z
Z
− 2ǫ ξ(∂1 g)(∧v α+2 )2 dx + ǫ ∂ 2 (ξ∂1 g)(∧v α+2 )2 dx
Z
ZR
ZR
2
∂(ξ∂0 f )(∧v α+2 )2 dx
− 2ǫ ξ(∂1 g)(∧v α+3 ) dx + 3 ∂(ξ∂0 f )(∧v α+1 )2 dx +
R
R
R
Divulgaciones Matem´
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138
Octavio Paulo Vera Villagr´an
Z
∂ 3 (ξ∂(∂1 g))(∧v α )2 dx − 3(α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+1 )2 dx
R
Z
ZR
2
(α+2)
2
)(∧v α )2 dx
− (α + 2)ǫ ∂(ξ∂(∂1 g))(∧vα+2 ) dx + ∂ (ξh
+ (α + 2)ǫ
Z
R
R
−2
+2
Z
Z
(α+2)
ξh
2
(∧v α+1 ) dx − 2
Z
ξh
(α+2)
2
(∧v α+2 ) dx + 2
R
R
ξq(∧w)vα ∧wα+2 dx + 2
α+1
XZ
j=2
Z
ξh(j) vα ∧v j dx
R
ξvα p(∧wα+1 , . . .)dx,
R
R
then we have
Z
Z
ξt vα2 dx
ξvα2 dx = − (3δ∂ξ + 2ǫξ(∂1 g))(∧v α+3 )2 dx +
R
R
Z R
Z
Z
2
3
2
+ δ ∂ ξ(∧v α+2 ) dx − 5δ ∂ξ(∧v α+2 ) dx + ǫ ∂ 2 (ξ(∂1 g))(∧v α+2 )2 dx
R
R
Z
ZR
∂(ξ(∂0 f ))(∧v α+2 )2 dx
− 2ǫ ξ(∂1 g)(∧v α+2 )2 dx +
R
R
Z
Z
− (α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+2 )2 dx − 2 ξh(α+2) (∧v α+2 )2 dx
R
R
Z
Z
2
3
2
+ 5δ ∂ ξ(∧v α+1 ) dx + 4ǫ ∂ (ξ∂1 g)(∧v α+1 )2 dx
R
Z
Z R
2
+ 3 ∂(ξ∂0 f )(∧v α+1 ) dx − 3(α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+1 )2 dx
R
Z
Z
ZR
5
(α+2)
2
(∧v α+1 ) dx − δ ∂ ξ(∧v α )2 dx − ǫ ∂ 4 (ξ∂1 g)(∧v α )2 dx
− 2 ξh
R
R
Z
Z R
3
3
2
∂ (ξ∂0 f )(∧v α ) dx + (α + 2)ǫ ∂ (ξ∂(∂1 g))(∧v α )2 dx
−
∂t
Z
+
Z
R
R
∂ 2 (ξh(α+2) )(∧v α )2 dx + 2
R
+2
α+1
XZ
j=2
Z
R
ξq(∧w)vα ∧wα+2 dx + 2
Z
ξh(j) vα ∧v j dx
R
ξvα p(∧wα+1 , . . .)dx.
R
The first term in the righthand side is nonpositive, hence
Divulgaciones Matem´
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Existence and Uniqueness for a Nonlinear Dispersive Equation
139
Z
Z
Z
ξt vα2 dx + δ ∂ 3 ξ(∧v α+2 )2 dx − 5δ ∂ξ(∧v α+2 )2 dx
ξvα2 dx ≤
R
R
R
R
Z
Z
+ ǫ ∂ 2 (ξ∂1 g)(∧v α+2 )2 dx − 2ǫ ξ(∂1 g)(∧v α+2 )2 dx
R
Z
Z R
2
+ ∂(ξ∂0 f )(∧v α+2 ) dx − (α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+2 )2 dx
R
R
Z
Z
3
(α+2)
2
(∧v α+2 ) dx + 5δ ∂ ξ(∧v α+1 )2 dx
− 2 ξh
R
ZR
Z
+ 4ǫ ∂ 2 (ξ∂1 g)(∧v α+1 )2 dx + 3 ∂(ξ∂0 f )(∧v α+1 )2 dx
R
R
Z
Z
− 3(α + 2)ǫ ∂(ξ∂(∂1 g))(∧v α+1 )2 dx − 2 ξh(α+2) (∧v α+1 )2 dx
R
R
Z
Z
4
2
5
2
− δ ∂ ξ(∧v α ) dx − ǫ ∂ (ξ∂1 g)(∧v α ) dx
R
Z
Z R
3
2
− ∂ (ξ∂0 f )(∧v α ) dx + (α + 2)ǫ ∂ 3 (ξ∂(∂1 g))(∧v α )2 dx
∂t
Z
+
Z
R
R
∂ 2 (ξh(α+2) )(∧v α )2 dx + 2
R
+2
α+1
XZ
j=2
Z
ξq(∧w)vα ∧wα+2 dx + 2
Z
ξh(j) vα ∧v j dx
R
ξvα p(∧wα+1 , . . .)dx.
R
R
In the last term we have
¯ Z
¯
¯Z
¯
¯Z
¯
¯
¯
¯
¯
¯
¯
¯2 ξ(q∧wα+2 + p)vα dx¯ ≤ 2 ¯ ξq∧wα+2 vα dx¯ + 2 ¯ ξpvα dx¯ ,
¯
¯
¯
¯
¯
¯
R
R
R
but ∧wα+2 = ∧wα − wα then
¯
¯Z
¯
¯Z
¯
¯ Z
¯
¯
¯
¯
¯
¯
¯2 ξ(q∧wα+2 + p)vα dx¯ ≤ 2 ¯ ξqvα (∧wα − wα )dx¯ + 2 ¯ ξpvα dx¯
¯
¯
¯
¯
¯
¯
R
R
R
¯
¯
¯Z
¯Z
µZ
¶1/2 µZ
¶1/2
¯
¯
¯
¯
ξp2 dx
ξvα2 dx
≤ 2 ¯¯ ξq∧wα vα dx¯¯ + 2 ¯¯ ξqwα vα dx¯¯ + 2
R
¯ZR
¯
¯ZR
¯ Z R
Z
¯
¯
¯
¯
2
2
≤ 2 ¯¯ ξq∧wα vα dx¯¯ + 2 ¯¯ ξqwα vα dx¯¯ +
ξvα dx
ξp dx +
R
R
R
R
and since p = p(∧wα+1 , . . .) then
¯ Z
¯
¯
¯
¯2 ξ (q∧wα+2 + p) vα dx¯ ≤ r(kwkα )[kwk2α + kvk2α ] + kvkα + s(kwkα )
¯
¯
R
Divulgaciones Matem´
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140
Octavio Paulo Vera Villagr´an
in the same form, using that ∧v n = ∧v n−2 − vn−2 . By standard estimates,
the Lemma now follows.
We define a sequence of approximations to equation (3.3) as
(n)
−vt
(n)
(n)
(n)
(n)
+ δ∧v 5 − ǫ(∂1 g)∧v 4 + (∂0 f )∧v 3 − 2ǫ∂(∂1 g)∧v 3
(n)
(n−1)
−δ∧v 3 + O(∧v 2
(n−1)
, ∧v 1
, . . .) = 0
(3.7)
(n−1)
where g = g(∧v 1
), f = f (∧v (n−1) ) and where the initial condition is
(n)
given by v (x, 0) = ϕ(x) − ∂ 2 ϕ(x). The first approximation is given by
v (0) (x, 0) = ϕ(x)−∂ 2 ϕ(x). Equation (3.7) is a linear equation at each iteration
which can be solved in any interval of time in which the coefficients are defined.
This equation has the form
∂t v = δ∧v 5 − ǫ∧v 4 + b(1) ∧v 3 + b(0)
(3.8)
T
∞
N
Lemma 3.2. Given initial data in ϕ ∈ H (R) = N ≥0 H (R) there exists a
unique solution of (3.8). The solution is defined in any time interval in which
the coefficients are defined.
Proof. See [26].
4
Uniqueness and existence theorem
In this section, we study uniqueness and local existence of strong solutions for
problem (2.5). Specifically, we show that for initial ϕ(x) ∈ H N (R), for N ≥ 3,
there exists a unique u ∈ L∞ ([0, T ]; H N (R)) where the time of existence
depends of the norm of ϕ(x) ∈ H 3 (R). First we address the question of
uniqueness.
Theorem 4.1. (Uniqueness). Let ϕ ∈ H 3 (R) and 0 < T < +∞. Assume f
satisfies A.1-A.3. and g satisfies B.1-B.4, then there is at most one solution
u ∈ L∞ ([0, T ]; H 3 (R)) of (2.5) with initial data u(x, 0) = ϕ(x).
Proof. Assume u, v ∈ L∞ ([0, T ]; H 3 (R)) are two solutions of (2.5) with
ut , vt ∈ L∞ ([0, T ]; L2 (R)) and with the same initial data. Then
(u − v)t + δ(u − v)3 − ǫ[g(u1 ) − g(v1 )]1 + [f (u) − f (v)]1 = 0
(4.1)
with (u − v)(x, 0) = 0. Using the Mean Value Theorem there are smooth
functions d(1) and d(2) depending smoothly on u1 , x, t; v1 , x, t and u, x, t; v, x, t
respectively such that (4.1) has the form
(u − v)t + δ(u − v)3 − ǫ[d(1) ]1 (u − v)1 − ǫd(1) (u − v)2
+ [d(2) ]1 (u − v) + d(2) (u − v)1 = 0
Divulgaciones Matem´
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(4.2)
Existence and Uniqueness for a Nonlinear Dispersive Equation
We multiply equation (4.2) by 2ξ(u − v), integrate over x ∈ R
Z
Z
2 ξ(u − v)(u − v)t dx + 2δ ξ(u − v)(u − v)3 dx
R
R
Z
Z
(1)
− 2ǫ ξ[d ]1 (u − v)(u − v)1 dx − 2ǫ ξd(1) (u − v)(u − v)2 dx
R
Z
Z R
(2)
(2)
2
+ 2 ξ[d ]1 (u − v) dx + 2 ξd (u − v)(u − v)1 dx = 0.
141
(4.3)
R
R
Each term is treated separately. In the first term we have
Z
Z
Z
ξt (u − v)2 dx.
2 ξ(u − v)(u − v)t dx = ∂t ξ(u − v)2 dx −
R
R
R
The second term, integrating by parts, yields
Z
2δ ξ(u − v)(u − v)3 dx
R
Z
Z
= −2δ ∂ξ(u − v)(u − v)2 dx − 2δ ξ(u − v)1 (u − v)2 dx
Z R
Z
ZR
2
= 2δ ∂ ξ(u − v)(u − v)1 dx + 2δ ∂ξ(u − v)21 dx + δ ∂ξ(u − v)21 dx
R
R
R
Z
Z
2
3
2
(4.4)
= − δ ∂ ξ(u − v) dx + 3δ ∂ξ(u − v)1 dx.
R
R
Replacing over (4.3) we have
Z
Z
Z
∂t ξ(u − v)2 dx − ξt (u − v)2 dx − δ ∂ 3 ξ(u − v)2 dx
R
R
R
Z
Z
(1)
2
+ 3δ ∂ξ(u − v)1 dx + ǫ ∂(ξ[d ]1 )(u − v)2 dx
R
Z
Z R
2
(1)
2
− ǫ ∂ (ξd )(u − v) dx + 2ǫ ξd(1) (u − v)21 dx
R
Z
ZR
(2)
2
∂(ξd(2) )(u − v)2 dx = 0.
+ 2 ξ[d ]1 (u − v) dx −
R
R
Then
Z
Z
∂t ξ(u − v)2 dx + (3δ∂ξ + 2ǫξd(1) )(u − v)21 dx
R
Z
Z
ZR
3
2
2
ξt (u − v) dx + δ ∂ ξ(u − v) dx − ǫ ∂(ξ[d(1) ]1 )(u − v)2 dx
=
R
R
R
Z
Z
Z
(2)
2
(1)
2
∂(ξd(2) )(u − v)2 dx.
+ ǫ ∂ (ξd )(u − v) dx − 2 ξ[d ]1 (u − v)2 dx +
R
R
R
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142
Octavio Paulo Vera Villagr´an
Using B-4, the assumptions on f and g and for a suitably chosen constant c,
we have
Z
Z
Z
∂t ξ(u − v)2 dx + (3δ∂ξ + 2ǫξd(1) )(u − v)21 dx ≤ c ξ(u − v)2 dx.
R
R
R
By Gronwall’s inequality and the fact that (u − v) vanishes at t = 0 it follows
that u = v. This proves uniqueness.
We construct the mapping T : L∞ ([0, T ]; H s (R)) → L∞ ([0, T ]; H s (R)) by
defining that
u(0)
u(n)
=
=
ϕ(x)
T (u(n−1) )
n≥1
(n−1)
where u
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 3.2., u(n) exists and
is unique in C((0, +∞); H N (R)). A choice of c0 and the use of the a priori
estimate in §3 show that T : Bc0 (0) → Bc0 (0) with Bc0 (0) a bounded ball in
L∞ ([0, T ]; H s (R)).
Theorem 4.2. (Local existence). Assume f satisfies A.1 - A.4, and g satisfies
B.1 - B.4. Let N be an integers ≥ 3. If ϕ ∈ H N (R), then there is T > 0 and
u such that u is a strong solution of (2.5). u ∈ L∞ ([0, T ]; H N (R)) with initial
data u(x, 0) = ϕ(x).
T
Proof. We prove that for ϕ ∈ H ∞ (R) = k≥0 H k (R) there exists a solution
u ∈ L∞ ([0, T ]; H N (R)) with initial data u(x, 0) = ϕ(x) and which a time of
existence T > 0 which only depends ϕ.
We define a sequence of approximations to equation (3.2) as
− vt (n) + δ∧v 5 (n) − ǫ(∂1 g)∧v 4 (n) + (∂0 f )∧v 3 (n) − 2ǫ∂(∂1 g)∧v 3 (n)
+ 2∂(∂0 f )∧v 2 (n) − ǫ∂ 2 (∂1 g)∧v 2 (n) + ∂ 2 (∂0 f )∧v 1 (n)
− [δ∧v 3 −
(n)
ǫ(∂1 g)∧v 2
+
(n)
(∂0 f )∧v 1 ]
(4.5)
=0
(n−1)
where g = g(∧v 1
) and f = f (∧v (n−1) ) and whith initial data v (n) (x, 0) =
2
ϕ(x) − ∂ ϕ(x).
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.
By Lemma 3.1. it follows that
Z
∂t ξ[vα(n) ]2 dx ≤ G(kv (n−1) kλ )kv (n) k2α + E(kv (n−1) kλ )kv (n−1) k2α
R
+ F (kv (n−1) kα ).
Divulgaciones Matem´
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(4.6)
Existence and Uniqueness for a Nonlinear Dispersive Equation
143
Choose α = 1. Let c0 ≥ kϕ − ∂ 2 ϕk1 ≥ kϕk3 . For each iterate n, kv (n) (·, t)k is
c2 2
c +1. Let T (n)
continuous in t ∈ [0, T ], and kv (n) (·, 0)k1 ≤ c0 . Define c3 = 2c
1 0
be the maximum time such that kv (k) (·, t)k2 ≤ c3 for 0 ≤ t ≤ T (n) , 0 ≤ k ≤ n.
Integrating (4.5) over [0, t], we have for 0 ≤ t ≤ T (n) and j = 0, 1, . . .
¶
Z t
Z tµ Z
(n)
∂s ξ[vj ]2 dx ds ≤
G(kv (n−1) k1 )kv (n) k2j ds
0
+
0
R
Z
t
E(kv
(n−1)
0
k1 )kv (n−1) k2j ds
+
Z
t
Z
t
F (kv (n−1) kj )ds
0
and it follows that
Z t
Z
Z
(n)
(n)
G(kv (n−1) k1 )kv (n) k2j ds
ξ(x, 0)[vj (x, 0)]2 dx+
ξ(x, t)[vj (x, t)]2 dx ≤
0
R
R
+
Z
t
0
E(kv (n−1) k1 )kv (n−1) k2j ds +
hence
Z
(n)
c1 [vj (x, t)]2 dx
≤
R
≤
Z
F (kv (n−1) kj )ds,
0
(n)
R
ξ(x, t)[vj (x, t)]2 dx
Z t
(n)
G(kv (n−1) k1 )kv (n) k2j ds
ξ(x, 0)[vj (x, 0)]2 dx +
0
R
Z t
Z t
F (kv (n−1) kj )ds.
E(kv (n−1) k1 )kv (n−1) k2j ds +
+
Z
0
0
In this way,
Z
Z
c2
G(c3 ) 2
E(c3 ) 2
F (c3 )
(n) 2
(n)
[vj ] dx ≤
[vj (x, 0)]2 dx +
c3 t +
c3 t +
t
c
c
c
c1
1 R
1
1
R
and we obtain for j = 0, 1.
kv (n) k1 ≤
E(c3 ) 2
F (c3 )
c2 2 G(c3 ) 2
c0 +
c3 t +
c3 t +
t.
c1
c1
c1
c1
Claim. T (n) 6→ 0.
Proof. We suppose T (n) → 0. Since kv (n) (·, t)k is continuous in t > 0, there
exists τ ∈ [0, T ] such that kv (k) (·, τ )k1 = c3 for 0 ≤ τ ≤ T (n) , 0 ≤ k ≤ n.
Then
c23 ≤
c2 2 G(c3 ) 2 (n) E(c3 ) 2 (n) F (c3 ) (n)
c +
c T
+
c T
+
T
c1 0
c1 3
c1 3
c1
Divulgaciones Matem´
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144
Octavio Paulo Vera Villagr´an
we do n → +∞ follows
µ
¶2
c2
c2 2
c0 + 1 ≤ c20
2c1
c1
thus
c22 2
c +1≤0
4c21 0
(contradiction)
this way T (n) →
6 0. Choosing T = T (c0 ) sufficiently small, but T not depending on n, one concludes that
kv (n) k1 ≤ c
(4.7)
for 0 ≤ t ≤ T . This shows that T (n) ≥ T .
def
Hence of (4.6) there exists a subsequence v (nj ) = v (n) such that
∗
v (n) ⇀ v
weakly in L∞ ([0, T ]; H 1 (R))
(4.8)
Claim. u = ∧v is the solution we are looking for.
Proof. In the linearized equation (4.4) we have
(n)
∧v 5
(n)
=
∧(I − (I − ∂ 2 ))v3
=
∧v 3 − v3
=
∂ 2 ( ∧v 1 ) − ∂ 2 (v1 )
| {z }
| {z }
(n)
(n)
(n)
∈L2 (R)
(n)
∈H −2 (R)
(n)
since ∧ = (I − ∂ 2 )−1 is bounded in H 1 (R) then ∧v 5 belong to H −2 (R),
so still v (n) is bounded in L∞ ([0, T ]; H 1 (R)) ֒→ L2 ([0, T ]; H 1 (R)) and since
(n)
(n)
∧ : L2 (R) −→ H 2 (R) is a bounded operator, k∧v 1 kH 2 (R) ≤ ckv1 kL2 (R) ≤
(n)
(n)
c, kv1 kH 1 (R) hence ∧v 1 is bounded in L2 ([0, T ]; H 2 (R)) ֒→ L2 ([0, T ]; L2 (R)),
(n)
follows ∂ 2 (∧v 1 ) is bounded in L2 ([0, T ]; H −2 (R)). This way
(n)
∧v 5
is bounded in L2 ([0, T ]; H −2 (R))
(4.9)
(n)
Similarly all other terms are bounded. By equation (4.4), vt is a sum of
terms each of which is the product of a coefficient, bounded uniformly in n
(n)
and a function in L2 ([0, T ]; H −2 (R)) bounded uniformly n such that vt is
Divulgaciones Matem´
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Existence and Uniqueness for a Nonlinear Dispersive Equation
145
c
1/2
1
bounded in L2 ([0, T ]; H −2 (R)) for on the other hand Hloc
(R) ֒→ Hloc (R) ֒→
−2
H (R). By Lions-Aubin’s compactness Theorem there is a subsequence
def
1/2
v (nj ) = v (n) such that v (n) −→ v strongly in L2 ([0, T ]; Hloc (R)). Hence for
def
1/2
a subsequence v (nj ) = v (n) we have v (n) −→ v a. e. in L2 ([0, T ]; Hloc (R)).
(n)
Moreover from (4.8) ∧v 5 ⇀ ∧v 5 weakly in L2 ([0, T ]; H −2 (R)). Similarly
(n)
∧v 4 ⇀ ∧v 4 weakly in L2 ([0, T ]; H −2 (R)). Since k∧v (n) kH 4 (R) ≤ ckv (n) kH 1 (R)
1/2
≤ c, kv (n) kH 1/2 (R) and v1 (n) −→ v1 strongly in L2 ([0, T ]; Hloc (R)), then ∧v (n)
4
(R)), thus ∂(∧v (n) ) −→ ∂(∧v) strongly
−→ ∧v strongly in L2 ([0, T ]; Hloc
(n)
3
2
in L2 ([0, T ]; Hloc
(R)) ֒→ L2 ([0, T ]; Hloc
(R)), then ∧v 1 −→ ∧v 1 strongly
(n)
2
3
2
2
in L ([0, T ]; Hloc (R)) ֒→ L ([0, T ]; Hloc (R)). In this way ∂1 g(∧v 1 ) −→
2
3
2
2
∂1 g(∧v 1 ) strongly in L ([0, T ]; Hloc (R)) ֒→ L ([0, T ]; Hloc (R)). Thus the
(n−1)
(n)
third term on the right hand side of (4.4), ∂g(∧v 1
)∧v 4 ⇀ ∂g(∧v 1 )∧v 4
(n)
weakly in L2 ([0, T ]; L1loc (R)) as ∧v 4 ⇀ ∧v 4 weakly in L2 ([0, T ]; H −2 (R)),
(n−1)
2
and ∂g(∧v 1
) −→ ∂g(∧v 1 ) strongly in L2 ([0, T ]; Hloc
(R)). Similarly all
(n)
other terms in (4.4) converge to their correct limits, implying vt ⇀ vt weakly
in L2 ([0, T ]; L1loc (R)). Passing to limits,
vt
=
δ∧v 5 − ǫ∂1 g(∧v 1 )∧v 4 + ∂0 f (∧v)∧v 3 − 2ǫ∂(∂1 g(∧v 1 ))∧v 3
+ O(∧v 2 , ∧v 1 , . . .) − [δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 ],
then
vt
=
=
∂ 2 (δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 )
− (δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 )
−(I − ∂ 2 )(δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 ).
Thus vt + (I − ∂ 2 )(δ∧v 3 − ǫ∂1 g(∧v 1 )∧v 2 + ∂0 f (∧v)∧v 1 ) = 0. This way we
have (2.5) for u = ∧v.
We prove that there exists a solution of equation (2.5), u ∈ L∞ ([0, T ]; H N (R)),
with N ≥ 4, where T depends only on ϕ. We already know that there
is a solution (previously) u ∈ L∞ ([0, T ]; H 3 (R)). It suffices to prove that
the approximating sequence v (n) is bounded in L∞ ([0, T ]; H N −2 (R)). 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) depending on the norm of ϕ but independent
n such that kv (n) kα ≤ c for all 0 ≤ t ≤ T (α) . Thus v ∈ L∞ ([0, T (α) ]; H α (R)).
Now denote by 0 ≤ T ∗(α) ≤ +∞ the maximal number such that for all
0 < T ≤ T ∗(α) , u = ∧v ∈ L∞ ([0, T ]; H N (R)) with T (1) ≤ T ∗(α) for all α ≥ 2.
Thus T can be chosen depending only on norm of ϕ. Approximating ϕ by
Divulgaciones Matem´
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146
Octavio Paulo Vera Villagr´an
j−→+∞
{ϕj } ∈ C0∞ (R) such that kϕ − ϕj kH N (R) −→ 0. Let uj be a solution of
(2.5) with uj (x, 0) = ϕj (x). According to the above argument, there exists T
which is independent of n but depending only supj kϕj k such that uj exists on
j−→+∞
[0, T ] and a subsequence uj −→ u in L∞ ([0, T ]; H N (R)). As a consequence
of Theorem 4.1 and Theorem 4.2 and its proof one gets
Corollary 4.3. Let ϕ ∈ H N (R) with N ≥ 3 such that ϕ(γ) −→ ϕ in H N (R).
Let u and u(γ) be the corresponding unique solutions given by Theorems 4.1
and 4.2 in L∞ ([0, T ]; H N (R)) with T depending only on supγ kϕ(γ) kH 3 (R) then
∗
u(γ) ⇀ u
weakly in
L∞ ([0, T ]; H N (R))
and
u(γ) −→ u
strongly in L2 ([0, T ]; H N +1 (R)).
Theorem 4.4. (Persistence) Let i ≥ 1 and L ≥ 3 be non-negative integers,
0 < T < +∞. Assume that u is the solution to (2.5) in L∞ ([0, T ]; H 3 (R))
with initial data ϕ(x) = u(x, 0) ∈ H 3 (R). If ϕ(x) ∈ H L (W0i0 ) then
\
u ∈ L∞ ([0, T ]; H 3 (R) H L (W0i0 ))
(4.10)
where σ is arbitrary, η ∈ Wσ,i−1,0 for i ≥ 1.
Proof. Similar to Theorem 4.2.
Acknowledgments
The author is very grateful to professor Felipe Linares (IMPA) for his valuable
suggestions.
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