Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation
The paper deals with the global existence of weak solutions for a weakly dissipative μ-Hunter–Saxton equation. The problem is analyzed by using smooth data approximating the initial data and Helly’s theorem.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508392975499264 |
|---|---|
| author | Liu, Jianjun Лю, Жіанюнь |
| author_facet | Liu, Jianjun Лю, Жіанюнь |
| author_sort | Liu, Jianjun |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:16:44Z |
| description | The paper deals with the global existence of weak solutions for a weakly dissipative μ-Hunter–Saxton equation. The problem is analyzed by using smooth data approximating the initial data and Helly’s theorem. |
| first_indexed | 2026-03-24T02:24:29Z |
| format | Article |
| fulltext |
UDC 517.9
J. Liu (Zhengzhou Univ. Light Industry, China)
GLOBAL WEAK SOLUTIONS
FOR THE WEAKLY DISSIPATIVE µ-HUNTER – SAXTON EQUATION
ГЛОБАЛЬНI СЛАБКI РОЗВ’ЯЗКИ СЛАБКОДИСИПАТИВНОГО
µ-РIВНЯННЯ ХАНТЕРА – САКСТОНА
The paper deals with the global existence of weak solutions for a weakly dissipative µ-Hunter –Saxton equation by using
smooth data approximate to the initial data and Helly’s theorem.
Розглянуто проблему глобального iснування слабких розв’язкiв слабкодисипативного µ–рiвняння Хантера– Сакстона
за допомогою гладких даних, що є наближенням до початкових даних, та теорему Хеллi.
1. Introduction. Recently, Khesin et al. [7] derived and studied the following the µ-Hunter – Saxton
(also called µ-Camassa – Holm) equation:
µ(u)t − utxx = −2µ(u)ux + 2uxuxx + uuxxx,
which describes evolution of rotators in liquid crystals with external magnetic and self-interaction.
Here u(t, x) is a time-dependent function on the unit circle S = R/Z and µ(u) =
∫
S
udx denotes
its mean. The µ-Hunter – Saxton equation lies mid-way between the periodic Hunter – Saxton and
Camassa – Holm equations. Moreover, the equation describes the geodesic flow on Ds(S) with the
right-invariant metric given at the identity by the inner product [7]
(u, v) = µ(u)µ(v) +
∫
S
uxvxdx.
The Cauchy problem of the µ-Hunter – Saxton equation has been studied extensively. It has been
shown that the µ-Hunter – Saxton equation is locally well-posed [7] with the initial data u0 ∈ Hs(S),
s >
3
2
. Interestingly, it has global strong solutions [7] and also blow-up solutions in finite time
[3, 5, 7] with a different class of initial profiles in the Sobolev spaces Hs(S), s >
3
2
. On the
other hand, it has global dissipative weak solutions in H1(S) [15]. Moreover, the µ-Hunter – Saxton
equation admits both periodic one-peakon solution and the multi-peakons [7, 9].
In general, it is difficult to avoid energy dissipation mechanisms in a real world. So, it is reason-
able to study the model with energy dissipation. In [4] and [13], the authors discussed the energy
dissipative KdV equation from different aspects. Weakly dissipative Camassa – Holm equation and
weakly dissipative Degasperis – Procesi equation have been studied in [17, 19] and [2, 6, 18, 20]
respectively. Recently, Wei and Yin [16] discussed the global existence and blow-up phenomena of
the weakly dissipative periodic Hunter – Saxton equation.
In this paper, we will discuss global existence of weak solutions of the following weakly dissi-
pative µ-Hunter – Saxton equation:
c© J. LIU, 2013
1092 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
GLOBAL WEAK SOLUTIONS FOR THE WEAKLY DISSIPATIVE µ-HUNTER – SAXTON EQUATION 1093
yt + uyx + 2uxy + λy = 0, t > 0, x ∈ R,
y = µ(u)− uxx, t > 0, x ∈ R,
u(0, x) = u0(x), x ∈ R,
u(t, x+ 1) = u(t, x), t ≥ 0, x ∈ R,
(1.1)
or in the equivalent form:
µ(u)t − utxx + 2µ(u)ux − 2uxuxx − uuxxx + λ(µ(u)− uxx) = 0, t > 0, x ∈ R,
u(0, x) = u0(x), x ∈ R,
u(t, x+ 1) = u(t, x), t ≥ 0, x ∈ R.
(1.2)
Here the constant λ is assumed to be positive and λy = λ(µ(u) − uxx) is the weakly dissipative
term. The Cauchy problem (1.1) has been discussed in [10] recently. The author established the local
well-posedness, derived the precise blow-up scenario for Eq. (1.1) and proved that Eq. (1.1) has
global strong solutions and also finite time blow-up solutions. However, the existence of global weak
solutions to Eq. (1.1) has not been studied yet. The aim of this paper is to present a global existence
result of weak solutions to Eq. (1.1).
Throughout the paper, we denote by ∗ the convolution. Let ‖ · ‖Z denote the norm of Banach
space Z and let 〈·, ·〉 denote the H1(S), H−1(S) duality bracket. Let M(S) be the space of Radon
measures on S with bounded total variation and M+(S) (M−(S)) be the subset of M(S) with positive
(negative) measures. Finally, we write BV (S) for the space of functions with bounded variation, V(f)
being the total variation of f ∈ BV (S).
Before giving the precise statement of the main result, we first introduce the definition of weak
solution to the problem (1.2).
Definition 1.1. A function u(t, x) ∈ C(R+× S)∩L∞(R+;H1(S)) is said to be an admissible
global weak solution to (1.2) if u satisfies the equations in (1.2) and z(t, ·) → z0 as t → 0+ in the
sense of distributions on R+ × R. Moreover,
µ(u) = µ(u0)e
−λt and ‖ux(t, ·)‖L2(S) = e−λt‖u0,x‖L2(S).
The main result of this paper can be stated as follows.
Theorem 1.1. Let u0 ∈ H1(S). Assume that y0 = (µ(u0) − u0,xx) ∈ M+(S), then the
equation (1.2) has an admissible global weak solution in the sense of Definition 1.1. Moreover,
u ∈ L∞loc(R+;W
1,∞(S)) ∩H1
loc(R+ × S).
Furthermore, y =
(
µ(u)− uxx(t, ·)
)
∈M+(S) for a.e. t ∈ R+ is uniformly bounded on S.
Remark 1.1. If y0 = (µ(u0)− u0,xx) ∈M−(S), then the conclusions in Theorem 1.1 also hold
with y = (µ(u)− uxx(t, ·)) ∈M−(S).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1094 J. LIU
The paper is organized as follows. In Section 2, we recall some useful lemmas and derive some
priori estimates on global strong solutions to (1.2). In Section 3, we obtain the global existence of
approximate solutions to (1.2) with smooth approximate initial data. In Section 4, we show that the
conclusions in Theorem 1.1 hold by using Helly’s theorem.
2. Preliminaries. On one hand, with y = µ(u) − uxx, the first equation in (1.2) takes the form
of a quasi-linear evolution equation of hyperbolic type
ut + uux = −∂xA−1
(
2µ(u)u+
1
2
u2x
)
− λu, (2.1)
where A = µ − ∂2x is an isomorphism between Hs and Hs−2 with the inverse v = A−1w given
explicitly by [1, 7]
v(x) =
(
x2
2
− x
2
+
13
12
)
µ(w) +
(
x− 1
2
) 1∫
0
y∫
0
w(s)dsdy−
−
x∫
0
y∫
0
w(s)dsdy +
1∫
0
y∫
0
s∫
0
w(r)drdsdy.
Since A−1 and ∂x commute, the following identities hold:
A−1∂xw(x) =
(
x− 1
2
) 1∫
0
w(x)dx−
x∫
0
w(y)dy +
1∫
0
x∫
0
w(y)dydx
and
A−1∂2xw(x) = −w(x) +
1∫
0
w(x)dx. (2.2)
On the other hand, integrating both sides of the first equation in (1.2) with respect to x on S, we
obtain
d
dt
µ(u) = −λµ(u),
it follows that
µ(u) = µ(u0)e
−λt := µ0e
−λt, (2.3)
where
µ0 := µ(u0) =
∫
S
u0(x)dx.
Using (2.1) and (2.3), the equation (1.2) can be rewrited as
ut + uux = −∂xA−1
(
2µ0e
−λtu+
1
2
u2x
)
− λu, t > 0, x ∈ R,
u(0, x) = u0(x), x ∈ R,
u(t, x+ 1) = u(t, x), t ≥ 0, x ∈ R.
(2.4)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
GLOBAL WEAK SOLUTIONS FOR THE WEAKLY DISSIPATIVE µ-HUNTER – SAXTON EQUATION 1095
If we rewrite the inverse of the operator A = µ − ∂2x in terms of a Green’s function, we find
(A−1m)(x) =
∫ 1
0
g(x− x′)m(x′)dx′ = (g ∗m)(x). So, we get another equivalent form
ut + uux = −∂xg ∗
(
2µ0e
−λtu+
1
2
u2x
)
− λu, t > 0, x ∈ R,
u(0, x) = u0(x), x ∈ R,
u(t, x+ 1) = u(t, x), t ≥ 0, x ∈ R.
(2.5)
where the Green’s function g(x) is given [9] by
g(x) =
1
2
x(x− 1) +
13
12
for x ∈ S, (2.6)
and is extended periodically to the real line. In other words,
g(x− x′) = (x− x′)2
2
− |x− x
′|
2
+
13
12
, x, x′ ∈ S.
In particular, µ(g) = 1.
Given u0 ∈ Hs with s >
3
2
, Theorem 2.2 in [10] ensures the existence of a maximal T > 0 and
a solution u to (1.2) such that
u = u(·, u0) ∈ C
(
[0, T );Hs(S)
)
∩ C1
(
[0, T );Hs−1(S)
)
.
Consider now the following initial value problem:
qt = u(t, q), t ∈ [0, T ),
q(0, x) = x, x ∈ R.
(2.7)
Lemma 2.1 [10]. Let u0 ∈ Hs with s >
3
2
, T > 0 be the maximal existence time. Then Eq. (2.7)
has a unique solution q ∈ C1([0, T )× R;R) and the map q(t, ·) is an increasing diffeomorphism of
R with
qx(t, x) = exp
t∫
0
ux(s, q(s, x))ds
> 0, (t, x) ∈ [0, T )× R.
Moreover, with y = µ(u)− uxx, we have
y(t, q(t, x))q2x = y0(x)e
−λt.
Lemma 2.2. If y0 = µ0−u0,xx ∈ H1(S) does not change sign, then the corresponding solution
u to (2.5) of the initial value u0 exists globally in time, that is u ∈ C
(
R+, H3(S))∩C1(R+, H2(S)
)
.
Moreover, the following properties hold:
(1) µ(u) = µ0e
−λt, t ∈ [0,∞),
(2) ‖ux‖2L2(S) = e−2λtµ21, t ∈ [0,∞), with µ1 =
(∫
S
u20,xdx
)1/2
,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1096 J. LIU
(3) ‖u(t, ·)‖L∞(S) ≤ |µ0|+
√
3
6
µ1,
(4) y(t, x), u(t, x) have the same sign with y0(x), and ‖ux‖L∞(R+×S) ≤ |µ0|,
(5) |µ0|e−λt = ‖y0‖L1(S)e
−λt = ‖y(t, ·)‖L1(S) = ‖u(t, ·)‖L1(S).
Proof. Except for (4) and (5), all of the conclusions in Lemma 2.2 can be found in [10]. So we
only need to prove (4) and (5) here.
Firstly, Lemma 2.1 and u = g ∗ y, g ≥ 0 imply y(t, x) and u(t, x) have the same sign with
y0(x). Moreover, from the proof of Theorem 5.1 in [10], we have ux(t, x) ≥ −|µ0|. Now note that
given t ∈ [0, T ), there is a ξ(t) ∈ S such that ux(t, ξ(t)) = 0 by the periodicity of u to x-variable. If
y0 ≥ 0, then y ≥ 0. For x ∈ [ξ(t), ξ(t) + 1], we have
ux(t, x) =
x∫
ξ(t)
∂2xu(t, x)dx =
x∫
ξ(t)
(µ(u)− y)dx =
= µ(u)(x− ξ(t))−
x∫
ξ(t)
ydx ≤ µ(u)(x− ξ(t)) ≤ |µ0|.
It follows that ux(t, x) ≤ |µ0|. On the other hand, if y0 ≤ 0, then y ≤ 0. Therefore, for x ∈
∈ [ξ(t), ξ(t) + 1], we obtain
ux(t, x) =
x∫
ξ(t)
∂2xu(t, x)dx =
x∫
ξ(t)
(µ(u)− y)dx ≤ µ(u)(x− ξ(t))−
∫
S
ydx =
= µ(u)(x− ξ(t))−
∫
S
(µ(u)− uxx)dx = µ(u)(x− ξ(t)− 1) ≤ |µ0|.
It follows that ux(t, x) ≤ |µ0|. So we have ‖ux‖L∞(R+×S) ≤ |µ0|, this completes the proof of (4).
By the first equation of (1.1), we get∫
S
y(t, x)dx =
∫
S
y0(x)dx
e−λt = µ0e
−λt.
If y0 ≥ 0, then y ≥ 0 and µ0 ≥ 0, we have
‖y‖L1(S) =
∫
S
y(t, x)dx =
∫
S
y0(x)dx
e−λt = ‖y0‖L1(S)e
−λt = µ0e
−λt.
If y0 ≤ 0, then y ≤ 0 and µ0 ≤ 0, we obtain
‖y‖L1(S) = −
∫
S
y(t, x)dx =
∫
S
(−y0(x))dx
e−λt = ‖y0‖L1(S)e
−λt = −µ0e−λt.
It follows from this two equalities that |µ0|e−λt = ‖y0‖L1(S)e
−λt = ‖y(t, ·)‖L1(S). A similar discus-
sion implies ‖y(t, ·)‖L1(S) = ‖u(t, ·)‖L1(S).
Lemma 2.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
GLOBAL WEAK SOLUTIONS FOR THE WEAKLY DISSIPATIVE µ-HUNTER – SAXTON EQUATION 1097
Lemma 2.3 [14]. Assume X ⊂ B ⊂ Y with compact imbedding X → B (X,B and Y are
Banach spaces), 1 ≤ p ≤ ∞ and (1) F is bounded in Lp(0, T ;X), (2) ‖τhf − f‖Lp(0,T−h;Y ) → 0
as h → 0 uniformly for f ∈ F. Then F is relatively compact in Lp(0, T ;B) (and in C(0, T ;B) if
p =∞), where (τhf)(t) = f(t+ h) for h > 0, if f is defined on [0, T ], then the translated function
τhf is defined on [−h, T − h].
Lemma 2.4 (Helly’s theorem [12]). Let an infinite family F of function f(x) be defined on the
segment [a, b]. If all functions of the family and the total variation of all functions of the family are
bounded by a single number |f(x)| ≤ K,
∨b
a(f) ≤ K, then there exists a sequence fn(x) in the
family F which converges at every point of [a, b] to some function ϕ(x) of finite variation.
Lemma 2.5 [11]. Let T > 0. If f, g ∈ L2((0, T );H1(R)) and
df
dt
,
dg
dt
∈ L2((0, T );H−1(R)),
then f, g are a.e. equal to a function continuous from [0, T ] into L2(R) and
〈f(t), g(t)〉 − 〈f(s), g(s)〉 =
t∫
s
〈
d(f(τ))
dτ
, g(τ)
〉
dτ +
t∫
s
〈
d(g(τ))
dτ
, f(τ)
〉
dτ
for all s, t ∈ [0, T ].
3. Global approximate solutions. In the section, we will prove the existence of global approx-
imate solutions and give some useful estimates to the approximate solutions. Now we consider the
approximate equation of (2.5) as follows:
unt + ununx = −∂xg ∗
(
2µn0e
−λtun +
1
2
(unx)
2
)
− λun, t > 0, x ∈ R,
un(0, x) = un0 (x), x ∈ R,
un(t, x+ 1) = un(t, x), t ≥ 0, x ∈ R,
(3.1)
where un0 (x) = φn ∗ u0 ∈ H∞(S), µn0 =
∫
S
un0 (x)dx and
φn(x) :=
∫
R
φ(ξ)dξ
−1 nφ(nx), x ∈ R, n ≥ 1,
where φ ∈ C∞c (R) is defined by
φ(x) =
e
1/(x2−1), |x| < 1,
0, |x| ≥ 1.
Obviously, ‖φn‖L1(R) = 1. Clearly, we have
un0 → u0 in H1(S), as n→∞ (3.2)
and
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1098 J. LIU
‖un0‖L2(S) ≤ ‖u0‖L2(S), ‖un0,x‖L2(S) ≤ ‖u0,x‖L2(S),
‖un0‖H1(S) ≤ ‖u0‖H1(S), ‖un0‖L1(S) ≤ ‖u0‖L1(S)
(3.3)
in view of Young’s inequality. Note that
µn0 = µ(un0 ) =
∫
S
un0 (x)dx =
∫
S
∫
R
φn(y)u0(x− y)dydx =
∫
R
∫
S
φn(y)u0(x− y)dxdy =
=
∫
R
φn(y)
∫
S
u0(x− y)dx
dy =
∫
R
φn(y)
∫
S
u0(z)dz
dy =
=
∫
R
φn(y)µ(u0)(x− y)dy = φn ∗ µ(u0) = µ(u0) = µ0.
We can rewrite (3.1) as follows:
unt + ununx = −∂xg ∗
(
2µ0e
−λtun +
1
2
(unx)
2
)
− λun, t > 0, x ∈ R,
un(0, x) = un0 (x), x ∈ R,
un(t, x+ 1) = un(t, x), t ≥ 0, x ∈ R.
(3.4)
Moreover, for all n ≥ 1, yn0 = µ(un0 )− un0,xx = µ0 − un0,xx ∈ H1(S) and
yn0 = µ(un0 )− un0,xx = φn ∗ µ(u0)− φn ∗ u0,xx = φn ∗ y0 ≥ 0.
Thus, by Lemma 2.2, we obtain the corresponding solution un ∈ C(R+;H3(S))∩C1(R+;H2(S)) to
Eq. (3.4) with initial data un0 (x) and yn = µ(un)−unxx ≥ 0, un = g∗yn ≥ 0 for all (t, x) ∈ R+×S).
Furthermore, combining Lemma 2.2 with (3.3), we have
µ(un) = µn0e
−λt = µ0e
−λt, t ∈ [0,∞), (3.5)
‖unx‖2L2(S) = e−2λt‖un0,x‖2L2(S) ≤ ‖u0,x‖
2
L2(S) = µ21, t ∈ [0,∞), (3.6)
‖un(t, ·)‖L∞(S) ≤ |µn0 |+
√
3
6
‖un0,x‖L2(S) ≤ |µ0|+
√
3
6
‖u0,x‖L2(S) = |µ0|+
√
3
6
µ1, (3.7)
‖unx‖L∞(R+×S) ≤ |µn0 | = |µ0|, (3.8)
|µ0|e−λt = ‖yn(t, ·)‖L1(S) = ‖un(t, ·)‖L1(S). (3.9)
4. Proof of Theorem 1.1. In this section, with the basic energy estimate in Section 3, we are
ready to obtain the necessary compactness of the approximate solutions un(t, x). Acquiring the
precompactness of approximate solutions, we prove the existence of the global weak solutions to the
equation (1.1).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
GLOBAL WEAK SOLUTIONS FOR THE WEAKLY DISSIPATIVE µ-HUNTER – SAXTON EQUATION 1099
Lemma 4.1. For any fixed T > 0, there exist a subsequence {unk(t, x)} of the sequence
{un(t, x)} and some function u(t, x) ∈ L∞(R+;H1(S)) ∩H1([0, T ]× S) such that
unk ⇀ u in H1([0, T ]× S) as nk →∞, (4.1)
and
unk → u in L∞([0, T ]× S) as nk →∞. (4.2)
Moreover, u(t, x) ∈ C(R+ × S).
Proof. Firstly, we will prove that the sequence {un(t, x)} is uniformly bounded in the space
H1([0, T ]× S). By (3.6), (3.7), we have
‖un‖2L2([0,T ]×S) =
T∫
0
∫
S
(un)2dxdt =
T∫
0
‖un‖2L2(S)dx ≤
(
|µ0|+
√
3
6
µ1
)2
T, (4.3)
‖unx‖2L2([0,T ]×S) =
T∫
0
∫
S
(unx)
2dxdt =
T∫
0
‖unx‖2L2(S)dx ≤ µ
2
1T. (4.4)
Moreover, by (3.8) and (4.3), we obtain
‖ununx‖L2([0,T ]×S) ≤ ‖un‖L2([0,T ]×S)‖unx‖L∞([0,T ]×S) ≤
(
|µ0|+
√
3
6
µ1
)
|µ0|
√
T , (4.5)
‖∂xg ∗ (2µ0e−λtun +
1
2
(unx)
2)‖L2([0,T ]×S) ≤
≤ ‖∂xg‖L2([0,T ]×S)‖2µ0e−λtun +
1
2
(unx)
2‖L1([0,T ]×S) ≤
≤ T
12
T∫
0
∫
S
(
2|µ0||un|+
1
2
(unx)
2
)
dxdt ≤
≤ T 2
12
[
µ20 + (|µ0|+
√
3
6
µ1)
2 + µ21
]
. (4.6)
Combining (4.3), (4.5), (4.6) with Eq. (3.4), we know that {unt (t, x)} is uniformly bounded in
L2([0, T ]× S). Thus, (4.3), (4.4) and this conclusion imply that
T∫
0
∫
S
((un)2 + (unx)
2 + (unt )
2)dxdt ≤ K,
where K = K(|µ0|, µ1, T, λ) ≥ 0. It follows that {un(t, x)} is uniformly bounded in the space
H1([0, T ]× S). Thus (4.1) holds for some u ∈ H1([0, T ]× S).
Observe that, for each 0 ≤ s, t ≤ T,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1100 J. LIU
‖un(t, ·)− un(s, ·)‖2L2(S) =
∫
S
t∫
s
∂un
∂τ
(τ, x)dτ
2 dx ≤ |t− s| T∫
0
∫
S
(unt )
2dxdt.
Note that {un(t, x)} is uniformly bounded in L∞([0, T ];H1(S)), {unt (t, x)} is uniformly bounded
in L2([0, T ]× S) and H1(S) ⊂ C(S) ⊂ L∞(S) ⊂ L2(S), then (4.2) and u(t, x) ∈ C(R+ × S) are a
consequence of Lemma 2.3.
Lemma 4.1 is proved.
Next, we will deal with unx and ∂xg ∗
(
2µ0e
−λtun +
1
2
(unx)
2
)
. For fixed t ∈ [0, T ], it follows
from (3.5), (3.7) – (3.9) that the sequence unk
x (t, ·) ∈ BV (S) with
V(unk
x (t, ·)) =
∥∥unk
xx(t, ·)
∥∥
L1(S) =
∥∥µ(unk)− ynk
∥∥
L1(S) ≤
≤
∥∥µ(unk)
∥∥
L1(S) +
∥∥unk
∥∥
L1(S) ≤ 2|µ0|+
√
3
6
µ1
and ∥∥unk
x (t, ·)
∥∥
L∞(S) ≤ |µ0| ≤ 2|µ0|+
√
3
6
µ1.
Applying Lemma 2.4, we obtain that there exists a subsequence, denoted again
{
unk
x (t, ·)
}
, which
converges at every point to some function v(t, x) of finite variation with V(v(t, ·)) ≤ 2|µ0|+
√
3
6
µ1.
Since for almost all t ∈ [0, T ], unk
x (t, ·) → ux(t, ·) in D′(S) in view of Lemma 4.1, it follows that
v(t, ·) = ux(t, ·) for a.e.t ∈ [0, T ]. So we have
unk
x (t, ·)→ ux(t, ·) a.e. on [0, T ]× S, as nk →∞, (4.7)
and for a.e. t ∈ [0, T ],
V(ux(t, ·)) = ‖uxx(t, ·)‖M(S) ≤ 2|µ0|+
√
3
6
µ1. (4.8)
Therefore,∥∥∥∥∂xg ∗ (2µ0e−λtunk +
1
2
(unk
x )2
)
− ∂xg ∗
(
2µ0e
−λtu+
1
2
(ux)
2
)∥∥∥∥
L∞([0,T ]×S)
≤
≤ ‖∂xg‖L1([0,T ]×S)
∥∥∥∥2µ0e−λt(unk − u) + 1
2
(unk
x )2 − (ux)
2
∥∥∥∥
L∞([0,T ]×S)
≤
≤ T
4
(
2|µ0|‖unk − u‖L∞([0,T ]×S) +
1
2
‖unk
x + ux‖L∞([0,T ]×S)‖unk
x − ux‖L∞([0,T ]×S)
)
.
Combining this inequality with (4.2), (4.7) and note that∥∥ux(t, ·)∥∥L∞(S) ≤ lim
nk→∞
∥∥unk
x (t, ·)
∥∥
L∞(S) ≤ |µ0|,
we obtain
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
GLOBAL WEAK SOLUTIONS FOR THE WEAKLY DISSIPATIVE µ-HUNTER – SAXTON EQUATION 1101
∂xg ∗
(
2µ0e
−λtunk +
1
2
(unk
x )2
)
→ ∂xg ∗
(
2µ0e
−λtu+
1
2
u2x
)
(4.9)
a.e. on [0, T ]×S. The relations (4.2), (4.7) and (4.9) imply that u satisfies Eq. (2.5) in D′([0, T ]×S).
Moreover, since ∥∥u(t, ·)∥∥
L∞(S) ≤ lim
nk→∞
‖unk(t, ·)‖L∞(S) ≤ |µ0|+
√
3
6
µ1,
we get u ∈ L∞loc
(
R+,W 1,∞(S)
)
in view of T in (4.2) and (4.7) being arbitrary.
Now, we prove that µ(u) = µ(u0)e
−λt, ‖ux‖2L2(S) = e−2λt‖u0,x‖2L2(S) and (µ(u) − uxx(t, ·)) ∈
∈M+(S) is uniformly bounded on S.
On one hand, by (4.2), we have∫
S
unk(t, x)dx→
∫
S
u(t, x)dx = µ(u) as nk →∞.
On the other hand, ∫
S
unk(t, x)dx = µ(unk) = µ0e
−λt.
We find that µ(u) = µ(u0)e
−λt by the uniqueness of limit.
By u satisfies (2.5) in the sense of distribution, we obtain
φn ∗ ut + φn ∗ (uux) = −φn ∗
(
∂xg ∗ (2µ(u)u+
1
2
u2x)
)
− λφn ∗ u. (4.10)
Differentiating (4.10) with respect to x, we get
(φn ∗ ux)t + φn ∗ (uuxx) = −φn ∗
(
2(µ(u))2 +
1
2
µ(u2x)− 2µ(u)u+
1
2
u2x
)
− λφn ∗ ux,
here we used the formula (2.2). Multiplying the equality above with φn ∗ux and integrating the result
with respect to x on S, we obtain
1
2
d
dt
∫
S
(φn ∗ ux)2dx+
∫
S
(φn ∗ ux)(φn ∗ (uuxx))dx =
= −
∫
S
(φn ∗ ux)
(
φn ∗ (2(µ(u))2 +
1
2
µ(u2x)− 2µ(u)u+
1
2
u2x)
)
dx− λ
∫
S
(φn ∗ ux)2dx.
Note that ∫
S
(φn ∗ ux)(φn ∗ (2(µ(u))2 +
1
2
µ(u2x))dx = 0
and ∫
S
(φn ∗ ux)
(
φn ∗ (−2µ(u)u)
)
dx = −2µ(u)
∫
S
(φn ∗ ux)(φn ∗ u)dx = 0,
we have
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1102 J. LIU
d
dt
∫
S
(φn ∗ ux)2dx+ 2λ
∫
S
(φn ∗ ux)2dx =
= −2
∫
S
(φn ∗ ux)(φn ∗ (uuxx))dx−
∫
S
(φn ∗ ux)(φn ∗ u2x)dx.
Let
fn(t) =
∫
S
(φn ∗ ux)2dx,
gn(t) = −2
∫
S
(φn ∗ ux)
(
φn ∗ (uuxx)
)
dx−
∫
S
(φn ∗ ux)(φn ∗ u2x)dx,
then we obtain
dfn(t)
dt
+ 2λfn(t) = gn(t), for a.e. t ∈ R+. (4.11)
Applying Lemma 2.5 to φn ∗ ux, it follows from (4.11) that
fn(t)− e−2λtfn(0) =
t∫
0
e−2λ(t−s)gn(s)ds. (4.12)
Note that gn(t) → 0 as n → ∞ for a.e. t ∈ R+. For any T > 0, there exists a constant K(T ) > 0
such that |gn(t)| ≤ K(T ), t ∈ [0, T ], n ≥ 1. An application of Lebesgue’s dominated convergence
theorem to (4.12), we get
lim
n→∞
[
fn(t)− e−2λtfn(0)
]
= 0.
Let t ∈ R+ be given. We have ‖ux‖2L2(S) = e−2λt‖u0,x‖2L2(S).
Note that L1(S) ⊂M(S). By (4.8) and µ(u) = µ0e
−λt, we obtain
∥∥µ(u)− uxx(t, ·)∥∥M(S) ≤ ‖µ(u)‖L1(S) + ‖uxx(t, ·)‖M(S) ≤ 3|µ0|+
√
3
6
µ1.
It follows that for all t ∈ R+, (µ(u)−uxx(t, ·)) ∈M(S) is uniformly bounded on S. In view of (4.2)
and (4.7), we have[
µ(unk)− unk
xx(t, ·)
]
→ [µ(u)− uxx(t, ·)] in D′(S) for nk →∞, t ∈ [0, T ].
Since µ(unk) − unk
xx(t, ·) = ynk(t, ·) ≥ 0 for all (t, x) ∈ R+ × S, we have (µ(u) − uxx(t, ·)) ∈
∈ L∞loc(R+,M+(S)).
Theorem 1.1 is proved.
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GLOBAL WEAK SOLUTIONS FOR THE WEAKLY DISSIPATIVE µ-HUNTER – SAXTON EQUATION 1103
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Received 02.05.12
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|
| id | umjimathkievua-article-2492 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:24:29Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fe/aefbbd46e64ffd56924e2f242b0214fe.pdf |
| spelling | umjimathkievua-article-24922020-03-18T19:16:44Z Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation Глобальні слабкі розв'язки слабкодисипативного μ-рівняння Хантера-Сакстона Liu, Jianjun Лю, Жіанюнь The paper deals with the global existence of weak solutions for a weakly dissipative μ-Hunter–Saxton equation. The problem is analyzed by using smooth data approximating the initial data and Helly’s theorem. Розглянуто проблему глобального існування слабких розв'язків слабкодисипативного μ-рівняння Хантера- Сакстона за допомогою гладких даних, що є наближенням до початкових даних, та теорему Хеллі. Institute of Mathematics, NAS of Ukraine 2013-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2492 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 8 (2013); 1092–1103 Український математичний журнал; Том 65 № 8 (2013); 1092–1103 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2492/1750 https://umj.imath.kiev.ua/index.php/umj/article/view/2492/1751 Copyright (c) 2013 Liu Jianjun |
| spellingShingle | Liu, Jianjun Лю, Жіанюнь Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation |
| title | Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation |
| title_alt | Глобальні слабкі розв'язки слабкодисипативного μ-рівняння Хантера-Сакстона |
| title_full | Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation |
| title_fullStr | Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation |
| title_full_unstemmed | Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation |
| title_short | Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation |
| title_sort | global weak solutions for the weakly dissipative μ-hunter–saxton equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2492 |
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