The Cauchy problem for a generalized Camassa–Holm equation with the velocity potential
Lin, Bohuan; Yin, Zhaoyang
Published in: Applicable Analysis
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10.1080/00036811.2016.1267342
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Lin, B., & Yin, Z. (2018). The Cauchy problem for a generalized Camassa–Holm equation with the velocity potential. Applicable Analysis, 97(3), 354–367. https://doi.org/10.1080/00036811.2016.1267342
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The Cauchy problem for a generalized
Camassa–Holm equation with the velocity
potential
Bohuan Lin & Zhaoyang Yin
To cite this article: Bohuan Lin & Zhaoyang Yin (2018) The Cauchy problem for a generalized Camassa–Holm equation with the velocity potential, Applicable Analysis, 97:3, 354-367, DOI: 10.1080/00036811.2016.1267342
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https://doi.org/10.1080/00036811.2016.1267342
The Cauchy problem for a generalized Camassa–Holm equation
with the velocity potential
Bohuan Linaand Zhaoyang Yina,b
aDepartment of Mathematics, Sun Yat-sen University, Guangzhou, China;bFaculty of Information Technology,
Macau Univercity of Science and Technology, Macau, China
ABSTRACT
In this paper, we discuss a generalized Camassa–Holm equation whose solutions are velocity potentials of the classical Camassa–Holm equation. By exploiting the connection between these two equations, we first establish the local well-posedness of the new equation in the Besov spaces and deduce several blow-up criteria and blow-up results. Then, we investigate the existence of global strong solutions and present a class of cuspon weak solutions for the new equation.
ARTICLE HISTORY Received 12 October 2016 Accepted 28 November 2016 COMMUNICATED BY A. Constantin KEYWORDS A generalized Camassa–Holm equation with the velocity potential; local well-posedness; blow-up; cuspon weak solutions
AMS SUBJECT CLASSIFICATIONS
35G25; 35L05
1. Introduction
Equations of the following form are quite common in physics and of great interests:
ut = F(u, u1,. . . , un). (1.1)
Among them, the Camassa–Holm (CH) Equation (1.2) and the Degasperis-Procesi (DP) Equation (1.3) have drawn much attention since the time they were proposed [1,2]:
ut− utxx+ 3uux = 2uxuxx+ uuxxx, (1.2)
(1 − D2
x)ut = 4uux− 3uxuxx− uuxxx. (1.3)
Both the CH and DP equations describe the motion of shallow water waves with one dimension in space and have the same asymptotic accuracy [3,4]. They are integrable and possess bi-Hamitonian structures, which makes them to be important mathematical objects for study [5–9]. It is worth pointing out that both CH and DP equations admit peaked solutions [1,10,11]. The peakons are of hydrodynamical relevance since the Stokes waves of greatest height (extreme wave solutions of the governing equations for water waves) are peaked [12–14], and this type of patterns also arise as solutions to the governing equations for gravity water waves [15,16] as well as for equatorial ocean waves [17–20].
The local well-posedness of the Cauchy problem for the CH equation in Sobolev spaces and Besov spaces was proved in [21–24], and some blow-up criteria were proposed in [21,25–27]. The existence
CONTACT Zhaoyang Yin mcsyzy@mail.sysu.edu.cn
of the global strong solutions and finite time blow-up solutions were shown in [21,22,26,28], while the global existence and uniqueness of the weak solutions were discussed in [29,30].
As for the DP equation, the local well-posedness of its Cauchy problem was discussed in [31–33]. Like the CH equation, it also possesses global strong solutions, finite time blow-up solutions and global weak solutions [33–40].
As we know, when talking about the shallow water waves, the fluid is always considered adiabatic and inviscid, and acted by the gravity. This indicates that the flow should preserve the circulation of its velocity field according to Kelvin’s theorem. So it is interesting to consider the case where it is a potential flow. Besides, when considering the waves in the space with one dimension, we are actually referring to motions of fluid in a two-dimensional space. The vertical component of the velocity is always considered negligible compared to the horizontal component and thus is omitted, while the latter one is assumed to be constant along the vertical direction. So the potential we are looking for is actually a function with two variables, t ∈ R+ and x ∈ R. In 2009, Novikov used the perturbative symmetry approach to deduce a series of generalized CH equations [41], which are integrable and possess an infinite hierarchy of quasi-local higher symmetries. Among them, the following two equations happen to be the ones that we are interested in:
(1 − D2 x)ut = 1 2(3u 2 x− 2uxuxxx− u2xx), (1.4) (1 − D2 x)ut = (4 − D2x)u2x. (1.5)
This paper mainly aims at solving (1.4) in Besov spaces by exploiting its connection with the classical CH equation. More explicitly, we are dealing with the following Cauchy problem:
⎧ ⎨ ⎩ (1 − D2 x)ut = 12(3ux2− 2uxuxxx− u2xx), u t=0 = u0. (1.6)
As we will see in the paper, applying our method in Section 3for establishing the local well-posedness of the Cauchy problem (1.6), we can easily obtain the blow-up criteria and some blow-up results for (1.6) in Section4. As pointed out in the later discussion, a similar argument can be applied to (1.5). In the last section, a class of weak solutions to (1.6) is presented.
Notation 1.1: In the following, since all spaces of functions are over R, for simplicity, we drop R
in our notations of function spaces if there is no ambiguity.
2. Preliminaries
The main tools of this paper are based on the Littlewood–Payley decomposition and theories of the Besov spaces. Let’s denote by C the annulusξ ∈ R34 <|ξ| <83. Then there exist radial functions
χ ∈ D(B(0,4
3)) and ϕ ∈ D(C) which take value in [0, 1] such that
χ(ξ) +
j≥0
ϕj(ξ) = 1. (2.1)
Denote by F the Fourier transform. For∀u ∈ S, we have the following definition: −1u= F−1(χFu),
ϕj(ξ) = ϕ(2−jξ),
Sju=
k≤j−1
ku.
Then for 1≤ p, r ≤ ∞ and any real number s, the nonhomogeneous Besov spaces can be defined as follows: Bsp,r = u∈ S 2js ju p r <∞ .
When p, r= 2, the corresponding Besov spaces are actually the Sobolev spaces Hs. The following propositions and lemmas are fundamental and necessary to our work.
Proposition 2.1 [42]: 1 ≤ p ≤ q ≤ ∞, 1 ≤ r ≤ ∞, then Bs p,r → B s−(1 p−1q) q,r . Proposition 2.2 [42]: If s > 1 p, then Bsp,ris an algebra.
Lemma 2.3 [42]: Suppose that p, p, r, rsatisfy : 1
p+p1 = 1, 1r +r1 = 1. Then B−sp,ris isomorphic to
the dual space of C0∞ Bsp,r ⊂ Bsp,r. More specifically, the duality can be expressed in the following way
(u, φ) =
|j−j|≤1
ju,jφ .
Lemma 2.4 [42]: Let C be an annulus. Suppose that u ∈ Sandˆu is supported on λC. Let σ be a smooth
function onR, and possess the following property: for any α ∈ N,
|∂ασ(ξ)| ≤ Cα|ξ|m−|α|.
Then there exists some constant C such that
σ (D)u p≤ Cλm u p.
The operators mentioned above are called Smmultipliers. We have the following important fact for the this kind of operators.
Lemma 2.5 [42]: Assume that σ(D) is an Smmultiplier, then it is a bounded linear map from Bs−m p,r
to Bsp,r.
We would like to point out here thatσ(D) = (1 − D2x)−1is an S−2multiplier withσ = 1+|ξ|1 2.
Denote by p(x) = 12e−|x|. Then p= F−1(σ).
Lemma 2.6 (Commutator Estimates [42]): Let v be a vector field, and Rj= [v · ∇, j]f . Then there
exists some constant C dependent of s, p such that: if s > 0, then
2js R j p r≤ C ∇v ∞ f Bs p,r+ ∇v Bsp,r−1 ∇f ∞ ; if−1p< s < 1+1p, then 2js R j p r≤ C ∇v B 1 p p,r∩L∞ f Bs p,r.
Lemma 2.7 [42]: Let E be a Banach space, and be an open set in E. If v ∈ C([0, T], C0,1( , E)), then
there exists a mapψ ∈ C([0, T], C0,1( , E)) ∩ C1([0, T] × , E) such that
ψ(t, x) = x +
t
Lemma 2.8 [42]: Assume s > 1 + 1
p, v ∈ C([0, T], Bsp,r) ∩ C1([0, T], Bsp,r−1), g ∈ C([0, T], S) ∩
L1([0, T], Bsp,r). Then for any f0∈ Bsp,r, the Cauchy problem
∂tf + v · ∇f = g,
f(0, x) = f0,
(2.2)
has a unique solution in
C([0, T], Bs p,r), r <∞, t<sC([0, T], Btp,∞) Cw([0, T], Bsp,r), r = ∞.
Besides, the following estimate holds:
f (t) Bs p,rexp( − CV(t)) ≤ f0 Bsp,r+ t 0 exp( − CV(τ))g(τ)dτ, (2.3) where V(t) =t 0 ∇v Bsp,r−1dτ.
Lemma 2.9 [42]: Suppose f1(t, x), f2(t, x) are two solutions for the Cauchy problem (2.2) with the initial
data h1, h2, respectively. Then the following inequality holds:
f1− f2 L∞t (Bs
p,r)≤ h1− h2 Bsp,rexp CV(t). (2.4) Before recalling the basic results for the CH equation, we first introduce some notations:
Ep,rs (T) C([0, T]; Bsp,r) C1([0, T]; Bsp,r−1), if r < ∞, Ep,s∞(T) Cw([0, T]; Bsp,∞) C0,1([0, T]; Bsp,−1∞), if r = ∞.
The CH equation (1.2) can also be written as:
ut+ u∂xu= −∂xp∗ u2+1 2u 2 x .
Lemma 2.10 [42,43]: Assume s > 1 + max{1 p,
1
2}. Then for any u0 ∈ Bsp,r, there is some T > 0 such
that the Cauchy problem
ut+ u∂xu= −∂xp∗ (u2+12u2x),
u(0, x) = u0,
(2.5)
has a unique solution in Esp,r(T). When r < ∞, the solution map is continuous with respect to the initial data.
Lemma 2.11 [42]: Suppose that T∗is the lifespan of the solution for the Cauchy problem (2.5) in Bs p,r. Then we have T∗ 0 ux ∞dt= ∞, (2.6) T∗ 0 u B1∞,∞ dt= ∞. (2.7)
Lemma 2.12 [25]: Assume s > 3
2. If there exists some point x0such that∂xu0(x0) < −|u0(x0)|, then
the corresponding solution for the Cauchy problem (2.5) will blow up in finite time. Suppose thatγ is
the streamline of the velocity field u which starts at x0. Then
lim
t→T∗∂xu0◦ γ (t) = −∞. (2.8) Lemma 2.13 [27]: Denote by m = u − uxx. Assume that u0∈ H3. Then the corresponding solution to
(2.5) blows up in finite time if and only if there exist x1< x2such that m0(x1) > 0 > m0(x2).
3. Local well-posedness
In this section, we will establish the local well-posedness of the Cauchy problem (1.6). Rewrite (1.4) in the following form
ut− 1 2u 2 x = (1 − D2x)−1 u2x+ 1 2u 2 xx . (3.1)
Applying Dxto both sides of the above equation yields
(ux)t− ux∂xux = ∂xp∗ u2x+1 2u 2 xx ,
which indicates that−uxsatisfies the CH equation (1.2). Thus, solutions to (3.1) or (1.4) are actually
the velocity potentials of the solutions to the CH equation. By exploiting such a relation between the two equations, we give the following theorem.
Theorem 3.1: Let s > 2 + max{1
p,12} and u0 ∈ Bsp,r. Then there exists some T > 0 such that the
following Cauchy problem
⎧ ⎨ ⎩ ut −12u2x= (1 − D2x)−1(u2x+12u2xx), u t=0 = u0, (3.2)
has a unique solution in Es
p,r(T). When r < ∞, the continuity dependence of the solution with respect
to the initial data holds.
Proof: Let’s focus on the existence first, and the uniqueness and the continuity dependence with
respect to the initial data will be proved later. Consider the following Cauchy problem:
⎧ ⎨ ⎩ vt− vvx= ∂xp∗ (v2+12vx2), v t=0 = ∂xu0. (3.3)
According to Lemma 2.8, there exists some T > 0 such that it has a unique solution in Ep,rs−1(T). Then
when r <∞, v ∈ C([0, T], Bsp,r−1); when r = ∞, for any t < s, v ∈ C([0, T], Btp,1−1)
L∞([0, T], Bsp,−1∞).
Then, consider the next Cauchy problem: ⎧ ⎨ ⎩ ut−12vux= (1 − D2x)−1(v2+12v2x), u t=0 = u0. (3.4)
Notice that(1 − Dx2)−1is an S−2-multiplier. Since s > 2+1p, Bsp,rand Bsp,r−1are Banach algebras, we
have(1 − D2
problem has a solution u∈ C([0, T], Bs−1 p,r ), r <∞, C([0, T], Bt−1 p,r ) L∞([0, T], Bsp,r−1), r = ∞.
Next, we are going to prove that u is a solution to (3.2). In order to do so, we just need to prove that ux= v. Applying Dxto (3.4) and then subtracting (3.3) from it, we get
(ux− v)t−12v∂x(ux− v) −12vx(ux− v) = 0,
(ux− v)t=0= 0.
Denote ux− v by w. Since ux ∈ C([0, T], Btp,r−2) and v ∈ C([0, T], Btp,r−1), we have w ∈ C([0, T], Btp,r−2),
where s > t > 2+max{12,1p}. Thus, what we need descends to the uniqueness of the following Cauchy problem:
wt−12v∂xw−12vxw= 0,
wt=0= 0.
This is a fact ensured by the following lemma:
Lemma 3.2: Assume t > max{1
2,1p}, v ∈ C([0, T], B t+1
p,r ) and f , g ∈ L1([0, T], Btp,r). Suppose that
w∈ C([0, T], Btp,r) is the solution to the following Cauchy problem:
wt + v∂xw+ fw = g,
wt=0= w0.
(3.5)
Then for any 0 < < min{t − 1p, 1}, the following inequality holds with some constant C:
w ˜ Lt∞(B 1 p + p,r ) ≤ w0 B 1 p + p,r + t 0 g B 1 p + p,r ds exp C t 0 vx B 1 p + p,r + f B 1 p + p,r ds .
Proof: Applying jto (3.5) yields
∂tjw+ v∂xjw+ [j, v∂x]w + j(fw) = jg.
If p <∞, multiplying both sides of the above equation by sgn(jw)|jw|p−1and integrating with
respect to x and t, we have
jw(t) pp≤ jw0 pp+ t 0 vx ∞ j w ppds + t 0 ( Rj p+ j g p+ j(fw) p) jw pp−1ds, jw(t) p sup s∈(0,t) jw(s) pp−1 ≤ jw0 pp+ sup s∈(0,t) jw(s) pp−1 t 0 vx ∞ j w pds + sup s∈(0,t) jw(s) p−1 p t 0 ( Rj p+ jg p+ j(fw) p) ds.
Then jw(t) p≤ jw0 p+ t 0 vx ∞ jw pds + t 0 Rj p+ j g p+ j(fw) pds. (3.6)
If p= ∞, since v ∈ C([0, T], Btp,r+1) → C([0, T], C0,1), we can utilize its flow ψ(t, x) in [0, T] × R to
get
d
dtjw= −j(fw) + jg+ Rj.
Integrating with respect to t and taking L∞− norm with respect to x, we obtain jw(t) ∞≤ jw0 ∞+
t
0 j(fw) ∞+ j
g ∞+ Rj ∞ds. (3.7)
Multiplying (3.6) and (3.7) by 2j(p1+), and taking the lr− norm with respect to j, we get w(t) B 1 p + p,r ≤ w0 B 1 p + p,r + t 0 vx ∞ w B 1 p + p,r ds + t 0 2 j(1 p+) Rj p r+ g B 1 p + p,r + (fw) B 1 p + p,r ds.
Exploiting the commutator estimate yields: w(t) B 1 p + p,r ≤ w0 B 1 p + p,r + t 0 vx ∞ w B 1 p + p,r ds + t 0 vx B 1 p + p,r L∞ w B 1 p + p,r + g B 1 p + p,r + f B 1 p + p,r w B 1 p + p,r ds.
Then the lemma is proved by applying the Gronwall inequality. Now let’s go onto prove Theorem 3.1.
From Lemma 3.2, we know that u∈
C([0, T], Bs−1
p,r ), r <∞,
t<sC([0, T], Btp,r−1), r = ∞,
is the solution to the Cauchy problem (3.2). Due to the fact that
ux = v ∈ C([0, T], Bs−1 p,r ), r <∞, t<sC([0, T], Btp,−1∞) L∞([0, T], Bsp,−1∞), r = ∞, we know actually u∈ C([0, T], Bs p,r), r <∞, t<sC([0, T], Btp,∞) L∞([0, T], Bsp,∞), r = ∞.
Because Bsp,r−1and Bsp,r−2are both Banach algebras, we have
ut ∈ C([0, T], Bs−1 p,r ), r <∞, t<sC([0, T], Btp,−1∞) L∞([0, T], Bsp,−1∞), r = ∞.
It still remains to prove u ∈ Cw([0, T], Bsp,∞) in the case r = ∞. For any φ ∈ B−sp,1, u(t), φ =
Sju(t), φ + u(t), (id − Sj)φ . Since u ∈ C([0, T], Btp,∞), we have Sju ∈ C([0, T], Bsp,∞), and thus
Sju(t), φ is continuous with respect to t. On the other hand, (id − Sj)φ B−sp,1 → 0, j → ∞, and
| u(t), (id − Sj)φ | ≤ u L∞([0,T],Bs
p,∞) (id − Sj)φ B−sp,1, then together we can draw the conclusion
that u(t), φ is continuous with respect to t, and thus u ∈ Cw([0, T], Bsp,∞).
Thanks to the result for the CH equation, the argument for the uniqueness is quite simple. Suppose that u1, u2are two solutions to (3.2), then∂xu1,∂xu2are solutions to (3.3) with the same initial data
∂xu0. Due to the uniqueness of the Cauchy probelm for the CH equation, we have∂xu1 = ∂xu2.
Plugging into (3.2) yields∂tu1= ∂tu2. Since u1, u2share a common initial data, they are actually the
same.
Now let’s prove the continuity dependence with respect to the initial data in the case r < 0. Suppose that u1, u2 ∈ Esp,r are two solutions to (3.2) with respect to the initial data h1, h2, respectively. Then
∂xu1,∂xu2 are the two solutions to (3.3) with respect to∂xh1,∂xh2, respectively. Denote u1− u2by
w. And now we need to show that w Bs
p,r → 0, as h1− h2 Bp,rs → 0. Plugging u1, u2into equation, respectively, and subtracting one from the other, we get
∂tw− 1 2(∂xu1+ ∂xu2)∂xw = (1 − D2 x)−1 (u1− u 2)(u 1+ u 2) + 1 2(u 1− u 2)(u 1+ u 2) = P(u1, u 2). Since w∈ Es p,r(T) → C [0, T], W2,p C2and ∂xu1+ ∂xu2∈ L∞([0, T], Bsp,r−1) → L∞([0, T], C1,),
then it follows that: if p < ∞ , multiplying both sides of the above equation by sgn(w)|w|p−1 and integrating with respect to x and t, or if p= ∞, utilizing the flow of ∂xu1+ ∂xu2, we get
w(t) p≤ w0 p+ t 0 ∂ 2 xu1(s) + ∂x2u2(s) ∞ w(s) pds+ t 0 P(u 1, u 2) pds.
Applying the Gronwall inequality yields w(t) p≤ w0 p+ C t 0 u1− u 2 Bsp,r−1 u1 Bsp,r−1+ u 2 Bsp,r−1 ds × exp C t 0 ∂2 xu1(s) + ∂x2u2(s) ∞ds .
Thanks to the continuity dependence with respect to the initial data of the CH equation, we know that when h2 → h1in Bsp,r, w0 p+ C t 0 u1− u2 Bs−1 p,r u1 Bs−1 p,r + u 2 Bsp,r−1
ds tends to zero and expC0t ∂x2u1(s) + ∂x2u2(s) ∞ds
is bounded. Then w L∞t (Lp) → 0, and thus S0w L∞
t (Bsp,r) ≤
C w L∞t (Lp) → 0. On the other hand, due to the Bernstein inequality, and again by the continuity dependence with respect to the initial data of the CH equation, we deduce
(id − S0)w L∞t (Bs
p,r)≤ C ∂xu1− ∂xu2 L∞t (Bsp,r−1)→ 0.
4. Blow-up
By exploiting the method used to prove the existence of the solution to (3.2) in Section 3, we have the following theorem:
Theorem 4.1: For any u0 ∈ Bsp,r, s > 2+ max{21,1p}, the corresponding solutions to (3.2) and (3.3)
with the same initial data u0have the same lifespan.
Proof: It is clear that the lifespan of the solution to (3.3) is at least as long as that of (3.2). This is because that once u∈ Esp,r(T) is a solution to (3.2),∂xu∈ Esp,r−1(T) is a solution to (3.3) on[0, T].
So we need to show that if v∈ Ep,rs−1(T∗) is the solution to (3.3) for any T∗ > 0, then there is some
u∈ Esp,r(T∗) satisfying (3.2). This is also true. In the proof of Theorem 3.1, we find u by solving (3.4). Note that v∈ Esp,r−1(T∗) implies (1−Dx2)−1(v2+12v2x) ∈ L∞([0, T∗], Bp,rs ) and v ∈ L∞([0, T∗], Bsp,r−1).
So by Lemma 2.8, u∈ Ep,rs−1(T∗). As is shown in the proof, we have v = ux, which yields u∈ Ep,rs (T∗).
Thus the solutions to (3.2) and (3.3) have the same lifespan.
Thanks to Theorem 4.1 and Lemmas 2.11–2.13, the blow-up criteria and some blow-up results of the CH equation can be easily transferred into the ones that fit (1.6).
Theorem 4.2: Suppose that T∗is the lifespan of the solution to (3.2) and T∗<∞. Then
T∗ 0 uxx ∞ dt= ∞, T∗ 0 ux B1∞,∞ dt= ∞.
Theorem 4.3: Assume that s > 5
2, and u0 ∈ Hs. If there is some point x0 such that∂x2u0(x0) >
|∂xu0(x0)|, then the solution to the Cauchy problem (3.2) blows up in finite time. Ifγ is the streamline
of the velocity field which starts from x0, then
lim
t→T∗∂ 2
xu0◦ γ (t) = +∞.
Theorem 4.4: Let m = u − uxx. Assume that u0∈ H4. Then the solution to (3.2) blows up in finite
time if and only of there exist x1< x2such that∂xm0(x1) < 0 < ∂xm0(x2).
5. A few remarks
Although the local well-posedness has been proved in Section 3, we still want to introduce the following proposition which implies some kind of stability of (3.2), and also yields the uniqueness in a different way.
Proposition 5.1: Suppose that s > 2 + max{1
2,1p}, and u1, u2 ∈ Esp,r(T) are solutions to (3.4) with
the initial data h1, h2∈ Bsp,r, respectively. Then the following estimate holds:
u1− u2 Bs−1 p,r exp −C t 0 ∂xx u1− ∂xxu2 Bs−2 p,r dτ ≤ h1− h2 Bs−1 p,r exp C t 0 ∂x u1+ ∂xu2 Bs−1 p,r dτ , (5.1)
Proof: Denotet
0 ∂xxu1− ∂xxu2 Bs−2
p,r dø by V
(t). Plugging u
1, u2 into equation, respectively, and
subtracting one from another, we obtain
∂t(u1− u2) −1 2(∂xu1+ ∂xu2)(∂xu1− ∂xu2) = p ∗ (∂xu1− ∂xu2)(∂xu1+ ∂xu2) +1 2(∂xxu1− ∂xxu2)(∂xxu1+ ∂xxu2) .
By (2.3) and the product law of the Besov spaces, we have u1− u2 Bs−1 p,r exp − V(t) ≤ h1− h2 Bs−1 p,r + t 0 exp− V(τ) u1− u2 Bs−1 p,r u1+ u2 B s p,rdτ.
Applying the Gronwall inequality, we get (5.1).
As we know, if u∈ Bsp,r+1, then ux ∈ Bp,rs . Conversely, if v∈ Bsp,r, we may not have some u∈ Bsp,r+1
such that v= ux. Let’s denote by∂Bsp,r+1the subspace
v∈ Bsp,r∃u∈ Bsp,r+1such that v= ux
. Theorem 4.1 actually tells us that once u0belongs to the subspace∂Bsp,r+1, then the solution remains in it until
blowing up. And that the velocity field shares the same lifespan with its potential in corresponding spaces seems to be a reasonable result. We would like to point out that if the initial data u0takes on
the following shape, then (3.2) has a global solution.
The local well-posedness of (3.2) has been proved under the assumption that s > 2+ max{12,1p} by exploiting the results for the CH equation in the Besov space Btp,rwhere t > 1+ max{12,
1
p}. However,
according to [24], the local well-posedness of the CH equation also holds for B
3 2
2,1. So actually we can
deduce the results for (3.2) in B
5 2
2,1by applying the same argument in Theorem 3.1.
As mentioned before, the DP equation (1.3) is the other well-known equation which also describes the shallow water waves and is integrable and (1.5) is the other equation proposed by Novikov [41]. Rewrite it in the following form
ut − ux2= (1 − D2x)−1(3u2x).
Applying Dxto both sides of equation yields
(ux)t− 2ux∂xux = ∂xp∗ (3u2x),
which is actually the DP equation. Obviously, the same discussions in Sections 3-4 can be applied to the above two equations to get desired results.
6. Cuspon weak solutions
In this section, we intend to give a class of cuspon weak solutions to (1.4). Let’s first state the definition of weak solutions:
Definition 6.1: u ∈ C1([0, T], C1) C([0, T], C2) is called a weak solution to (1.4) if for any
φ(x, t) ∈ C1[0, T], C∞ 0 (R)
, it satisfies the following condition: RuT(1 − ∂ 2 x)φTdx− Ru0(1 − ∂ 2 x)φ0dx− T 0 Ru(1 − ∂ 2 x)φtdxdt (6.1) =1 2 T 0 Ru 2 x(1 − ∂x2)φdxdt + T 0 Ru 2 xφdxdt + 1 2 T 0 Ru 2 xxφdxdt. (6.2)
Theorem 6.2: (1.4) has a class of weak solutions of the following form:
u=
c− cect−x, ct < x,
cex−ct− c, x ≤ ct. (6.3)
Proof: Plugging (6.3) into both sides of (6.2), we get Ru 2 x(1 − ∂x2)φdxdt = ∞ −∞u 2 xφdxdt − ∞ ct c2e2ct−2x∂x2φdxdt − ct −∞c 2e2x−2ct∂2 xφdxdt = ∞ −∞u 2 xφdxdt + ∞ ct ( − 2)c 2e2ct−2x∂ xφdxdt + ct −∞2c 2e2x−2ct∂ xφdxdt − u2x∂xφ ct− ct+ = ∞ −∞u 2 xφdxdt − ∞ ct 4c2e2ct−2xφdxdt − ct −∞4c 2e2x−2ctφdxdt + 4c2φ(ct, t) = − ∞ ct 3c2e2ct−2xφdxdt − ct −∞3ce 2x−2ctφdxdt + 4c2φ(ct, t), and Ru 2 xφdx + 1 2 Ru 2 xxφdx = ∞ ct 3 2c 2e2ct−2xφdx + 3 2ce 2x−2ctφdx. Thus 1 2 T 0 Ru 2 x(1 − ∂x2)φdxdt + T 0 Ru 2 xφdxdt + 1 2 T 0 Ru 2 xxφdxdt = T 0 2c2φ(ct, t)dt.
On the other hand, RuT(1 − ∂ 2 x)φTdx− Ru0(1 − ∂ 2 x)φ0dx = RuTφTdx− Ru0φ0dx+ R∂xuT∂xφTdx− R∂xu0∂xφ0dx − uT∂xφT cT− cT+ − u0∂xφ0 0− 0+ = RuTφTdx− Ru0φ0dx+ ∂xuTφT cT− cT+ − ∂xu0φ0 0− 0+ − R∂ 2 xuTφTdx− R∂ 2 xu0φ0dx = − ∞ ct c(φT− φ0)dx + ct −∞c(φT− φ0)dx. Similarly, Ru(1 − ∂ 2 x)φtdx = − c ∞ ct ∂tφ(t, x)dx + c ct −∞∂tφ(t, x)dx = − cd dt ∞ ct φ(t, x)dx − c 2φ(t, ct) + cd dt ct −∞φ(t, x)dx − c 2φ(t, ct). Then T 0 Ru(1 − ∂ 2 x)φtdxdt = − c ∞ ct (φT− φ0)dx + c ct −∞(φT− φ0)dx − 2c 2 T 0 φ(t, ct)dt. Thus, we have RuT(1 − ∂ 2 x)φTdx− Ru0(1 − ∂ 2 x)φ0dx− T 0 Ru(1 − ∂ 2 x)φtdxdt = 2c2 T 0 φ(t, ct)dt =1 2 T 0 Ru 2 x(1 − ∂x2)φdxdt + T 0 Ru 2 xφdxdt + 1 2 T 0 Ru 2 xxφdxdt.
This completes the proof of the theorem.
Acknowledgement
The authors thank the referees for their valuable comments and suggestions.
Disclosure statement
Funding
This work was partially supported by NNSFC [grant number 11671407], [grant number 11271382]; FDCT [grant number 098/2013/A3]; Guangdong Special Support Program [grant number 8-2015]; the key project of NSF of Guangdong province [grant number 2016A030311004].
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