Abstract
A generalized Camassa–Holm equation proposed by Novikov is considered. The existence and uniqueness of a positive weak solution for the equation is established by using a classical method.
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1 Introduction
Recently, Novikov [22] proposed the following integrable quasi-linear scalar evolution equation of order 2:
where \(\epsilon \neq 0\) is a real constant.
Letting \(v(t,x)=u(\epsilon t,\epsilon x)\), one can transform Eq. (1) into the following form:
It was shown in [22] that Eq. (2) possesses a hierarchy of local higher symmetries. Equation (2) is regarded as a generalized Camassa–Holm equation [22]. In [17], Li and Yin establish the local existence and uniqueness of strong solutions for Eq. (2) in nonhomogeneous Besov spaces by using the Littlewood–Paley theory. Under some assumptions, a blow-up criterion and a global existence result for the equation are also presented in [17]. The well-posedness of (2) is studied in [11] for the periodic and the nonperiodic cases in the sense of Hadamard. In addition, nonuniform dependence is proved by using the method of approximate solutions and well-posedness estimates. To the best of our knowledge, up to now the weak solutions for Eq. (2) have not been investigated yet.
The equation closest to the relevant problem (2) is the Degasperis–Procesi equation,
Degasperis, Holm and Hone [12] proved the formal integrability of Eq. (2) by constructing a Lax pair. They showed that it has a bi-Hamiltonian structure and there is an infinite sequence of conserved quantities. Since the Degasperis–Procesi equation was born, much attention has been attracted by the study its dynamics. Yin proved local well-posedness of Eq. (2) on the line [24] and on the circle [25]. In addition, the precise blow-up scenario and blow-up structure for the equation were derived in [24, 25]. Lenells [16] classified all weak traveling wave solutions. Matsuno [20] obtained multisolutions of Eq. (2). Escher et al. [13] investigated the blow-up phenomena and global weak solutions for Degasperis–Procesi equation. In a different direction, Coclite and Karlsen [3–5], and Lundmark [19] initiated a study of discontinuous solutions (shock wave) to the Degasperis–Procesi equation (2). It is shown in [2] that a new blow-up quantity along the characteristics is established for the Degasperis–Procesi equation (2). The other equations related to Eq. (2), such as the Camassa–Holm equation, the Novikov equation and the Modified Camassa–Holm equation with cubic nonlinearity, can be found in [1, 6–10, 14, 15, 18, 22, 23] and the references therein.
Inspired by the ideas from [13, 26], in this paper, we investigate the weak solutions for the following Cauchy problem:
More precisely, we focus on the existence and uniqueness of positive weak solutions to the problem (4) using the method from [11] under the condition \(y_{0}=u_{0}-u_{0xx}\in \mathcal{M}^{+}\). One of the difficult issues in our proof is how to prove that there is a subsequence of \(\{u^{n}\}\) which converges pointwise a.e. to a function \(u\in H^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times \mathbb{R})\) that satisfies (4) in the sense of distributions, and how to show that \(u\in C_{w}(\mathbb{R}_{+};H^{1}(\mathbb{R}))\), the space of continuous functions from \(\mathbb{R}_{+}\) with values in \(H^{1}( \mathbb{R})\) when the latter space is equipped with its weak topology. Luckily, using \(y_{0}=u_{0}-u_{0xx}>0\) and the estimate \(\Vert u(t, \cdot ) \Vert _{L^{\infty }(\mathbb{R})}\leq \frac{3}{2} \Vert u _{0} \Vert ^{2}_{L^{3}(\mathbb{R})}t+ \Vert u_{0} \Vert _{L^{\infty }}\), we successfully overcome the problems.
Notations
The space of all infinitely differentiable functions \(\phi (t,x)\) with compact support in \([0,+\infty )\times \mathbb{R}\) is denoted by \(C^{\infty }_{0}\). Let \(1\leq p<+\infty \) and \(L^{p}=L^{p}( \mathbb{R})\) be the space of all measurable functions \(h(t,x)\) such that \(\Vert h \Vert ^{P}_{L^{P}}=\int _{\mathbb{R}} \vert h(t,x) \vert ^{p} \,dx< \infty \). We define \(L^{\infty }=L^{\infty }(\mathbb{R})\) with the standard norm \(\Vert h \Vert _{L^{\infty }}=\inf_{m(e)=0} \sup_{x\in \mathbb{R}\setminus e} \vert h(t,x) \vert \). For any real number s, let \(H^{s}=H^{s}(\mathbb{R})\) denote the Sobolev space with the norm defined by \(\Vert h \Vert _{H^{s}}= (\int _{\mathbb{R}}(1+ \vert \xi \vert ^{2})^{s} \vert \hat{h}(t,\xi ) \vert ^{2}\,d\xi )^{\frac{1}{2}}<\infty \), where \(\hat{h}(t,\xi )=\int _{\mathbb{R}}e^{-ix\xi }h(t,x)\,dx\).
We denote by ∗ the convolution. Let \(\Vert \cdot \Vert _{X}\) denote the norm of Banach space X and \(\langle \cdot ,\cdot \rangle \) denote the \(H^{1}(\mathbb{R})\), \(H^{-1}(\mathbb{R})\) duality bracket. Let \(\mathcal{M}(\mathbb{R})\) be the space of Radon measures on \(\mathbb{R}\) with bounded total variation and \(\mathcal{M}^{+}( \mathbb{R})\) be the subset of positive measures. Finally, we write \(\operatorname{BV}(\mathbb{R})\) for the space of functions with bounded variation, \(V(f)\) being the total variation of \(f\in \operatorname{BV}(\mathbb{R})\).
2 Preliminaries
Throughout this paper, let \(\{\rho _{n}\}_{n\geq 1}\) denote the mollifiers
where \(\rho \in C^{\infty }_{c}(\mathbb{R})\) is defined by
Thus, we get
Note that if \(G(x):=\frac{1}{2}e^{- \vert x \vert }\), \(x\in \mathbb{R}\). Then \((1-\partial ^{2}_{x})^{-1}f=G\ast f\) for all \(f\in L^{2}(\mathbb{R})\) and \(G\ast (u- u_{xx})=u\). Using this identity, we rewrite problem (4) in the form
which is equivalent to
Next, we give some useful results.
Lemma 2.1
Let\(f:\mathbb{R}\rightarrow \mathbb{R}\)be uniformly continuous and bounded. If\(\mu \in \mathcal{M}(\mathbb{R})\), then
Lemma 2.2
Let\(f: \mathbb{R}\rightarrow \mathbb{R}\)be uniformly continuous and bounded. If\(g\in L^{\infty }(\mathbb{R})\), then
Lemma 2.3
Let\(T>0\). If\(f,g\in L^{2}((0,T);H^{1}( \mathbb{R}))\)and\(\frac{df}{dt}, \frac{dg}{dt}\in L^{2}((0,T);H^{-1}( \mathbb{R})) \), thenf, gare a.e. equal to a function continuous from\([0,T]\)into\(L^{2}(\mathbb{R})\)and
for all\(s,t\in [0,T]\).
Lemma 2.4
Assume that\(u(t,\cdot )\in W^{1,1}(\mathbb{R})\)is uniformly bounded in\(W^{1,1}(\mathbb{R})\)for all\(t\in \mathbb{R} _{+}\). Then for a.e. \(t\in \mathbb{R}_{+}\)
and
3 Global weak solution
Lemma 3.1
Let\(u_{0}\in L^{3}(\mathbb{R})\cap H^{s}( \mathbb{R})\), \(s>\frac{3}{2}\)and\(y_{0}=(1-\partial ^{2}_{x})u_{0} \geq 0\)for all\(x\in \mathbb{R}\). Then the solutions of problem (6) exist globally in time. Moreover, if\(y(t,\cdot )=u-u_{xx}\), then, for all\(t\in \mathbb{R}_{+}\),
- (i)
\(y(t,\cdot )\geq 0\), \(u(t,\cdot )\geq 0\), \(\vert u_{x}(t,\cdot ) \vert \leq u(t,\cdot )\)and\(\Vert u(t,\cdot ) \Vert _{L^{\infty }(\mathbb{R})}\leq \frac{3}{2} \Vert u_{0} \Vert ^{2}_{L^{3}(\mathbb{R})}t+ \Vert u_{0} \Vert _{L^{ \infty }}\)on \(\mathbb{R}\).
- (ii)
\(\Vert u \Vert _{H^{1}}\leq \Vert u_{0} \Vert _{H^{1}} \exp [\frac{3}{2} \Vert u_{0} \Vert ^{2}_{L ^{3}(\mathbb{R})}t^{2}+2 \Vert u_{0} \Vert _{L^{\infty }}t]\).
Proof
The proof of (i) may be found in [17]. Now, we prove (ii).
Multiplying the first equation of problem (6) by u and integrating by parts, we find
which yields
where the Gronwall inequality and (i) were used. This proves (ii) and completes the proof of the lemma. □
Theorem 3.1
Let\(u_{0}\in H^{1}(\mathbb{R})\cap L^{3}( \mathbb{R})\), and\(y_{0}=(u_{0}-u_{0xx})\in \mathcal{M}^{+}( \mathbb{R})\). Then problem (6) has a unique solution\(u\in W^{1,\infty }(\mathbb{R}_{+}\times \mathbb{R})\cap L^{\infty }_{\mathrm{loc}}(\mathbb{R}_{+};H^{1}(\mathbb{R}))\)with initial value\(u_{0}\)and such that\((u-u_{xx})\in \mathcal{M}^{+}\), a.e. \(t\in \mathbb{R}_{+}\)is uniformly bounded on\(\mathbb{R}\).
Proof
We split the proof of Theorem 3.1 in two parts.
Let \(u_{0}\in H^{1}(\mathbb{R})\) and \(y_{0}=u_{0}-u_{0,xx}\in \mathcal{M}^{+}(\mathbb{R})\). Note that \(u_{0}=G\ast y_{0}\). Thus, for \(\varphi \in L^{\infty }(\mathbb{R})\), we have
Let us define \(u^{n}_{0}:=\rho _{n}\ast u_{0}\in H^{\infty }( \mathbb{R})\) for \(n\geq 1\). Obviously, we get
and
Note that, for all \(n\geq 1\),
Referring to the proof of (9), we have
From Lemma 3.1, we know that there exists a global strong solution,
and \(u^{n}(t,x)-u^{n}_{xx}(t,x)\geq 0\) for all \((t,x)\in \mathbb{R} _{+}\times \mathbb{R}\).
Note that for all \((t,x)\in \mathbb{R}_{+}\times \mathbb{R}\)
From Lemma 3.1 and (10), we obtain
From the Hölder inequality, Lemma 3.1 and (10), for all \(t\geq 0\) and \(n\geq 1\), we have
Using the Young inequality, we get
where \(\Vert \partial _{x}G \Vert _{L^{2}(\mathbb{R})}\) is bounded.
Applying (13)–(14) and problem (6), we have
For fixed \(T> 0\), from (12) and (15), we deduce
where M is a positive constant depending only on \(\Vert G_{x} \Vert _{L^{2}(\mathbb{R})}\), \(\Vert u_{0} \Vert _{H^{1}(\mathbb{R})}\), \(\Vert u_{0} \Vert _{L^{3}(\mathbb{R})}\), \(\Vert y_{0} \Vert _{\mathcal{M}(\mathbb{R})}\) and T. It follows that the sequence \(\{ u^{n}\}_{n\geq 1}\) is uniformly bounded in the space \(H^{1}((0, T)\times \mathbb{R})\). Thus, we can extract a subsequence such that
and
for some \(u\in H^{1}((0,T)\times \mathbb{R})\). From Lemma 3.1 and (10), for fixed \(t\in (0,T)\), we see that the sequence \(u^{n_{k}}_{x}(t,\cdot )\in \operatorname{BV}(\mathbb{R})\) satisfies
and
Applying Helly’s theorem [21], we infer that there exists a subsequence, denoted again \(\{u^{n_{k}}_{x}(t,\cdot )\}\), which converges at every point to some function \(\nu (t,\cdot )\) of finite variation with
From (18), we get, for almost all \(t\in (0,T)\), \(u^{n_{k}}_{x}(t, \cdot )\rightarrow u_{x}(t,\cdot )\) in \(D'(\mathbb{R})\). It follows that \(\nu (t,\cdot )=u_{x}(t,\cdot )\) for a.e.\(t\in (0,T)\). Therefore, we have
and for a.e.\(t\in (0,T)\),
By Lemma 3.1 and (12), we have
Note that, for fixed \(t\in (0,T)\), the sequence \(\{(u^{n})^{2}+((u ^{n})^{2})_{x}\}_{n\geq 1}\) is uniformly bounded in \(L^{2}(\mathbb{R})\). Therefore, it has a subsequence \(\{(u^{n_{k}})^{2}+((u^{n_{k}})^{2})_{x} \}_{n_{k}\geq 1}\) which converges weakly in \(L^{2}(\mathbb{R})\). From (18), we infer that the weak \(L^{2}(\mathbb{R})\)-limit is \(\{(u)^{2}+(u^{2})_{x}\}\). It follows from \(G_{x}\in L^{2}(\mathbb{R})\) that
From (18), (19) and (21), we see that u solves Eq. (6) in \(D'((0,T)\times \mathbb{R})\).
For fixed \(T> 0\), noticing that \(u^{n_{k}}_{t}\) is uniformly bounded in \(L^{2}(\mathbb{R})\) as \(t\in [0,T)\) and \(\Vert u^{n_{k}}(t) \Vert _{H^{1}(\mathbb{R})}\) is uniformly bounded for all \(t\in [0,T)\) and \(n\geq 1\), we infer that the family \(t\rightarrow u ^{n_{k}}\in H^{1}(\mathbb{R})\) is weakly equicontinuous on [0,T]. An application of the Arzela–Ascoli theorem shows that \(\{u^{n_{k}}\}\) has a subsequence, denoted again \(\{u^{n_{k}}\}\), which converges weakly in \(H^{1}(\mathbb{R})\), uniformly in \(t\in [0,T)\). The limit function is u. T being arbitrary, we see that u is locally and weakly continuous from \([0,\infty )\) into \(H^{1}(\mathbb{R})\), i.e., \(u\in C_{w,\mathrm{loc}}(\mathbb{R}_{+};H^{1}(\mathbb{R}))\).
Since for a.e. \(t\in \mathbb{R}_{+}\), \(u^{n_{k}}(t,\cdot )\rightharpoonup u(t,\cdot )\) weakly in \(H^{1}(\mathbb{R})\), from Lemma 3.1, we get
Inequality (22) shows that
From (10), for \(t\in \mathbb{R}_{+}\), we obtain
Combining with (18), we have
Finally, we prove \((u(t,\cdot )-u_{xx}(t,\cdot ))\in \mathcal{M}^{+}\) is uniformly bounded on \(\mathbb{R}\) and \(u(t,x)\in W^{1,\infty }( \mathbb{R}_{+}\times \mathbb{R})\).
We have
From (20), we get for a.e. \(t\in \mathbb{R}_{+}\)
The above inequality implies that, for a.e. \(t\in \mathbb{R}_{+}\), \((u(t,\cdot )-u_{xx}(t,\cdot ))\in \mathcal{M}(\mathbb{R})\) is uniformly bounded on \(\mathbb{R}\). For fixed \(T\geq 0\), applying (17) and (18), we have
Since \((u^{n_{k}}(t,\cdot )-u^{n_{k}}_{xx}(t,\cdot ))\geq 0\) for all \((t,x)\in \mathbb{R}_{+}\times \mathbb{R}\), we obtain, for a.e. \(t\in \mathbb{R}_{+}\), \((u(t,\cdot )-u_{xx}(t,\cdot ))\in \mathcal{M} ^{+}(\mathbb{R})\).
Note that \(u(t,x)=G\ast (u(t,x)-u_{xx}(t,x))\). Then we get
Combining with (23), it implies that \(u(t,x )\in W^{1,\infty }( \mathbb{R}_{+}\times \mathbb{R})\).
This completes the proof of the existence of Theorem 3.1.
Next, we present the uniqueness proof of Theorem 3.1.
Let \(u,v\in W^{1,\infty }(\mathbb{R}_{+}\times \mathbb{R})\cap L^{ \infty }_{\mathrm{loc}}(\mathbb{R}_{+}; H^{1}(\mathbb{R}))\) be two global weak solutions of problem (6) with the same initial data \(u_{0}\). Assume that \((u(t,\cdot )-u_{xx}(t,\cdot ))\in \mathcal{M} ^{+}(\mathbb{R})\) and \((v(t,\cdot )-v_{xx}(t,\cdot ))\in \mathcal{M} ^{+}(\mathbb{R})\) are uniformly bounded on \(\mathbb{R}_{+}\) and set
From the assumption, we know that \(N< \infty \). Then, for all \((t,x)\in \mathbb{R}_{+}\times \mathbb{R}\),
and
Similarly
Following the same procedure as in (9), we may also get
and for all \((t,x)\in \mathbb{R}_{+}\times \mathbb{R}\)
We define
Convoluting Eq. (6) for u and v with \(\rho _{n}\), we get, for a.e. \(t\in \mathbb{R}_{+}\) and all \(n\geq 1\),
and
Subtracting (31) from (30) and using Lemma 2.4, integration by parts shows that for a.e. \(t\in \mathbb{R}_{+}\) and all \(n\geq 1\)
Using (24)–(26) and the Young inequality to the first term on the right-hand of (32) yields
Similarly, we obtain
and
For the last term on the right-hand side of (32), we have
and
From (33)–(37), for a.e. \(t\in \mathbb{R}_{+}\) and all \(n\geq 1\), we find
where
where K is a positive constant depending on N and the \(H^{1}( \mathbb{R})\)-norms of \(u(0)\) and \(v(0)\).
In the same way, convoluting Eq. (6) for u and v with \(\rho _{n,x}\) and using Lemma 2.4, we see that for a.e. \(t\in \mathbb{R}_{+}\) and all \(n\geq 1\)
Using the identity \(\partial ^{2}_{x}(G\ast g)=G\ast g-g\) for \(g\in L^{2}(\mathbb{R})\) and the Young inequality, we estimate the fourth term of the right-hand side of (40):
Using (24)–(26) and the Young inequality to the first term on the right-hand of (40) gives rise to
To treat the second term of the right-hand side of (40), we note that
Applying Lemma 2.1, the second expression of right-hand of (43) can be estimated by a function \(R_{n}(t)\) belonging to (39). Making use of the Hölder inequality and (9), for a.e. \(t\in \mathbb{R}_{+}\) and all \(n\geq 1\), we have
It follows from (43) and (44) that
Now, we deal with the third term on the right-hand side of (40)
Therefore, (46) implies that for a.e. \(t\in \mathbb{R}_{+}\) and all \(n\geq 1\)
From (41), (42), (45) and (47), for a.e. \(t\in \mathbb{R}_{+}\) and all \(n\geq 1\), we deduce
Combining with (38) and (48), we find
It follows from the Gronwall inequality that for a.e. \(t\in \mathbb{R}_{+}\) and all \(n\geq 1\)
Fix \(t>0\) and let \(n\rightarrow \infty \) in (50). Since \(w=u-v\in W^{1,1}(\mathbb{R})\) and Eq. (39) holds, making use of Lebesgue’s dominated convergence theorem yields
Note that \(w(0)=w_{x}(0)=0\), therefore, we obtain \(u(t,x)=v(t,x)\) for a.e. \((t,x)\in \mathbb{R}_{+}\times \mathbb{R}\). This completes the proof of the theorem. □
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Acknowledgements
The author thanks the referees for their valuable comments and suggestions. This work is supported by Zunyi Normal University Doctoral Program project [grant number BS[2017]10], Department of Guizhou Province Education project [grant number QianJiaoHe KY Zi[2019]124 and the GuiZhou Province Science and Technology Plan Project [Grant number QianKeHe Platform Talents [2018]5784].
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Guo, Y. On weak solutions to a generalized Camassa–Holm equation with solitary wave. Bound Value Probl 2020, 15 (2020). https://doi.org/10.1186/s13661-020-01326-3
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DOI: https://doi.org/10.1186/s13661-020-01326-3