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The local well-posedness and stability to a nonlinear generalized Degasperis-Procesi equation
Boundary Value Problems volume 2015, Article number: 170 (2015)
Abstract
A nonlinear generalized Degasperis-Procesi equation is investigated. The local well-posedness of a strong solution for the equation in the Sobolev space \(H^{s}(R)\) with \(s>\frac{3}{2}\) is established. The \(L^{1}(R)\) stability is obtained under certain assumptions on the initial data.
1 Introduction
Degasperis and Procesi [1] investigated a family of third order dispersive nonlinear equations
where constants \(\alpha_{i}\) (\(i=0,1,2,3\)), β and \(\gamma\in R\).
Applying the method of asymptotic integrability to Eq. (1), it is found in [1] that only three equations satisfy asymptotic integrability conditions: the KdV equation, the Camassa-Holm (CH) equation and one new equation of the form
which can be transformed to the dispersionless form
Equation (3) is called Degasperis-Procesi (DP) equation which represents a model for shallow water dynamics, and its asymptotic accuracy is similar to the CH equation.
Here, we set the coefficients of Eq. (1) \(\gamma=-\alpha_{0}\beta^{2}\), \(\alpha_{2}=\alpha_{3}\) and \(\beta>0\). The objective of this work is to study the following nonlinear equation:
Obviously, DP equation (3) is the special case of Eq. (4).
Degasperis et al. [2] proved the formal integrability of Eq. (3), derived the infinite sequence of conserved quantities with a bi-Hamiltonian structure and obtained the existence of the non-smooth solutions by constructing a Lax pair. Lenells [3] classified all weak traveling wave solutions of Eq. (3), while Coclite and Karlsen [4] established the well-posedness of periodic weak solutions for Eq. (3) and studied their long-time asymptotic behavior. It was shown in [5] that several numerical schemes were constructed and proved that they converged to weak solutions of Eq. (3). Escher et al. [6] discussed several qualitative properties of the DP equation. The existence and uniqueness of global weak solutions for Eq. (3) have been established provided that the initial data satisfy appropriate conditions in [6].
In fact, the KdV equation modeling weakly nonlinear unidirectional long waves and the CH equation modeling the unidirectional shallow water waves have been extensively investigated (see [7–11]). For other approaches to study related partial differential equations, the reader is referred to [12–20] and the references therein.
In this paper, assuming that the initial value \(u(0,x)\) of Eq. (4) belongs to \(H^{s}(R)\) (\(s>\frac{3}{2}\)), we will prove the existence and uniqueness of the local solution for Eq. (4) in the space \(C ([0,\infty); H^{s}(R) )\cap C^{1} ([0,\infty);H^{s-1}(R) )\) by using the Kato theorem (see [20]) and use the approaches presented in Kruzkov [19] to establish the \(L^{1}(R)\) local stability of the solution for this nonlinear equation (4). From our knowledge, Eq. (4) has not been discussed in the literature.
In the following, we will give the conservation law in some case for Eq. (4) in Section 2. The proof of existence for the local strong solution and its stability to Eq. (4) are given in Section 3 and Section 4, respectively. For simplicity, we let c denote any positive constants.
2 Preliminaries
Applying the operator \(\Lambda^{-2}=(1-\beta^{2}\partial _{x}^{2})^{-1}\) to Eq. (4), we have
where \(b_{1}=\frac{\alpha_{2}}{\beta^{2}}\), \(b_{2}=-(\alpha_{1}+\frac {\alpha_{2}}{2\beta^{2}})\).
Here we consider the Cauchy problem of Eq. (4)
which is equivalent to the problem
Lemma 2.1
Let \(y_{1}=u-\beta^{2}\partial_{x}^{2}u\) and \(\bar{y}=(1-\beta^{2}\partial_{x}^{2})^{-1}u\). The solutions of Eq. (4) in case \(\alpha_{1}=-\frac{\alpha _{2}}{2\beta^{2}}\) conform with the following conservation law:
where \(u_{0}(x)=u(0,x)\), \(\hat{u}(t,\xi)\) is the Fourier transform of \(u(t,x)\) with respect to x.
Proof
For \(y_{1}=u-\beta^{2}\partial_{x}^{2}u\), \(\bar{y}=(1-\beta^{2}\partial _{x}^{2})^{-1}u\), we have \(\frac{\partial y_{1}}{\partial t}=u_{t}-\beta^{2}\partial _{x}^{2}u_{txx}\) and \(u=\bar{y}-\beta^{2}\bar{y}_{xx}\). If \(\alpha_{1}=-\frac{\alpha_{2}}{2\beta^{2}}\), it has
This completes the proof of Lemma 2.1. □
3 Existence of the local solution for problem (6)
Firstly, we need to state some notions. Let \(H^{s}(R)\) (where s is a real number) denote the Sobolev space with the norm defined by
where \(\hat{h}(t,\xi)=\int_{R}e^{-ix\xi}h(t,x)\, dx\). For \(T>0\) and \(s\geq 0\), we let \(C ([0,\infty); H^{s}(R) )\) denote the Fréchet space of all continuous \(H^{s}\)-valued functions on \([0,T)\).
We introduce the abstract quasi-linear evolution equation
Let X and Y be Hilbert spaces, where Y is continuously and densely embedded in X, and \(W:Y\rightarrow X\) be a topological isomorphism. We define \(L(Y,X)\) to be the space of all bounded linear operators from Y to X. We denote \(L(X,X)\) by \(L(X)\). Note that \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3} \) and \(\rho_{4}\) in the following depend on \(\max\{\|y\|_{Y},\|z\|_{Y}\}\).
-
(I)
\(Q(y)\in L(Y,X)\) for \(y\in X\) with
$$ \bigl\Vert \bigl(Q(y)-Q(z) \bigr)w\bigr\Vert _{X}\leq \rho_{1}\|y-z\|_{X}\|w\|_{Y},\quad y,z,w\in Y, $$\(Q(y)\in G(X,1,\alpha)\) (\(\alpha>0\)) and uniformly on bounded sets in Y.
-
(II)
\(WQ(y)W^{-1}=Q(y)+\bar{Q}(y)\), in which \(\bar{Q}\in L(X)\) is bounded and uniformly on bounded sets in Y and
$$ \bigl\Vert \bigl(\bar{Q}(y)-\bar{Q}(z) \bigr)w\bigr\Vert _{X}\leq \rho_{2}\|y-z\|_{X}\|w\| _{Y},\quad y,z\in Y, w\in X. $$ -
(III)
\(g:Y\rightarrow Y\) extends to a map from X into X, is bounded on bounded sets in Y and satisfies
$$\begin{aligned} \begin{aligned} &\bigl\Vert g(y)-g(z)\bigr\Vert _{Y}\leq\rho_{3}\|y-z \|_{Y},\quad y,z\in Y, \\ &\bigl\Vert g(y)-g(z)\bigr\Vert _{X}\leq\rho_{4}\|y-z \|_{X},\quad y,z\in Y. \end{aligned} \end{aligned}$$
Kato theorem
([20])
Assume that (I), (II) and (III) hold. If \(\phi_{0}\in Y\), there is a maximal \(T>0\) depending only on \(\|\phi_{0}\|_{Y}\) and a unique solution u to problem (9) such that
Moreover, the map \(\phi_{0}\rightarrow\phi(\cdot,\phi_{0})\) is a continuous map from Y to the space \(C ([0,T);Y )\cap C^{1} ([0,T);X )\).
For problem (7), we set \(Q(u)=(\alpha _{0}+b_{1}u)\partial_{x}\), \(Y=H^{s}(R)\), \(X=H^{s-1}(R)\), \(g(u)=-b_{2}\partial_{x}\Lambda^{-2}u^{2}\) and \(W=\Lambda\). Then we will verify that \(Q(u)\) and \(g(u)\) satisfy conditions (I)-(III). We cite several conclusions presented in [11].
Lemma 3.1
The operator \(Q(u)=(\alpha _{0}+b_{1}u)\partial_{x}\) with \(\phi\in H^{s}(R)\) (\(s>\frac{3}{2}\)) belongs to \(G(H^{s-1},1,\alpha)\).
Lemma 3.2
For \(u,z,w\in H^{s}(R)\) with \(s>\frac {3}{2}\), \(Q(\phi)\in L(H^{s}(R),H^{s-1}(R))\), it holds that
Lemma 3.3
For \(u,z\in H^{s}(R)\) and \(w\in H^{s-1}(R)\) (\(s>\frac{3}{2}\)), it holds that \(\bar{Q}(u)= [\Lambda,(1-b_{1}u)\partial_{x} ]\Lambda^{-1}\in L(H^{s-1})\) and
Lemma 3.4
([20])
Let r and q be real numbers such that \(-r< q\leq r\). Then
Lemma 3.5
Let \(u,z\in H^{s}(R)\) with \(s>\frac {3}{2}\) and \(g(u)=-b_{2}\partial_{x}\Lambda^{-2}u^{2}\). Then g is bounded on bounded sets in \(H^{s}\) and satisfies
Proof
For \(s>\frac{3}{2}\), we have \(\|u\|_{L^{\infty }}\leq c\|u\|_{H^{s}}\) and \(\|u\|_{H^{s-1}}\leq c\|u\|_{H^{s}}\). Applying the algebra property of \(H^{s}(R)\) and Lemma 3.4, we get
which completes the proof of (10). Similarly, we get (11). □
Now we give the following theorem.
Theorem 3.6
Let \(u_{0}(x) \in H^{s}(R)\) with \(s>\frac{3}{2}\). There exists \(T>0\) depending on \(\|u_{0}\|_{H^{s}(R)}\) such that problem (6) or (7) has a unique solution \(u(t,x)\in C ([0,T);H^{s}(R) )\cap C^{1} ([0,T);H^{s-1}(R) )\).
Proof
Using Lemmas 3.1-3.3 and 3.5, we know that conditions (I), (II) and (III) hold. Applying the Kato theorem, we find that problem (6) or (7) has a unique local solution
where \(T>0\) depends on \(\|u_{0}\|_{H^{s}}\). □
Remark 3.7
Let \(T>0\) be described in Theorem 3.6. Using the Sobolev embedding theorem, we ensure the boundedness of solution \(u(t,x)\) to problem (7) in the domain \([0,T)\times R\). Namely, provided that \(u_{0} \in H^{s}(R)\) with \(s>\frac{3}{2}\), we have \(\|u\|_{L^{\infty}(R)}\leq M_{T}\), where \(M_{T}\) is a positive constant.
4 \(L^{1}\) Stability of the local solution for problem (6)
Letting \(f'(u)=\alpha_{0}+b_{1}u\), \(\Psi (t,x,u)=b_{2}\Lambda^{-2}\partial_{x}u^{2}\) in the first equation of (7), we get
Assume that \(u_{1}(t,x)\) and \(u_{2}(t,x)\) are solutions of problem (12) in the domain \([0,T)\times R\) with initial functions \(u_{10}(x)\) and \(u_{20}(x)\in H^{s}(R)\) (\(s>\frac{3}{2}\)). Here we give several lemmas.
Lemma 4.1
Let \(u(t,x)\) be the solution of problem (12) and \(u_{0}(x)\in H^{s}(R)\) with \(s>\frac{3}{2}\). Then \(\|\Psi(t,x,u)\| _{L^{\infty}(R)}\leq M_{T}\) and
where positive constant c depends on \(b_{2}\), β, \(\|u_{0}\| _{L^{\infty}}\).
Proof
We have
Applying Remark 3.7 and the integral \(\int_{R}e^{-\frac{1}{\beta }|x-y|}\, dy=2\beta\), we complete the proof. □
Lemma 4.2
Assume that \(u_{1}(t,x)\) and \(u_{2}(t,x)\) are solutions of problem (12) in the domain \([0,T)\times R\) with initial functions \(u_{10}(x)\) and \(u_{20}(x)\in H^{s}(R)\) (\(s>\frac{3}{2}\)), respectively. Then
where \(c>0\) depends on \(b_{2}\), β, \(\|u_{10}\|_{L^{\infty}(R)}\), \(\| u_{20}\|_{L^{\infty}(R)}\) and T.
Proof
Using the property of the operator \(\Lambda ^{-2}\) and Remark 3.7, we get
in which we apply the Tonelli theorem to complete the proof. □
We introduce a function \(\delta(\sigma)\) which is infinitely differential on \((-\infty,+\infty)\) and \(\delta(\sigma)\geq0\), \(\delta(\sigma)\equiv0\) for \(|\sigma|\geq1\), \(\int_{-\infty}^{+\infty}\delta(\sigma)\, d\sigma=1\). Let \(\delta_{\varepsilon}(\sigma)=\delta(\varepsilon^{-1}\sigma)\diagup \varepsilon\), where ε is an arbitrary positive constant. It is found that \(\delta_{\varepsilon}(\sigma)\in C^{\infty}_{0}(-\infty ,+\infty)\) and
Let the function \(\phi(x)\) be defined and locally integrable on \((-\infty,+\infty)\). Set \(\phi^{\varepsilon}(x)\) to denote the approximation function of \(\phi(x)\) as
We call \(x_{0}\) a Lebesgue point of the function \(\phi(x)\) if
At any Lebesgue point \(x_{0}\), we get
Since the set of points which are not Lebesgue points of \(\phi(x)\) has measure zero, we have \(\phi^{\varepsilon}(x)\rightarrow\phi(x)\) as \(\varepsilon\rightarrow0\) almost everywhere.
For any \(T_{0}\in[0,T)\), we denote the band \(\{ (t,x):[0,T_{0}]\times R\}\) by \(\pi_{T_{0}}\). Let \(K_{r}=\{x:|x|\leq r\}\) and
where \(r>0\), \(\rho>0\).
Lemma 4.3
([19])
Let the function \(\phi (t,x)\) be bounded and measurable in cylinder \([0,T_{0}]\times K_{r}\). If for any \(\rho\in(0,\min[r,T_{0}])\) and any \(\varepsilon\in(0,\rho)\), the function
satisfies
Lemma 4.4
([19])
If \(\frac{\partial\Phi (\phi)}{\partial\phi}\) is bounded, then the function \(H(\phi_{1},\phi_{2})=\operatorname{sign}(\phi_{1}-\phi_{2})(\Phi(\phi_{1})-\Phi(\phi _{2}))\) satisfies the Lipschitz condition in \(\phi_{1}\) and \(\phi _{1}\).
We state the concept of a characteristic cone. Let T be described in Theorem 3.6 and \(\|\phi\|_{L^{\infty}(R)}\leq M_{T}\). For any \(T_{0}\in[0,T)\) and \(R_{0}>0\), we define
Let Ω represent the cone \(\{(t,x):|x|\leq R_{0}-Nt, 0\leq t\leq T_{1}=\min(T_{0},R_{0}N^{-1}) \}\) and \(S_{\tau}\) designate the cross-section of the cone Ω by the plane \(t=\tau\), \(\tau\in[0,T_{1}]\).
Lemma 4.5
Let \(u(t,x)\) be the solution of problem (12) on \(\pi_{T_{0}}\), \(q(t,x)\in C^{\infty}_{0}(\pi_{T_{0}})\). It holds that
where k is an arbitrary constant.
Proof
Suppose that \(F(u)\) is a twice differential function. Multiplying the first equation of problem (12) by \(F'(u)q(t,x)\) and integrating over \(\pi_{T_{0}}\), we get
The application of the method of integration by parts yields
Since
we have
Substitute Eqs. (18) and (19) into Eq. (17). Let \(F^{\varepsilon}(u)\) be an approximation of the function \(F(u)=|u-k|\). When \(\varepsilon \rightarrow0\), \(F^{\varepsilon}(u)\rightarrow F(u)\), we obtain Eq. (16). □
Set function \(q(t,x)\in C^{\infty}_{0}(\pi_{T_{0}})\), \(q(t,x)\equiv0\) outside the cylinder \(\varpi=\{(t,x)\}=[\rho,T_{0}-2\rho]\times K_{r-2\rho}\), where \(K_{r-2\rho} =\{|x|:|x|\leq r-2\rho\}\), \(r>0\), \(0<2\rho<\min(T_{0},r)\). Now we give the proof of the local stability for problem (6) or (7).
Theorem 4.6
Assume that \(u_{1}(t,x)\) and \(u_{2}(t,x)\) are two local strong solutions of problem (6) or (7) with initial data \(u_{0}(x), v_{0}(x)\in L^{1}(R)\cap H^{s}(R)\) (\(s>\frac{3}{2}\)), respectively. Let \(T>0\) be the maximum existence time of \(u_{1}(t,x)\) and \(u_{2}(t,x)\). For any \(t\in[0,T)\), it holds that
where c is a positive constant depending on \(\|u_{10}\|_{L^{\infty }(R)}\) and \(\|u_{20}\|_{L^{\infty}(R)}\).
Proof
We define
in which \((\cdots)=(\frac{t+\tau}{2},\frac{x+y}{2})\), \((\vdots)=(\frac {t-\tau}{2},\frac{x-y}{2})\). Thus, we obtain
Using Lemma 4.5 and the Kruzkov’s device of doubling the variables [19], we get
Similarly, we have
Adding (21) and (22), we obtain
We note that the first two terms of the integrand of (23) have the form
where G satisfies the Lipschitz condition in all its variables. Then
As \(G_{\varepsilon}=0\) outside the region Π, applying the estimate \(|\lambda_{\varepsilon}(\vdots)|\leq \frac{c}{\varepsilon^{2}}\) and Lemma 4.4, we get
where c is a positive constant independent of ε. Using Lemma 4.3, we know \(A_{11}(\varepsilon)\rightarrow0\) as \(\varepsilon\rightarrow0\).
For the term \(A_{12}\), we substitute \(t=\eta_{1}\), \(\frac{t-\tau }{2}=\mu_{1}\), \(x=\eta_{2}\), \(\frac{x-y}{2}=\mu_{2}\). Combining with the identity
we obtain
Thus, we have
Similarly, the integrand of the third term in (23) can be represented as
Then
Applying Lemma 4.4, we get
By using Lemma 4.3, it yields \(A_{21}(\varepsilon)\rightarrow0\) as \(\varepsilon\rightarrow0\). Repeating the steps as before, we have
Set
and
Take two numbers \(\rho, \tau\in(0,T_{1})\) and \(\rho<\tau\). In (28), we let
in which
where θ is a small positive constant and \(\chi(t,x)=0\) outside the cone Ω. When \(\theta\rightarrow0\), \(R_{0}\rightarrow+\infty\), we observe that \(\chi_{\theta}\rightarrow1\). By the definition of the number N, we have
According to (28)-(32), we get
For (33), sending \(\theta\rightarrow0\), \(R_{0}\rightarrow+\infty\) combining with Lemma 4.2, we obtain
where c is independent of ε.
Applying the properties of the function \(\delta_{\varepsilon}\) for \(\varepsilon\leq \min(\rho,T_{1}-\rho)\), we get
Consequently,
Now set
It is seen that
Letting \(\varepsilon\rightarrow0\), we derive
and
Hence, we get
Applying (34)-(36), we obtain the inequality
As \(\rho\rightarrow0\), \(\tau\rightarrow t\), we get the following result:
Finally, we complete the proof of Theorem 4.6 by using the Gronwall inequality. □
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This work is supported by the National Natural Science Foundation of China (11471263).
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Chen, J., Li, R. The local well-posedness and stability to a nonlinear generalized Degasperis-Procesi equation. Bound Value Probl 2015, 170 (2015). https://doi.org/10.1186/s13661-015-0433-2
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DOI: https://doi.org/10.1186/s13661-015-0433-2