Abstract
In this paper, we establish a result of unique continuation for a special two-dimensional nonlinear system that models the evolution of long water waves with small amplitude in the presence of surface tension. More precisely, we will show that if \((\eta ,\Phi ) = (\eta (x,y, t),\Phi (x,y, t))\) is a solution of the nonlinear system, in a suitable function space, and \((\eta ,\Phi )\) vanishes on an open subset \(\Omega \) of \(\mathbb {R}^2 \times [-T,T],\) then \((\eta ,\Phi )\equiv 0\) in the horizontal component of \(\Omega .\) To state such property, we use a Carleman-type estimate for a differential operator \(\mathcal {L}\) related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality.
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1 Introduction
The focus of the present work is the following two-dimensional system
that describes the evolution of long water waves with small amplitude in the presence of surface tension (see Quintero and Montes 2013). Here, \(\epsilon \) is the amplitude parameter (nonlinearity coefficient), \(\mu \) is the long-wave parameter (dispersion coefficient), \(\sigma \) is the inverse of the Bond number (associated with the surface tension) and the functions \(\eta =\eta (x,y,t)\) and \(\Phi =\Phi (x,y,t)\) denote the wave elevation and the potential velocity on the bottom \(z = 0,\) respectively.
As happens in water wave models, there is a Hamiltonian type structure which is clever to find the appropriate space for special solutions (solitary waves for example) and also provide relevant information for the study of the Cauchy problem. For the system (1), the Hamiltonian functional \(\mathcal {H}=\mathcal {H}(t)\) is defined as
and the Hamiltonian type structure is given by
We see directly that the functional \(\mathcal {H}\) is well defined when \(\eta , \nabla \Phi \in H^1(\mathbb R^2)\), for t in some interval. These conditions already characterize the natural space for the study of solutions of the system (1). Certainly, in Quintero and Montes (2013) showed for the model (1) the existence of solitary wave solutions which propagate with speed of wave \(\theta >0,\)
in the energy space \(H^{1}(\mathbb {R}^2)\times {\mathcal {V}}(\mathbb {R}^2)\), where \(H^{1}(\mathbb {R}^2)\) is the usual Sobolev space of order 1 and the space \({\mathcal {V}}(\mathbb {R}^2)\) is defined with respect to the norm given by
In Quintero and Montes (2016), it was proved the local well-posedness for the Cauchy problem associated to the system (1) in the Sobolev type space \(H^{s-1}(\mathbb {R}^2)\times {\mathcal {V}}^{s}(\mathbb {R}^2)\), \(s\ge 2,\) where \(H^{s}(\mathbb {R}^2)\) is the usual Sobolev space of order s defined as the completion of the Schwartz class with respect to the norm
and \({\mathcal {V}}^{s}(\mathbb {R}^2)\) denotes the completion of the Schwartz class with respect to the norm
where \(\widehat{w}\) is the Fourier transform of w defined on \(\mathbb {R}^2\) by
and in the work (Montes and Quintero 2015), using a general result established by Grillakis et al. (1987) to analyze the orbital stability of solitary waves for a class of abstract Hamiltonian systems, Quintero and Montes showed the orbital stability of the solutions of the form (2). The existence of x-periodic solitary wave solutions for the system (1) can be seen in Quintero and Montes (2017).
In the present work we will prove a unique continuation result for the system (1). More precisely, we show that if \((\eta ,\Phi ) = (\eta (x,y, t),\Phi (x,y, t))\) is a solution of the system (1) in a suitable function space,
and \((\eta ,\Phi )\) vanishes on an open subset \(\Omega \) of \(\mathbb {R}^2 \times [-T,T],\) then \((\eta ,\Phi )\equiv 0\) in the horizontal component of \(\Omega .\) The horizontal component \(\Omega _1\) of an open subset \(\Omega \subseteq \mathbb {R}^2\times \mathbb {R}\) is the set defined by
The unique continuation property has been intensively studied for a long time. An important work on the subject was done by Saut and Scheurer (1987). They showed a unique continuation result for a general class of dispersive equations including the well known KdV equation,
and various generalizations. In a similar way, Shang (2007) showed a unique continuation result for the symmetric regularized long wave equation,
In the previous equations, a Carleman estimate is established to prove that if a solution u vanishes on an open subset \(\Omega \), then \(u\equiv 0\) in the horizontal component of \(\Omega \).
By using the inverse scattering transform and some results from the Hardy function theory, Zhang (1992) established that if u is a solution of the KdV equation, then it cannot have compact support at two different moments unless it vanishes identically. In the work (Bourgain 1997), Bourgain introduced a different approach and prove that if a solution u to the KdV equation has compact support in a nontrivial time interval \(I = [t_1, t_2]\), then \(u\equiv 0.\) His argument is based on an analytic continuation of the Fourier transform via the Paley–Wiener Theorem and the dispersion relation of the linear part of the equation. It also applies to higher order dispersive nonlinear models, and to higher spatial dimensions; in particular, Panthee (2005) showed that if u is a smooth solution of the Kadomtsev–Petviashvili (KP) equation,
such that, for some \(B>0\),
then \(u\equiv 0\).
More recently, Kenig et al. (2002) proposed a new method and proved that if a sufficiently smooth solution u to a generalized KdV equation is supported in a half line at two different instants of time, then \(u\equiv 0.\) Moreover, Escauriaza et al. (2007) established uniqueness properties of solutions of the k-generalized Korteweg–de Vries equation,
They obtained sufficient conditions on the behavior of the difference \(u_1 - u_2\) of two solutions \(u_1,\) \(u_2\) of (3) at two different times \(t_0 = 0\) and \(t_1 = 1\) which guarantee that \(u_1\equiv u_2.\) This kind of uniqueness results has been deduced under the assumption that the solutions coincide in a large sub-domain of \(\mathbb {R}\) at two different times. In a similar fashion, Bustamante et al. (2011) proved that if u is a smooth solution of the Zakharov–Kuznetsov equation,
such that, for some \(B>0\),
then \(u\equiv 0\). Moreover, in Bustamante et al. (2013) it was proved that if the difference of two sufficiently smooth solutions of the Zakharov–Kuznetsov equation decays as \(e^{-a\left( x^2+y^2\right) ^{3/4}}\) at two different times, for some \(a>0\) large enough, then both solutions coincide. More unique continuation results can be seen in Carvajal and Panthee (2005), Carvajal and Panthee (2006), Iório (2003a, 2003b) and Kenig et al. (2003).
Following from close the works of Saut and Scheurer (1987), we base our analysis in finding an appropriate Carleman-type estimate for the linear operator \(\mathcal {L}\) associated to the system (1). In order to do this we use a particular version of the well known Treves’ inequality. For the operator \(\mathcal {L}\) we also prove that if a solution vanishes in a ball in the xyt space, which passes through the origin, then it also vanished in a neighborhood of the origin. The paper is organized as follows. In Sect. 2, using a particular version of the Treves inequality, we establish a Carleman estimate for a differential operator \(\mathcal L\) closely related to our problem. In Sect. 3, first we give some useful technical results. Later, we show the unique continuation result for the system (1).
2 Carleman Estimate
In this section, we will use the notation \(D=\left( \partial _x, \partial _y, \partial _t \right) \). If \(P=P(\xi _1, \xi _2,\xi _3)\) is a polynomial in three variables, has constant coefficients and degree m, then we consider the differential operator of order m associated to P,
where \(D^{\alpha } ={\partial ^{\alpha _1}_x}{\partial ^{\alpha _2}_y}{\partial ^{\alpha _3}_t}\) and \(|\alpha |=\alpha _{1}+\alpha _{2}+\alpha _3\). By definition
Using a particular version of the Treves’ inequality, we will establish a Carleman estimate for the differential operator \(\mathcal {L}\) defined as
where \(f_{j}=f_{j}(x,y,t),\) for \(j=1,2,3\) and the operators \(P_{j},\) \(j=1,2,3,4\) are defined by
and
Theorem 2.1
(Treves’ Inequality) Let \(P(D)=P\left( {\partial _x},\partial _y,{\partial _t}\right) \) be a differential operator of order m with constant coefficients. Then for all \(\xi =(\xi _1,\xi _2,\xi _3)\in \mathbb {R}^3,\) \(\alpha =(\alpha _{1},\alpha _{2},\alpha _3)\in \mathbb {N}^{3}\) and \(\Psi \in C^{\infty }_{0}(\mathbb {R}^{3})\) we have that
where
Proof
See Theorem 2.4 in Treves (1966). \(\square \)
Corollary 2.2
Let \(P(D)=P\left( {\partial _x},\partial _y,{\partial _t}\right) \) be a differential operator of order m with constant coefficients. Then for all \(\alpha =(\alpha _{1},\alpha _{2},\alpha _3)\in \mathbb {N}^{3}\), \(\delta >0,\) \(\tau >0\), \(\Psi \in C^{\infty }_{0}(\mathbb {R}^{3})\) and \(\psi (x,y,t)=(x-\delta )^{2}+(y-\delta )^{2}+\delta ^{2}t^{2}\) we have that
with
Proof
We will use the above theorem with the differential operator
where
Then, using inequality (5) we have that
for all \(\Psi \in C^{\infty }_{0}(\mathbb {R}^{3}),\) \(\alpha =(\alpha _1,\alpha _2,\alpha _3)\in \mathbb {N}^3\) and any \(\tau >0.\) Now, multiply both sides of the previous inequality by \(e^{4\tau \delta ^2}\) we obtain
In particular, we can choose \(\Psi =\widetilde{\Psi }e^{-2\tau \delta (x+y)}\) where \(\widetilde{\Psi }\in C^{\infty }_{0}(\mathbb {R}^3).\) Observing that
and also that
we obtain
\(\square \)
Now we present the Carleman estimate for the differential operator \(\mathcal L\).
Theorem 2.3
Let \(\mathcal L\) the differential operator defined in (4), where \(c_{1}, c_{2}, c_3, c_4\) are real constants and \(f_{1}, f_{2}, f_{3}, f_4, f_5 \in L^{\infty }_{loc}(\mathbb {R}^{3})\). Let \(\delta > 0\) and
Then, there exists \(C > 0\) such that for all \(\Psi = (\Psi _1, \Psi _2)\in C^{\infty }_{0}(B_{\delta })\times C^{\infty }_{0}(B_{\delta })\) and \(\tau >0\) with
and
we have that
Proof
Let \(\Psi =(\Psi _{1},\Psi _{2})\in C^{\infty }_{0}(B_{\delta })\times C^{\infty }_{0}(B_{\delta }).\) Consider the polynomial
and
the differential operator associated to \(P_1\). Then, if \(\alpha =(1,0,1)\) we have that
and
Thus, using Theorem 2.1 we see that
Now, if \(\alpha =(0,1,1)\) we have that
and also
So, using Theorem 2.1 we see that
Moreover,
Then, using again the Theorem 2.1 we obtain that
Now, by defining
we have that
and
In a similar fashion
Hence, we see that
and also that
By considering
and
we have that
Then, using Theorem 2.1 we obtain that
and
and also that
Finally, we see that
Then, using Theorem 2.1 we obtain that
From (8)–(17), there is \(C>0\) such that
Now, we note that
implies that
Then, using inequalities (8)–(9), we have that
In a similar way, for
we obtain, using (14) and (17), that
and also, using (15)–(16), we have that
Next, if we choose \(\tau > 0\) large enough such that
and
then from inequalities (19)–(21) we have that
what implies
where
Therefore
Hence, from previous inequality and (18) we obtain the estimate (7). \(\square \)
Remark 2.4
The estimate (7) is invariant under changes of signs on the components of \(\mathcal {L}\).
Corollary 2.5
Let \(T>0\). Assume that in addition to the hypotheses of the Theorem 2.3 we have that
and the support of \(\eta \) and support of \(\Phi \) are compact contained in \(B_{\delta }.\) Then, the inequality (7) holds if we replace \(\Psi =(\Psi _1, \Psi _2)\) by \(U=(\eta , \Phi ).\) Indeed,
Proof
Let \(\{\rho _{\epsilon }\}_{\epsilon >0}\) be a regularizing sequence (in three variables) and consider
where \(*\) denotes the usual convolution. Then we have that \(U_{\epsilon }\in C^{\infty }_{0}( B_{\delta })\times C^{\infty }_{0}( B_{\delta })\) and the inequality (7) holds for \(U_{\epsilon },\) that is,
Now, for \(n=0,1\) and \(m=0,1,2\) we have that
and
where C is a positive constant depending only on \(\tau \) and \(\delta \). Similarly we have that
which allows us to pass to the limit in (23) to conclude the proof of Corollary 2.5. \(\square \)
3 Unique Continuation
In this section, we prove the unique continuation result for the system (1). Before to do the proof, we establish the following results.
Lemma 3.1
Let \(T > 0\) and \(f_{1}, f_{2}, f_{3}, f_4, f_5 \in L^ {\infty }_{loc}(\mathbb {R}^2\times (-T,T))\). Let \(U=(\eta ,\Phi )\) with
be a solution of \(\mathcal {L}U = 0\) in \(\mathbb {R}^2\times (-T,T)\) where \(\mathcal {L}\) is the differential operator defined in (4). Let
Suppose that \(\widetilde{U}\equiv 0\) in the region \(\{(x,y,t)\, : \, x<t, \ y<t\}\) intercepted with a neighborhood of (0, 0, 0). Then there exists a neighborhood \(\mathcal {O}_1\) of (0, 0, 0) (in the space xyt) such that \(\widetilde{U}\equiv 0\) in \(\mathcal {O}_{1}.\)
Proof
By hypotheses there is \( 0<\delta <1\) such that \( \widetilde{U}\equiv 0\) in \(R_\delta =R_1\cup R_2\), where
Next, consider \(\chi \in C^{\infty }_{0}(B_{\delta })\) such that \(\chi = 1\) in a neighborhood \(\mathcal {O}\) of (0, 0, 0) and define
Then we have that
and
By using the definition of \(\chi \), we note that \(\mathcal {L}\Psi = 0\) in \(\mathcal {O}\). Thus, using the Corollary 2.5, we have for \(\psi (x,y,t)=(x-\delta )^{2}+(y-\delta )^{2}+\delta ^{2}t^{2}\) and \(\tau >0\) large enough that
Now, using again the definition of \(\chi \) and the fact that \( \widetilde{U}\equiv 0\) in \(R_\delta \), we see that
It follows that if \((x,y, t)\ne (0,0,0)\) and \((x,y,t)\in D\) then
Thus, there exists \(0<\epsilon <2\delta ^{2}\) such that
Moreover, since \(\psi (0,0,0)=2\delta ^{2}\), we can choose \(\mathcal {O}_1\subset \mathcal {O} \) a neighborhood of (0, 0, 0) such that
From the above construction and inequality (24), we have that there exists \(C_1>0\) such that
Therefore
Then, passing to the limit as \(\tau \rightarrow +\infty \), we have that \(\Psi \equiv 0\) in \(\mathcal {O}_1.\) Since \(\widetilde{U}=\Psi \) in \(\mathcal {O}\) and \(\mathcal {O}_1 \subset \mathcal {O}\), we see that \(\widetilde{U}=0\) in \(\mathcal {O}_1\). \(\square \)
Similarly, we also have the following result.
Lemma 3.2
Let \(T > 0\) and \(f_{1}, f_{2}, f_{3}, f_4, f_5 \in L^ {\infty }_{loc}(\mathbb {R}^2\times (-T,T))\). Let \(U=(\eta ,\Phi )\) with
be a solution of \(\mathcal {L}U = 0\) in \(\mathbb {R}^2\times (-T,T)\) where \(\mathcal {L}\) is the differential operator defined in (4). Let
Suppose that \(\,\widetilde{U}\equiv 0\,\) in the region \(\{\,(x,y,t) \ \, : \ \, x<- t, \ y<- t\,\}\) intercepted with a neighborhood of (0, 0, 0). Then there exists a neighborhood \(\,\mathcal {O}_{2}\,\) of \(\,(0,0, 0)\,\) (in the space xyt) such that \(\widetilde{U}\equiv 0\) in \(\,\mathcal {O}_{2}.\)
Corollary 3.3
Let \(T > 0\) and \(F_1, F_2, F_3, F_4, F_5 \in L^{\infty }_{loc}(\mathbb {R}^2\times (-T,T))\). Let \(U=(\eta ,\Phi )\) with
be a solution in \(\mathbb {R}^2\times (-T,T)\) of the system
Let \(\gamma \) be a sphere passing through the origin (0, 0, 0). Suppose that \(U\equiv 0\) in the interior of \(\gamma \) in a neighborhood of (0, 0, 0). Then, there exists a neighborhood of (0, 0, 0) where \(U\equiv 0\).
Proof
Let us assume that the sphere (a piece of it) \(\gamma \) is given by \((x,y) = (g_1(t),g_2(t))\). By using the hypotheses, we have that \(U\equiv 0\) in the region \(\{(x,y, t)\, : \, x<g_1(t), \, y<g_2(t)\}\) intercepted with a neighborhood of (0, 0, 0). Then, we can to see that there exists \(\omega _1, \omega _2\in \mathbb {R}\setminus \{0,1\}\) such that \(U\equiv 0\) in a neighborhood of (0, 0, 0) intercepted with the region \(\{(x,y, t)\, : \, x<h_1(t), \, y<h_2(t)\}\) where
Now, we consider the following change of variables \((x,y,t)\rightarrow (X,Y,T)\) with
Notice that in the new variables, if \(T\ge 0\) then the function
is a solution of the system
Then, \(U\equiv 0\) in the region \(\{(X,Y,T)\, : \,X<T , \, T<T \ T\ge 0 \}\) intercepted with a neighborhood of (0, 0, 0) and U satisfies
where
with
and
and also
So, using Lemma 3.1 with the previous differential operator \(\mathcal {L}\), we obtain that there exists a neighborhood \(\mathcal {O}_{1}\) of (0, 0, 0) in the space XYT where \(U\equiv 0.\)
In a similar fashion, \(U\equiv 0\) in the region \(\{(X,Y,T)\, : \,X<-T , \ Y<-T, \ T<0 \}\) intercepted with a neighborhood of (0, 0, 0) and U satisfies
where
Then, from Lemma 3.2 we have that there exists a neighborhood \(\mathcal {O}_{2}\) of (0, 0, 0) in the space XYT where \(U\equiv 0.\) Thus, returning to the original variables (x, y, t) we have the result. \(\square \)
Now we have the main result on the unique continuation property for the system (1).
Theorem 3.4
Let \(T > 0\) and \((\eta ,\Phi )=(\eta (x,y,t),\Phi (x,y,t))\) with
be a solution in \(\mathbb {R}^2\times (-T,T)\) of the system (1). If \((\eta ,\Phi )\equiv 0\) in an open subset \(\Omega \) of \(\mathbb {R}^2 \times (-T,T)\), then \((\eta ,\Phi )\equiv 0\) in the horizontal component of \(\Omega \).
Proof
By defining the functions
and
the system (1) takes the form
with \(F_1, F_2, F_3, F_4, F_5 \in L^{\infty }_{loc}(\mathbb {R}^2\times (-T,T))\) and \(a=c=\frac{\mu }{2},\) \(b=\frac{2\mu }{3},\) \(d=\mu \sigma .\) Then, we will show the result for the system (25).
Denote by \(\Omega _{1}\) the horizontal component of \(\Omega \) and let
Let \(Q\in \Omega _1\) arbitrary. Choose \(P\in \Lambda \) and let \(\Gamma \) be a continuous curve contained in \(\Omega _1\) joining P to Q, parametrized by a continuous function \(f: [0,1] \rightarrow \Omega _1\) with \(f(0) = P\) and \(f(1) = Q.\) Since \(P\in \Lambda \), there exists \(r>0\) such that
Taking \(0<r_0 <\min \{r, dist(\Gamma , \partial \Omega _{1})\},\) where \(\partial \Omega _1\) denotes the boundary of \(\Omega _1,\) we have that
Now, if \(r_1 < \frac{r_0}{4} \) we see that
in fact, if \(w\in B_{2r_1}(f(s))\) and \(w\notin \Omega _1\) then
which is a contradiction.
Next, let
and
We will prove that \(f(\ell _0)\in \Lambda _1.\) If \(w\in B_{r_1}(f(\ell _0))\) and \(r_2=\Vert w-f(\ell _0)\Vert \) then there exists \(0<\delta < \ell _0\) such that \( \Vert f(\ell _0)-f(\ell _0 -\delta )\Vert <r_1-r_2. \) Therefore
and so \(w\in B_{r_{1}}(f(\ell _0 -\delta )).\) Now, from the definition of \(\ell _0\) there exists \(\ell _{\delta }\in S\) such that \(\ell _{0}-\delta <\ell _{\delta }\le \ell _{0}\), what implies \(f(\ell _0 -\delta )\in \Lambda _1.\) Then, using (27) we see that
Consequently we obtain that \((\eta (w),\Phi (w))= 0\) and then
Hence, we have showed \(f(\ell _0)\in \Lambda _1.\)
If \(\ell _0=1\) then from previous analysis we have that \(Q=f(1)\in \Lambda _1\subset \Lambda \). Thus, since Q was arbitrarily chosen we obtain that \((\eta ,\Phi )\equiv 0\) in \(\Omega _1,\) which proves Theorem 3.4. Then to finish the proof of Theorem 3.4 remains to prove that \(\ell _{0}=1\). In fact, let us suppose that \(\ell _0 < 1\) and let
For \(w=(x_1,y_1,t_1)\in G\) fixed, we consider the change of variable \((x,y,t)\rightarrow (X,Y,T)\) where
Notice that \((0,0,0)\in G^*=\{Z=(X,Y,T)\, : \, \Vert Z-(f(\ell _{0})-w)\Vert = {r_{1}} \}\). Moreover, from (29) we see that
So that, by using Corollary 3.3, there exists \(r^{*}_w >0\) such that
Returning to the original variables we have that for each \(w\in G\) there exists \(r^*_w>0\) such that
Then, using (29) and the compactness of G, we have that there is \(\epsilon _{1}>0\) such that
Now, we note that there exists \(0<\delta _1<1-\ell _0\) such that if \(w\in B_{{r_1}}(f(\ell _{0}+\delta _1))\) then
Thus, \(w\in B_{{r_1}+\epsilon _{1}}(f(\ell _{0}))\) and so \( B_{{r_1}}(f(\ell _{0}+\delta _1))\subset B_{{r_1}+\epsilon _{1}}(f(\ell _{0})). \) Therefore, using (30) we have that \((\eta ,\Phi )\equiv 0\) in \(B_{{r_1}}(f(\ell _{0}+\delta _1)).\) Consequently \(f(\ell _0+\delta _1)\in \Lambda _1\), which contradicts the definition of \(\ell _0.\) So, \(\ell _0=1\) and the proof of Theorem 3.4 is complete. \(\square \)
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Acknowledgements
A. Montes and R. Córdoba were supported by University of Cauca under Grant ID 5845; Math Amsud and Minciencias-Colombia under Grant MATHAMSUD 21-MATH-03.
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Montes, A.M., Córdoba, R. A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves. Bull Braz Math Soc, New Series 54, 32 (2023). https://doi.org/10.1007/s00574-023-00336-w
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DOI: https://doi.org/10.1007/s00574-023-00336-w