A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves

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 (η,Φ)=(η(x,y,t),Φ(x,y,t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\eta ,\Phi ) = (\eta (x,y, t),\Phi (x,y, t))$$\end{document} is a solution of the nonlinear system, in a suitable function space, and (η,Φ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\eta ,\Phi )$$\end{document} vanishes on an open subset Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} of R2×[-T,T],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2 \times [-T,T],$$\end{document} then (η,Φ)≡0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\eta ,\Phi )\equiv 0$$\end{document} in the horizontal component of Ω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega .$$\end{document} To state such property, we use a Carleman-type estimate for a differential operator L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document} related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality.


Introduction
The focus of the present work is the following two-dimensional system ⎧ ⎨ ⎩ I − μ 2 η t + − 2μ 3 2 + ∇ · (η∇ ) = 0, that describes the evolution of long water waves with small amplitude in the presence of surface tension (see Quintero and Montes 2013). Here, is the amplitude parameter (nonlinearity coefficient), μ is the long-wave parameter (dispersion coefficient), σ is the inverse of the Bond number (associated with the surface tension) and the functions η = η(x, y, t) and = (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 H = H(t) is defined as H η = 1 2 R 2 η 2 + μσ |∇η| 2 + |∇ | 2 + 2μ 3 | | 2 + η|∇ | 2 dxdy, and the Hamiltonian type structure is given by We see directly that the functional H is well defined when η, ∇ ∈ H 1 (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 θ > 0, in the energy space H 1 (R 2 ) × V(R 2 ), where H 1 (R 2 ) is the usual Sobolev space of order 1 and the space V(R 2 ) is defined with respect to the norm given by w 2 x + w 2 y + w 2 x x + 2w 2 xy + w 2 yy dxdy.
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 (R 2 ) × V s (R 2 ), s ≥ 2, where H s (R 2 ) is the usual Sobolev space of order s defined as the completion of the Schwartz class with respect to the norm and V s (R 2 ) denotes the completion of the Schwartz class with respect to the norm where w is the Fourier transform of w defined on 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 (η, ) = (η(x, y, t), (x, y, t)) is a solution of the system (1) in a suitable function space, 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, u t + uu x + u x x x = 0, 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 , then u ≡ 0 in the horizontal component of . 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 ≡ 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, then u ≡ 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 ≡ 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 ≡ u 2 . This kind of uniqueness results has been deduced under the assumption that the solutions coincide in a large sub-domain of 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, then u ≡ 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 x 2 +y 2 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, b) 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 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 L we also prove that if a solution vanishes in a ball in the x yt 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 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).

Carleman Estimate
In this section, we will use the notation D = ∂ x , ∂ y , ∂ t . If P = P(ξ 1 , ξ 2 , ξ 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, x ∂ α 2 y ∂ α 3 t and |α| = α 1 + α 2 + α 3 . By definition Using a particular version of the Treves' inequality, we will establish a Carleman estimate for the differential operator 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

be a differential operator of order m with constant coefficients. Then for all
where Proof See Theorem 2.4 in Treves (1966).
Proof We will use the above theorem with the differential operator where Then, using inequality (5) we have that ∈ N 3 and any τ > 0. Now, multiply both sides of the previous inequality by e 4τ δ 2 we obtain In particular, we can choose = e −2τ δ(x+y) where ∈ C ∞ 0 (R 3 ). Observing that and also that Now we present the Carleman estimate for the differential operator L.
Hence, from previous inequality and (18) we obtain the estimate (7).

Remark 2.4
The estimate (7) is invariant under changes of signs on the components of 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 η and support of are compact contained in B δ . Then, the inequality (7) holds if we replace = ( 1 , 2 ) by U = (η, ). Indeed, Proof Let {ρ } >0 be a regularizing sequence (in three variables) and consider where * denotes the usual convolution. Then we have that U ∈ C ∞ 0 (B δ ) × C ∞ 0 (B δ ) and the inequality (7) holds for U , that is, Now, for n = 0, 1 and m = 0, 1, 2 we have that where C is a positive constant depending only on τ and δ. Similarly we have that which allows us to pass to the limit in (23) to conclude the proof of Corollary 2.5.

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.
Suppose that U ≡ 0 in the region {(x, y, t) : x < t, y < t} intercepted with a neighborhood of (0, 0, 0). Then there exists a neighborhood O 1 of (0, Next, consider χ ∈ C ∞ 0 (B δ ) such that χ = 1 in a neighborhood O of (0, 0, 0) and define Then we have that By using the definition of χ , we note that L = 0 in O. Thus, using the Corollary 2.5, we have for ψ(x, y, t) = (x − δ) 2 + (y − δ) 2 + δ 2 t 2 and τ > 0 large enough that Now, using again the definition of χ and the fact that U ≡ 0 in R δ , we see that It follows that if (x, y, t) = (0, 0, 0) and (x, y, t) ∈ D then Thus, there exists 0 < < 2δ 2 such that Moreover, since ψ(0, 0, 0) = 2δ 2 , we can choose O 1 ⊂ 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 Then, passing to the limit as τ → +∞, we have that Similarly, we also have the following result.
Proof Let us assume that the sphere (a piece of it) γ is given by (x, y) = (g 1 (t), g 2 (t)). By using the hypotheses, we have that U ≡ 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 ω 1 , ω 2 ∈ R \ {0, 1} such that U ≡ 0 in a neighborhood of (0, 0, 0) intercepted with the region {(x, y, t) : Now, we consider the following change of variables (x, y, t) → (X , Y , T ) with Notice that in the new variables, if T ≥ 0 then the function is a solution of the system intercepted with a neighborhood of (0, 0, 0) and U satisfies and and also So, using Lemma 3.1 with the previous differential operator L, we obtain that there exists a neighborhood O 1 of (0, 0, 0) in the space XY T where U ≡ 0. In a similar fashion, U ≡ 0 in the region {(X , Y , T ) : X < −T , Y < −T , T < 0} intercepted with a neighborhood of (0, 0, 0) and U satisfies Then, from Lemma 3.2 we have that there exists a neighborhood O 2 of (0, 0, 0) in the space XY T where U ≡ 0. Thus, returning to the original variables (x, y, t) we have the result. Now we have the main result on the unique continuation property for the system (1).
the system (1) takes the form Then, we will show the result for the system (25).
Let Q ∈ 1 arbitrary. Choose P ∈ and let be a continuous curve contained in 1 joining P to Q, parametrized by a continuous function f : [0, 1] → 1 with f (0) = P and f (1) = Q. Since P ∈ , there exists r > 0 such that (η, ) ≡ 0 in B r (P).
Returning to the original variables we have that for each w ∈ G there exists r * w > 0 such that (η, ) ≡ 0 in B r * w (w).