On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

Abstract

In this work we investigate unique continuation properties of solutions to the initial value problem associated to the Benjamin–Ono equation in weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb R )\cap L^2(|x|^{2r}dx)$ for $s\in \mathbb R$, and $s\ge 1$, $s\ge r$. More precisely, we prove that the uniqueness property based on a decay requirement at three times can not be lowered to two times even by imposing stronger decay on the initial data.

Appendix: Existence of Infinitely Many Data that Satisfy Given Conditions of Theorem 1

We will show that there exists data in the Schwartz class $u_0\in \mathcal S (\mathbb R )$ such that the following conditions hold (see Eqs. 3.8 and 4.6):

$$\begin{aligned} \Phi _2(u_0,t)=4t^2I_3(u_0)+6t\int xu_0^2(x)dx+12\int x^2 u_0(x)dx=0, \end{aligned}$$

and

$$\begin{aligned} \Phi _1(u_0,t)=\int xu_0(x)dx +\frac{t}{4}\int u_0^2(x)dx=0, \end{aligned}$$

for some time, $t^*\ne 0$ where

$$\begin{aligned} I_3(u_0)=\int |D^{1/2}u_0(x)|^2+\frac{u_0(x)^3}{3} ~dx. \end{aligned}$$

Consider linearly independent Schwartz class functions, $f_i\in \mathcal S (\mathbb R )$ for $i=1,2,3,4$ such that the following are true:

$$\begin{aligned} \begin{array}{lll} \int f_1(x)dx=1, &{} \int xf_1(x)dx=0, &{} \int x^2f_1(x)dx=0, \\ \int f_2(x)dx=0, &{} \int xf_2(x)dx=1, &{} \int x^2f_2(x)dx=0, \\ \int f_3(x)dx=0, &{} \int xf_3(x)dx=0, &{} \int x^2f_3(x)dx=1. \\ \end{array} \end{aligned}$$

An application of the Paley–Weiner theorem will guarantee that such functions satisfying these conditions exist. Next, we define $\Phi _i(x):\mathbb R ^5\longrightarrow \mathbb R$ for $i=0,1,2$ as:

$$\begin{aligned} \Phi _0(\alpha _1,\alpha _2,\alpha _3,\alpha _4,t)&= \int \sum \limits _{j=1}^4 \alpha _jf_j ~dx \\ \Phi _1(\alpha _1,\alpha _2,\alpha _3,\alpha _4,t)&= \int x\left( \sum \limits _{j=1}^4 \alpha _jf_j \right) ~dx +\frac{t}{4} \int \left( \sum \limits _{j=1}^4 \alpha _jf_j \right) ^2dx \\ \Phi _2(\alpha _1,\alpha _2,\alpha _3,\alpha _4,t)&= 8t^2 I_3\left( \sum \limits _{j=1}^4 \alpha _jf_j \right) +6t \int x\left( \sum \limits _{j=1}^4 \alpha _jf_j \right) ^2dx\\&+12\int x^2\left( \sum \limits _{j=1}^4 \alpha _jf_j \right) dx. \end{aligned}$$

We apply the Implicit Function Theorem (IFT) to $F:\mathbb R ^5\longrightarrow \mathbb R ^3$ where

$$\begin{aligned} F(\alpha _1,\alpha _2,\alpha _3,\alpha _4,t)=(\Phi _0,\Phi _1,\Phi _2). \end{aligned}$$

We observe that $F({0})={{0}}$ and that

$$\begin{aligned} DF(\vec {0})=\left. \left( \frac{\partial \Phi _i}{\partial \alpha _j}\right) \right| _{(\alpha _1,\alpha _2,\alpha _3\alpha _4,t)=\vec {0}}= \left( \begin{array}{ccc|cc} 1 &{} 0 &{} 0 &{}*&{}*\\ 0 &{} 1 &{} 0 &{}*&{}*\\ 0 &{} 0 &{} 12 &{}*&{}*\end{array} \right) , \end{aligned}$$

where the asterisks, $*$, denote values that are not relevant for the application of the IFT. Thus we can solve for $\alpha _j$ for $j=1,2,3$ implicitly in terms of $\alpha _4$ and $t$ in a neighborhood of the origin, i.e.,

$$\begin{aligned} F(\alpha _1(\alpha _4,t),\alpha _2(\alpha _4,t),\alpha _3(\alpha _4,t),\alpha _4,t)=\vec {0}, \end{aligned}$$

for values of $\alpha _4$ and $t$ that are sufficiently small. Furthermore, there are infinitely many data $u_0=\sum _{i=1}^4\alpha _if_i$ such that

$$\begin{aligned} \widehat{u_0}(0)&= 0,\\ \Psi _1(u_0,t^*)&= 6t\delta (\xi )\Phi _1(u_0,t^*)=0, \end{aligned}$$

and

$$\begin{aligned} \Psi _2(u_0,t^*)=it^2\delta (\xi )\Phi _2(u_0,t^*)=0, \end{aligned}$$

which is the desired result.