1 Introduction and Main Results

The one-dimensional fully dispersive Boussinesq system is governed by

$$\begin{aligned} \left\{ \begin{aligned} \partial _t\eta&=-\mathcal {K}\partial _xu-\partial _x(\eta u),\\ \partial _tu&=-\partial _x\eta -u\partial _xu, \end{aligned} \right. \end{aligned}$$
(1.1)

here \(t,x\in {\mathbb {R}}\), and the surface elevation \(\eta \) and the velocity \(u\) at the surface are real functions. We also consider the two-dimensional fully dispersive Boussinesq system which reads as:

$$\begin{aligned} \left\{ \begin{aligned} \partial _t\eta&=-\mathcal {K}\mathrm {div}\varvec{v}-\mathrm {div}(\eta \varvec{v}),\\ \partial _t\varvec{v}&=-\nabla \eta -\frac{1}{2}\nabla |\varvec{v}|^2, \end{aligned} \right. \end{aligned}$$
(1.2)

in which \(t\in {\mathbb {R}}, x=(x_1,x_2)\in {\mathbb {R}}^2\) and the unknowns \(\eta \in {\mathbb {R}},\varvec{v}=(v_1,v_2)\in {\mathbb {R}}^2\). The operator \(\mathcal {K}\) is a Fourier multiplier with symbol \(m\in S_\infty ^a({\mathbb {R}}^d),\ d=1\ \text {or}\ 2\) for some \(a\in {\mathbb {R}}\setminus \{0\}\), that is

$$\begin{aligned} \mathcal {F}(\mathcal {K}f)(\xi )=m(\xi )\hat{f}(\xi ),\quad \forall f\in \mathcal {S}({\mathbb {R}}^d), \end{aligned}$$

for some smooth even function \(m:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\) with the properties

$$\begin{aligned} m(\xi )\eqsim (1+|\xi |)^a, \end{aligned}$$
(1.3)

and

$$\begin{aligned} |m^{\prime }(\xi )|\lesssim (1+|\xi |)^{a-1}. \end{aligned}$$
(1.4)

The system (1.1) was formally derived in [1, 16] from the incompressible Euler equations to model fully dispersive water waves whose propagation is allowed to be both left- and rightward, and was also proposed in [13, 14, 19] (as well as the system (1.2)) as a full dispersion system in the Boussinesq regime with the dispersion of the water waves system. This model is the two-way equivalent of the one-dimensional Whitham equation

$$\begin{aligned} \partial _tu=-\mathcal {K}^{1/2}\partial _xu-u\partial _xu, \end{aligned}$$

with \(m(\xi )=\tanh (\xi )/\xi \), which has taken a vast of attractions (we refer to [7] for a fairly complete list of references). There have been several investigations on the system (1.1) with \(m(\xi )=\tanh (\xi )/\xi \) [which corresponds to \(a=-1\) in (1.3)]: local well-posedness [13, 18], a logarithmically cusped wave of greatest height [6], existence of solitary wave solutions [17], and numerical results [3, 4, 21]. We should point out that the assumption in [13, 18] on the initial surface elevation \(\eta _0\ge C>0\) is nonphysical, which only yields the well-posedness in homogeneous Sobolev spaces, however the Cauchy problem of (1.1) by this choice of symbol is probably ill-posed for negative initial surface elevation (see a heuristic argument in [13]).

When \(a>0\) in (1.3), because of the strong dispersion, the system (1.1) exhibits different phenomenon regarding well-posedness. Very recently, considering the effect of surface tension, i.e., \(m(\xi )=(\tanh (|\xi |)/|\xi |)(1+|\xi |^2)\) [which corresponds to \(a=1\) in (1.3)], the well-posedness in Sobolev spaces was shown in [10] for the Cauchy problem of the system (1.1), as well as (1.2) with curl free initial velocity \(\varvec{v}_0\). Due to the lack of symmetry of the nonlinearity, the usual energy estimates only give

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}}{\mathrm{d} t}(\Vert \eta \Vert _{H^s}^2+\Vert u\Vert _{H^{s+\frac{1}{2}}}^2)&\lesssim (1+\Vert \eta \Vert _{H^s}+\Vert u\Vert _{H^{s+\frac{1}{2}}})(\Vert \eta \Vert _{H^s}^2+\Vert u\Vert _{H^{s+\frac{1}{2}}}^2)\\&\quad +\left| \int \eta J^s\partial _xuJ^s\eta \,\mathrm {d}x\right| . \end{aligned} \end{aligned}$$
(1.5)

To close the resulting energy estimate, the crucial idea is to define a modified energy, i.e., adding a lower order cubic term \(\int \eta (J^s\partial _xu)^2\,\mathrm {d}x\) (which has no fixed sign) to the LHS of (1.5). Another important observation is that the non-cavitation assumption on \(\eta \) (which means that the surface elevation of the wave cannot touch the bottom of the fluid) allows \(\Vert u\Vert _{H^{s+\frac{1}{2}}}^2\) to control the part \(\Vert u\Vert _{H^{s+\frac{1}{2}}}^2+\int \eta (J^s\partial _xu)^2\,\mathrm {d}x\) of the modified energy from below. Besides, the authors also used quite a few tricks such as some operator decompositions (to reformulate the system (1.1)), and the estimate comparing the Bessel and Riesz potentials, and so on. Please refer to [5, 8, 9, 12] for other applications of modified energy approaches in some related contexts of dispersive equations. We also refer to [13, 15] for discussions and interesting issues about the influence of dispersion on the lifespan of solutions to dispersive perturbations of hyperbolic quasi-linear equations or systems which typically arise in water waves theory.

Inspired by [10, 13], our purpose is to understand the modified energy approach in a different point view. Our idea is to use symmetrization of hyperbolic systems, which will help us to find the modified energy easily. The energy estimates are straightforward and yield

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d} t}\int \left( (J^s\eta )^2+\eta (J^su)^2+(\sqrt{\mathcal {K}}J^su)^2\right) \, \mathrm {d}x\nonumber \\&\quad \lesssim (\Vert \eta \Vert _{H^s}+\Vert u\Vert _{H^s}+\Vert \eta \Vert _{H^s}\Vert u\Vert _{H^s})^3+\Vert u\Vert _{H^s}\Vert u\Vert _{H^{s+\frac{a}{2}}}^2. \end{aligned}$$
(1.6)

The integration in the LHS of (1.6) is exactly the modified energy that we need. Notice that

$$\begin{aligned} \int (\sqrt{\mathcal {K}}J^su)^2\, \mathrm {d}x\eqsim \Vert u\Vert _{H^{s+\frac{a}{2}}}^2, \end{aligned}$$

so we see that this modified energy is essentially same to the one in [10]. On the other hand, this zeroth-order [\(s=0\)] modified energy is corresponding to the Hamiltonian of the system (1.1) (up to a constant coefficient). We also refer to [20] for using the idea of symmetrization to study dispersive systems.

We start from recalling the definition of the non-cavitation:

Definition 1.1

([10]) Let \(d=1\) or \(2\) and \(k>\frac{d}{2}\). We say that the initial elevation \(\eta _0\in H^k({\mathbb {R}}^d)\) satisfies the non-cavitation condition if there exists \(h_0\in (0,1)\) such that

$$\begin{aligned} 1+\eta _0(x) \ge h_0, \quad x\in {\mathbb {R}}^d. \end{aligned}$$
(1.7)

We then have similar results to [10] under the non-cavitation assumption:

Theorem 1.2

Let \(a>0\) and \(s>\frac{3}{2}+a\). Assume that \((\eta _0,u_0)\in H^s({\mathbb {R}})\times H^{s+\frac{a}{2}}({\mathbb {R}})\) satisfy the non-cavitation condition (1.7). Then there exist a positive number

$$\begin{aligned} T=T(\Vert (\eta _0,u_0)\Vert _{H^s({\mathbb {R}})\times H^{s+\frac{a}{2}}({\mathbb {R}})}) \end{aligned}$$

and a unique solution

$$\begin{aligned} (\eta ,u)\in C\big ([0,T];H^s({\mathbb {R}})\times H^{s+\frac{a}{2}}({\mathbb {R}})\big ) \end{aligned}$$

to (1.1) with \((\eta ,u)(0,x)=(\eta _0,u_0)(x)\). In addition, the flow function that maps initial data to solutions is continuous.

Theorem 1.3

Let \(a>0\) and \(s>2+a\). Assume that \((\eta _0,\varvec{v}_0)\in H^s({\mathbb {R}}^2)\times H^{s+\frac{a}{2}}({\mathbb {R}}^2)^2\) satisfy the non-cavitation condition (1.7) and \(\mathrm {curl} \varvec{v}_0=0\). Then there exist a positive number

$$\begin{aligned} T=T(\Vert (\eta _0,\varvec{v}_0)\Vert _{H^s({\mathbb {R}}^2)\times H^{s+\frac{a}{2}}({\mathbb {R}}^2)^2}) \end{aligned}$$

and a unique solution

$$\begin{aligned} (\eta ,\varvec{v})\in C\big ([0,T];H^s({\mathbb {R}}^2)\times H^{s+\frac{a}{2}}({\mathbb {R}}^2)^2\big ) \end{aligned}$$

to (1.2) with \((\eta ,\varvec{v})(0,x)=(\eta _0,\varvec{v}_0)(x)\). In addition, the flow function that maps initial data to solutions is continuous.

To end this section, we include some notations frequently used throughout this paper. Let \(\langle x\rangle ^l =(1+|x|^2)^{\frac{l}{2}}\) and \(\widehat{J^lf}(\xi )=\langle \xi \rangle ^l\hat{f}(\xi )\), and denote by \(H^l\) the \(L^2\) based Sobolev space with the norm \(\Vert \cdot \Vert _{H^l}=\Vert J^l\cdot \Vert _{L^2}\). The notation \(C\) always denotes a nonnegative universal constant which may be different from line to line but is independent of the parameters involved. Otherwise, we will specify it by the notation \(C(a,b,\ldots )\). We write \(f\lesssim g\) [\(f\gtrsim g\)] when \(f\le Cg\) [\(f\ge Cg\)], and \(f \eqsim g\) when \(f \lesssim g \lesssim f\).

2 Proof of Theorem 1.2

2.1 Energy Estimates

Denote

$$\begin{aligned} E(\eta ,u)(t)=\int \left( (J^s\eta )^2+\eta (J^su)^2+(\sqrt{\mathcal {K}}J^su)^2\right) \, \mathrm {d}x. \end{aligned}$$
(2.1)

We priorly assume the solution \(\eta \) satisfies the non-cavitation assumption: there exists \(h_1\in (0,1)\) such that

$$\begin{aligned} 1+\eta (t,x) \ge h_1, \quad (x,t)\in {\mathbb {R}}\times [0,T], \end{aligned}$$
(2.2)

and the upper bound assumption: there exists \(h_2>0\) such that

$$\begin{aligned} \eta (t,x) \le h_2, \quad (x,t)\in {\mathbb {R}}\times [0,T], \end{aligned}$$
(2.3)

where the time \(T\) will be determined by the energy estimate below, and aim to show that there exists some appropriately small positive number \(T\) such that

$$\begin{aligned} E(\eta ,u)(t)\le C(T,a,h_1,h_2)E(\eta ,u)(0),\quad \forall t\in [0,T]. \end{aligned}$$
(2.4)

To prove (2.4), we will use the idea of symmetrization of hyperbolic systems, which will help us to find the modified energy \(E(\eta ,u)(t)\) in (2.1) easily. Let

$$\begin{aligned} A= \begin{bmatrix} 1&{}\quad 0\\ 0&{}\quad \eta \end{bmatrix},\ B= \begin{bmatrix} u&{}\quad \eta \\ \eta &{}\quad \eta u \end{bmatrix},\ U= \begin{bmatrix} \eta \\ u \end{bmatrix},\ G= \begin{bmatrix} \mathcal {K}\partial _xu\\ 0 \end{bmatrix}. \end{aligned}$$

Multiplying (1.1) by the matrix \(A\) yields the following system

$$\begin{aligned} \begin{aligned} A\partial _tU+B\partial _xU+G=0. \end{aligned} \end{aligned}$$
(2.5)

Applying \(J^s\) to (2.5) and multiplying it by \(J^sU\) give

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int J^sU\cdot A\cdot J^sU\, \mathrm {d}x\\&\quad =\underbrace{\frac{1}{2}\int J^sU\cdot \partial _tA \cdot J^sU\, \mathrm {d}x}_{I_1}-\underbrace{\int J^sU\cdot [J^s(A\partial _tU)-AJ^s\partial _tU]\, \mathrm {d}x}_{I_2}\\&\qquad -\underbrace{\int J^sU\cdot [J^s(B\partial _xU)-BJ^s\partial _xU]\, \mathrm {d}x}_{I_3}-\underbrace{\int J^sU\cdot B\cdot \partial _xJ^sU\, \mathrm {d}x}_{I_4}\\&\qquad -\underbrace{\int J^sU\cdot J^sG\, \mathrm {d}x}_{I_5}. \end{aligned} \end{aligned}$$
(2.6)

The terms \(I_2,I_3\) and \(I_4\) come from the hyperbolic parts [do not involve the Fourier multiplier \(\mathcal {K}\)], and therefore can be handled either by Kato–Ponce’s commutator estimates [11] or by integration by parts to obtain

$$\begin{aligned} \begin{aligned} |I_2|+|I_3|+|I_4|\lesssim (\Vert \eta \Vert _{H^s}+\Vert u\Vert _{H^s}+\Vert \eta \Vert _{H^s}\Vert u\Vert _{H^s})^3. \end{aligned} \end{aligned}$$
(2.7)

It remains to treat the terms \(I_1\) and \(I_5\). In view of (1.1)\(_1\), we estimate

$$\begin{aligned} \begin{aligned} |I_1|&=\frac{1}{2}\left| \int [\mathcal {K}\partial _xu+\partial _x(\eta u)](J^su)^2 \, \mathrm {d}x\right| \lesssim \Vert \mathcal {K}\partial _xu+\partial _x(\eta u)\Vert _{L^\infty }\Vert u\Vert _{H^s}^2\\&\lesssim (\Vert u\Vert _{H^{\frac{3}{2}+a+\delta }}+\Vert \eta \Vert _{H^{\frac{3}{2}+\delta }}\Vert u\Vert _{H^{\frac{3}{2}+\delta }})\Vert u\Vert _{H^s}^2 \lesssim (1+\Vert \eta \Vert _{H^s})\Vert u\Vert _{H^s}^3, \end{aligned} \end{aligned}$$
(2.8)

where we have used (1.3), and the embedding \(H^{\frac{1}{2}+\delta }({\mathbb {R}})\hookrightarrow L^\infty ({\mathbb {R}})\) with \(\delta \in (0,s-\frac{3}{2}-a)\). Using integration by parts, one calculates

$$\begin{aligned} \begin{aligned} I_5&=-\int \sqrt{\mathcal {K}}J^s\partial _x\eta \cdot \sqrt{\mathcal {K}}J^su\, \mathrm {d}x =\int \sqrt{\mathcal {K}}J^s(\partial _tu+u\partial _xu)\cdot \sqrt{\mathcal {K}}J^su\, \mathrm {d}x\\&=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int (\sqrt{\mathcal {K}}J^su)^2\, \mathrm {d}x+\underbrace{\int \sqrt{\mathcal {K}}J^s(u\partial _xu)\cdot \sqrt{\mathcal {K}}J^su\, \mathrm {d}x}_{I_6}, \end{aligned} \end{aligned}$$
(2.9)

where we have inserted the Eq. (1.1)\(_2\) into the second equality. To estimate the term \(I_6\), we commute \(u\) out as follows:

$$\begin{aligned} I_6=\underbrace{\int u\sqrt{\mathcal {K}}J^s\partial _xu\cdot \sqrt{\mathcal {K}}J^su\, \mathrm {d}x}_{I_7} +\underbrace{\int [\sqrt{\mathcal {K}}J^s,u]\partial _xu\cdot \sqrt{\mathcal {K}}J^su\, \mathrm {d}x}_{I_8}. \end{aligned}$$

The term \(I_7\), by integration by parts, can be easily controlled by

$$\begin{aligned} |I_7|=\frac{1}{2}\left| \int \partial _xu\big (\sqrt{\mathcal {K}}J^su\big )^2\, \mathrm {d}x\right| \lesssim \Vert u\Vert _{H^s}\Vert u\Vert _{H^{s+\frac{a}{2}}}^2. \end{aligned}$$
(2.10)

We next estimate the term \(I_8\). By the Plancherel’s theorem, we rewrite

$$\begin{aligned} \left\| [\sqrt{\mathcal {K}}J^s,u]\partial _xu\right\| _{L^2} =\left\| \int \left( \sqrt{m(\xi )}\langle \xi \rangle ^s-\sqrt{m(\eta )}\langle \eta \rangle ^s\right) \hat{u}(\xi -\eta )\widehat{\partial _xu}(\eta )\, \mathrm {d}\eta \right\| _{L_{\xi }^2}. \end{aligned}$$
(2.11)

Via the mean value theorem, there exists some \(\theta \in (0,1)\) such that

$$\begin{aligned} \sqrt{m(\xi )}\langle \xi \rangle ^s-\sqrt{m(\eta )}\langle \eta \rangle ^s=\left( \frac{m^{\prime }(\zeta )}{2\sqrt{m(\zeta )}}\langle \zeta \rangle ^s +s\sqrt{m(\zeta )}\langle \zeta \rangle ^{s-2}\zeta \right) (\xi -\eta ), \end{aligned}$$

with \(\zeta =\theta (\xi -\eta )+(1-\theta )\eta \). Then the above equality and the assumptions (1.3) and (1.4) imply

$$\begin{aligned} \begin{aligned}&\left| \sqrt{m(\xi )}\langle \xi \rangle ^s-\sqrt{m(\eta )}\langle \eta \rangle ^s\right| \lesssim \langle \zeta \rangle ^{s+\frac{a}{2}-1}|\xi -\eta |\\&\quad \lesssim \langle \xi -\eta \rangle ^{s+\frac{a}{2}-1}|\xi -\eta |+|\xi -\eta |\langle \eta \rangle ^{s+\frac{a}{2}-1}. \end{aligned} \end{aligned}$$
(2.12)

Notice that

$$\begin{aligned} \begin{aligned}&\left\| \int \big |\hat{f}(\xi -\eta )\widehat{g}(\eta )\big |\, \mathrm {d}\eta \right\| _{L_{\xi }^2} \le \int \left( \int \big |\hat{f}(\xi -\eta )\big |^2\, \mathrm {d}\xi \right) ^{1/2} \big |\widehat{g}(\eta )\big |\, \mathrm {d}\eta \\&\quad \lesssim \Vert f\Vert _{L^2} \left( \int (1+|\eta |^2)^{b}\big |\widehat{g}(\eta )\big |^2\, \mathrm {d}\eta \right) ^{1/2}\left( \int (1+|\eta |^2)^{-b}\, \mathrm {d}\eta \right) ^{1/2}\\&\quad \lesssim \Vert f\Vert _{L^2}\Vert g\Vert _{H^b}, \end{aligned} \end{aligned}$$

for any \(b>1/2\). Substituting (2.12) into (2.11) gives

$$\begin{aligned} \left\| [\sqrt{\mathcal {K}}J^s,u]\partial _xu\right\| _{L^2}\lesssim \Vert \partial _xu\Vert _{H^1}\Vert J^{s+\frac{a}{2}-1}\partial _xu\Vert _{L^2}, \end{aligned}$$

which leads to

$$\begin{aligned} |I_8|\lesssim \Vert u\Vert _{H^s}\Vert u\Vert _{H^{s+\frac{a}{2}}}^2. \end{aligned}$$
(2.13)

It follows from (2.6)–(2.10) and (2.13) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d} t}\int \left( (J^s\eta )^2+\eta (J^su)^2+(\sqrt{\mathcal {K}}J^su)^2\right) \, \mathrm {d}x\nonumber \\&\quad \lesssim (\Vert \eta \Vert _{H^s}+\Vert u\Vert _{H^s}+\Vert \eta \Vert _{H^s}\Vert u\Vert _{H^s})^3+\Vert u\Vert _{H^s}\Vert u\Vert _{H^{s+\frac{a}{2}}}^2. \end{aligned}$$
(2.14)

By the assumption (1.3), we have

$$\begin{aligned} c^{-1}\Vert u\Vert _{H^{s+\frac{a}{2}}}^2\le \int (\sqrt{\mathcal {K}}J^su)^2\, \mathrm {d}x\le c\Vert u\Vert _{H^{s+\frac{a}{2}}}^2. \end{aligned}$$
(2.15)

We then claim that

$$\begin{aligned} E(\eta ,u)(t)\eqsim _{a,h_1,h_2} \Vert \eta \Vert _{H^s}^2+\Vert u\Vert _{H^{s+\frac{a}{2}}}^2,\quad t\in [0,T]. \end{aligned}$$
(2.16)

The upper bound estimate of (2.16) is an easy consequence of (2.3). We now turn to the lower bound estimate of (2.16). The following observation is due to [10]. By the non-cavitation assumption (2.2), one may estimate

$$\begin{aligned} \begin{aligned}&\int \left( \eta (J^su)^2+(\sqrt{\mathcal {K}}J^su)^2\right) \, \mathrm {d}x\ge \int \left( \eta (J^su)^2+c^{-1}(J^{s+\frac{a}{2}}u)^2\right) \, \mathrm {d}x\\&\quad \ge h_1\int (1+\xi ^2)^s|\hat{u}(\xi )|^2\,\mathrm {d}\xi +\int \left( c^{-1}(1+\xi ^2)^{s+\frac{a}{2}}-(1+\xi ^2)^s\right) |\hat{u}(\xi )|^2\,\mathrm {d}\xi \\&\quad \ge h_1C(a)\int _{|\xi |\le \xi _0} (1+\xi ^2)^{s+\frac{a}{2}}|\hat{u}(\xi )|^2\,\mathrm {d}\xi +C(a)\int _{|\xi |\ge \xi _0} (1+\xi ^2)^{s+\frac{a}{2}}|\hat{u}(\xi )|^2\,\mathrm {d}\xi \\&\quad \ge C(a,h_1)\int (J^{s+\frac{a}{2}}u)^2\, \mathrm {d}x, \end{aligned} \end{aligned}$$

where \(\xi _0>0\) is a number depending on \(a\) and \(c\). This finishes the proof of (2.16). We then conclude from (2.14)–(2.16) that

$$\begin{aligned} \begin{aligned} E(\eta ,u)(t)\lesssim _{a,h_1,h_2} E(\eta ,u)(0)+\int _0^t \big (E(\eta ,u)(\tau )+E^3(\eta ,u)(\tau )\big )\, \mathrm {d}\tau , \end{aligned} \end{aligned}$$

which together with the Grönwall’s inequality completes the proof of (2.4).

2.2 Estimates for the Differences of Two Solutions

Assume \(i=1,2\). Let \((\eta _i,u_i)\) be the solutions of (1.1) with the initial data \(({\eta _i}_0,{u_i}_0)\) and satisfy the non-cavitation assumption (2.2) and the upper bound assumption (2.3). This subsection we instead consider the energy

$$\begin{aligned} \tilde{E}(t)=\int \left( (\eta _1-\eta _2)^2+\eta _1 (u_1-u_2)^2+\big (J^{\frac{a}{2}}(u_1-u_2)\big )^2\right) \, \mathrm {d}x, \end{aligned}$$

and aim to show

$$\begin{aligned} \tilde{E}(t)\le C\tilde{E}(0),\quad \forall t\in [0,T], \end{aligned}$$
(2.17)

where \(T\) is defined in (2.4) and \(C=C\big (T,a,h_1,h_2,E(\eta _1,u_1)(0),E(\eta _2,u_2)(0)\big )\).

We let

$$\begin{aligned} A_i= \begin{bmatrix} 1&{}\quad 0\\ 0&{}\quad \eta _i \end{bmatrix},\ B_i= \begin{bmatrix} u_i&{}\quad \eta _i\\ \eta _i&{}\quad \eta _i u_i \end{bmatrix},\ U_i= \begin{bmatrix} \eta _i\\ u_i \end{bmatrix},\ G_i= \begin{bmatrix} \mathcal {K}\partial _xu_i\\ 0 \end{bmatrix}, \end{aligned}$$

and take \(U=U_1\) and \(U=U_2\) in (2.5), and then find that \(\tilde{U}=U_1-U_2\) solves the following system

$$\begin{aligned} \begin{aligned} A_1\partial _t\tilde{U}+B_1\partial _x\tilde{U}+(A_1-A_2)\partial _tU_2+(B_1-B_2)\partial _xU_2+(G_1-G_2)=0. \end{aligned} \end{aligned}$$
(2.18)

Multiplying (2.18) by \(\tilde{U}\) yields

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int \tilde{U}\cdot A_1\cdot \tilde{U}\, \mathrm {d}x =\underbrace{\frac{1}{2}\int \tilde{U}\cdot \partial _tA_1 \cdot \tilde{U}\, \mathrm {d}x}_{II_1}-\underbrace{\int \tilde{U}\cdot B_1\cdot \partial _x\tilde{U}\, \mathrm {d}x}_{II_2}\nonumber \\&\quad -\underbrace{\int \tilde{U}\cdot (A_1-A_2)\cdot \partial _tU_2\, \mathrm {d}x}_{II_3}-\underbrace{\int \tilde{U}\cdot (B_1-B_2)\cdot \partial _xU_2\, \mathrm {d}x}_{II_4}\nonumber \\&\quad -\underbrace{\int \tilde{U}\cdot (G_1-G_2)\, \mathrm {d}x}_{II_5}. \end{aligned}$$
(2.19)

Since \(II_2,II_3\) and \(II_4\) are usual hyperbolic terms, standard estimates give

$$\begin{aligned} \begin{aligned}&|II_2|+|II_3|+|II_4|\\&\quad \lesssim (\Vert \eta _1\Vert _{H^s}+\Vert u_1\Vert _{H^s}+\Vert \eta _1\Vert _{H^s}\Vert u_1\Vert _{H^s}+\Vert u_1\Vert _{H^s}^2)\\&\qquad \times (\Vert \eta _1-\eta _2\Vert _{L^2}^2+\Vert u_1-u_2\Vert _{L^2}^2+\Vert \eta _1-\eta _2\Vert _{L^2}\Vert u_1-u_2\Vert _{L^2}). \end{aligned} \end{aligned}$$
(2.20)

In light of the Eqs. (1.1)\(_1\) and the assumption (1.3), we have

$$\begin{aligned} |II_1|= & {} \frac{1}{2}\left| \int [\mathcal {K}\partial _xu_1+\partial _x(\eta _1 u_1)](u_1-u_2)^2 \, \mathrm {d}x\right| \nonumber \\\lesssim & {} (\Vert u_1\Vert _{H^s}+\Vert \eta _1\Vert _{H^s}\Vert u_1\Vert _{H^s})\Vert u_1-u_2\Vert _{L^2}^2. \end{aligned}$$
(2.21)

We now handle the term \(II_5\) as follows:

$$\begin{aligned} II_5= & {} -\int \sqrt{\mathcal {K}}\partial _x(\eta _1-\eta _2)\cdot \sqrt{\mathcal {K}}(u_1-u_2)\, \mathrm {d}x\nonumber \\= & {} \int \sqrt{\mathcal {K}}\big (\partial _t(u_1-u_2)+u_1\partial _xu_1-u_2\partial _xu_2\big )\cdot \sqrt{\mathcal {K}}(u_1-u_2)\, \mathrm {d}x\nonumber \\= & {} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int \big (\sqrt{\mathcal {K}}(u_1-u_2)\big )^2\, \mathrm {d}x+\underbrace{\int \sqrt{\mathcal {K}}\big (u_1\partial _xu_1-u_2\partial _xu_2\big )\cdot \sqrt{\mathcal {K}}(u_1-u_2)\, \mathrm {d}x}_{II_6},\nonumber \\ \end{aligned}$$
(2.22)

where we have inserted the Eq. (1.1)\(_2\) into the second equality. The term \(II_6\) can be dominated by

$$\begin{aligned} \begin{aligned} |II_6|\lesssim (\Vert u_1\Vert _{H^s}^2+\Vert u_2\Vert _{H^s}^2)\Vert u_1-u_2\Vert _{H^{\frac{a}{2}}}. \end{aligned} \end{aligned}$$
(2.23)

It follows from (2.19)–(2.23) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d} t}\int \left( (\eta _1-\eta _2)^2+\eta _1 (u_1-u_2)^2+\big (\sqrt{\mathcal {K}}(u_1-u_2)\big )^2\right) \, \mathrm {d}x\nonumber \\&\quad \le C\big (E(\eta _1,u_1)(t),E(\eta _2,u_2)(t)\big ) \bigg (\Vert \eta _1-\eta _2\Vert _{L^2}^2+\Vert \eta _1-\eta _2\Vert _{L^2}\Vert u_1-u_2\Vert _{L^2}\nonumber \\&\qquad +\Vert u_1-u_2\Vert _{L^2}^2+\Vert u_1-u_2\Vert _{H^{\frac{a}{2}}}\bigg ). \end{aligned}$$
(2.24)

Since \(\eta _1\) satisfies the non-cavitation assumption (2.2), like (2.16), the following equivalence holds:

$$\begin{aligned} \tilde{E}(t)\eqsim _{a,h_1,h_2} \Vert \eta _1-\eta _2\Vert _{L^2}^2+\Vert u_1-u_2\Vert _{H^{\frac{a}{2}}}^2,\quad t\in [0,T]. \end{aligned}$$

We then see from (2.24) that

$$\begin{aligned} \begin{aligned} \tilde{E}(t)\lesssim _{a,h_1,h_2} \tilde{E}(0)+\int _0^t C\big (E(\eta _1,u_1)(\tau ),E(\eta _2,u_2)(\tau )\big )\big (\tilde{E}(\tau )+\tilde{E}^{1/2}(\tau )\big )\, \mathrm {d}\tau . \end{aligned} \end{aligned}$$

This estimate completes the proof of (2.17).

2.3 Conclusion

To verify the non-cavitation assumption (2.2), one first rewrites

$$\begin{aligned} 1+\eta (t,x)=1+\eta _0(x)+\int _0^t\partial _t\eta (\tau ,x)\, \mathrm {d}\tau \ge h_0-t\sup _{\tau \in [0,t]}\Vert \partial _t\eta (\tau ,\cdot )\Vert _{L_x^\infty }, \end{aligned}$$

and then uses the Eq. (1.1)\(_1\) together with the energy estimate (2.4) to control \(\Vert \partial _t\eta (\tau ,\cdot )\Vert _{L_x^\infty }\), and finally obtains

$$\begin{aligned} 1+\eta (t,x)\ge h_0/2=:h_1, \quad (x,t)\in {\mathbb {R}}\times [0,T], \end{aligned}$$

by carefully choosing the time \(T\). The upper bound assumption (2.3) is a consequence of the energy estimate (2.4) and Sobolev embedding. We refer to [10] for the full details.

Based on the energy estimate (2.4) and the estimate for the differences of two solutions (2.17), it is a standard process following a compactness argument to construct a solution to the Cauchy problem of (1.1) and show the uniqueness of solutions in the solution class stated in Theorem 1.2. Finally one can apply the Bona–Smith argument [2] to verify the continuity of the flow map.

3 Proof of Theorem 1.3

3.1 Energy Estimates

Taking \(\mathrm {curl}\) on the equation of \(\varvec{v}\) in (1.2) we see that \(\mathrm {curl} \varvec{v}=\mathrm {curl} \varvec{v}_0=0\), i.e.,

$$\begin{aligned} \partial _{x_1}v_2=\partial _{x_2}v_1. \end{aligned}$$
(3.1)

we then use (3.1) to rewrite the system (1.2) into the following form

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\eta =-\mathcal {K}(\partial _{x_1}v_1+\partial _{x_2}v_2)-(v_1\partial _{x_1}\eta +v_2\partial _{x_2}\eta )-\eta (\partial _{x_1}v_1+\partial _{x_2}v_2),\\&\partial _tv_1 =-\partial _{x_1}\eta -v_1\partial _{x_1}v_1-v_2\partial _{x_2}v_1,\\&\partial _tv_2 =-\partial _{x_2}\eta -v_1\partial _{x_1}v_2-v_2\partial _{x_2}v_2. \end{aligned} \right. \end{aligned}$$
(3.2)

Now the system (3.2) allows us to perform the idea of symmetrization as the system (1.1). Let

$$\begin{aligned} \mathcal {A}= & {} \begin{bmatrix} 1&{}\quad 0 &{}\quad 0\\ 0&{}\quad \eta &{}\quad 0\\ 0&{}\quad 0 &{}\quad \eta \end{bmatrix},\ \mathcal {B}_1= \begin{bmatrix} v_1&{}\quad \eta &{}\quad 0\\ \eta &{}\quad \eta v_1&{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 \end{bmatrix},\ \mathcal {B}_2= \begin{bmatrix} v_2&{}\quad 0 &{}\quad \eta \\ 0&{}\quad \eta v_2&{}\quad 0\\ \eta &{}\quad 0 &{}\quad \eta v_2 \end{bmatrix}, \\ \mathcal {U}= & {} \begin{bmatrix} \eta \\ v_1\\ v_2 \end{bmatrix},\ \mathcal {G}= \begin{bmatrix} \mathcal {K}(\partial _{x_1}v_1+\partial _{x_2}v_2)\\ 0\\ 0 \end{bmatrix}. \end{aligned}$$

Multiplying (3.2) by the matrix \(\mathcal {A}\) yields the following system

$$\begin{aligned} \begin{aligned} \mathcal {A}\partial _t\mathcal {U}+\mathcal {B}_1\partial _{x_1}\mathcal {U}+\mathcal {B}_2\partial _{x_2}\mathcal {U}+\mathcal {G}=0. \end{aligned} \end{aligned}$$
(3.3)

Applying \(J^s\) to (3.3) and multiplying it by \(J^s\mathcal {U}\) show

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int J^s\mathcal {U}\cdot \mathcal {A}\cdot J^s\mathcal {U}\, \mathrm {d}x\nonumber \\&\quad =\underbrace{\frac{1}{2}\int J^s\mathcal {U}\cdot \partial _t\mathcal {A} \cdot J^s\mathcal {U}\, \mathrm {d}x}_{III_1}-\underbrace{\int J^s\mathcal {U}\cdot [J^s(\mathcal {A}\partial _t\mathcal {U})-\mathcal {A}J^s\partial _t\mathcal {U}]\, \mathrm {d}x}_{III_2}\nonumber \\&\qquad -\underbrace{\int J^s\mathcal {U}\cdot [J^s(\mathcal {B}_1\partial _{x_1}\mathcal {U})-\mathcal {B}_1J^s\partial _{x_1}\mathcal {U}]\, \mathrm {d}x}_{III_3}-\underbrace{\int J^s\mathcal {U}\cdot \mathcal {B}_1\cdot \partial _{x_1}J^s\mathcal {U}\, \mathrm {d}x}_{III_4}\nonumber \\&\qquad -\underbrace{\int J^s\mathcal {U}\cdot [J^s(\mathcal {B}_2\partial _{x_2}\mathcal {U})-\mathcal {B}_2J^s\partial _{x_2}\mathcal {U}]\, \mathrm {d}x}_{III_5}-\underbrace{\int J^s\mathcal {U}\cdot \mathcal {B}_2\cdot \partial _{x_2}J^s\mathcal {U}\, \mathrm {d}x}_{III_6}\nonumber \\&\qquad -\underbrace{\int J^s\mathcal {U}\cdot J^s\mathcal {G}\, \mathrm {d}x}_{III_7}. \end{aligned}$$
(3.4)

Notice \(s>2+a\) and \(H^{1+\delta }({\mathbb {R}}^2)\hookrightarrow L^\infty ({\mathbb {R}}^2)\) with \(\delta \in (0,s-2-a)\). Proceed as Sect. 2.1 by using Kato–Ponce’s commutator estimates or integration by parts, we then can estimate

$$\begin{aligned} \begin{aligned} \sum _{j=1}^6|III_j| \lesssim (\Vert \eta \Vert _{H^s}+\Vert \varvec{v}\Vert _{H^s}+\Vert \eta \Vert _{H^s}\Vert \varvec{v}\Vert _{H^s})^3. \end{aligned} \end{aligned}$$
(3.5)

We are left to handle the term \(III_7\). Using integration by parts and the Eqs. (3.2)\(_2\) and (3.2)\(_3\), we calculate

$$\begin{aligned} III_7= & {} -\int \sqrt{\mathcal {K}}J^s\partial _{x_1}\eta \cdot \sqrt{\mathcal {K}}J^sv_1\, \mathrm {d}x-\int \sqrt{\mathcal {K}}J^s\partial _{x_2}\eta \cdot \sqrt{\mathcal {K}}J^sv_2\, \mathrm {d}x\nonumber \\= & {} \int \sqrt{\mathcal {K}}J^s(\partial _tv_1+v_1\partial _{x_1}v_1+v_2\partial _{x_2}v_1)\cdot \sqrt{\mathcal {K}}J^sv_1\, \mathrm {d}x\nonumber \\&\quad +\int \sqrt{\mathcal {K}}J^s(\partial _tv_2+v_2\partial _{x_2}v_2+v_1\partial _{x_1}v_2)\cdot \sqrt{\mathcal {K}} J^sv_2\, \mathrm {d}x\nonumber \\= & {} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int \left( (\sqrt{\mathcal {K}}J^sv_1)^2+(\sqrt{\mathcal {K}}J^sv_2)^2\right) \, \mathrm {d}x +\underbrace{\int \sqrt{\mathcal {K}}J^s(v_1\partial _{x_1}v_1)\cdot \sqrt{\mathcal {K}}J^sv_1\, \mathrm {d}x}_{III_8}\nonumber \\&\quad +\underbrace{\int \sqrt{\mathcal {K}}J^s(v_2\partial _{x_2}v_1)\cdot \sqrt{\mathcal {K}}J^sv_1\, \mathrm {d}x}_{III_{9}}+\underbrace{\int \sqrt{\mathcal {K}}J^s(v_2\partial _{x_1}v_2)\cdot \sqrt{\mathcal {K}}J^sv_2\, \mathrm {d}x}_{III_{10}}\nonumber \\&\quad +\underbrace{\int \sqrt{\mathcal {K}}J^s(v_1\partial _{x_1}v_2)\cdot \sqrt{\mathcal {K}}J^sv_2\, \mathrm {d}x}_{III_{11}}. \end{aligned}$$
(3.6)

To estimate the term \(III_8\), we commute \(v_1\) out as:

$$\begin{aligned} III_8=\underbrace{\int v_1\sqrt{\mathcal {K}}J^s\partial _{x_1}v_1\cdot \sqrt{\mathcal {K}}J^sv_1\, \mathrm {d}x}_{III_{12}}+\underbrace{\int [\sqrt{\mathcal {K}}J^s,v_1]\partial _{x_1}v_1\cdot \sqrt{\mathcal {K}}J^sv_1\, \mathrm {d}x}_{III_{13}}. \end{aligned}$$

Integration by parts yields

$$\begin{aligned} |III_{12}|\lesssim \Vert \varvec{v}\Vert _{H^s}\Vert v_1\Vert _{H^{s+\frac{a}{2}}}^2. \end{aligned}$$

Using the assumptions (1.3) and (1.4), in a similar fashion to (2.13), we obtain

$$\begin{aligned} |III_{13}|\lesssim \Vert \varvec{v}\Vert _{H^s}\Vert v_1\Vert _{H^{s+\frac{a}{2}}}^2. \end{aligned}$$

One can similarly treat \(III_{9},III_{10}\) and \(III_{11}\) and finally conclude that

$$\begin{aligned} \sum _{j=8}^{11}|III_j|\lesssim \Vert \varvec{v}\Vert _{H^s}(\Vert v_1\Vert _{H^{s+\frac{a}{2}}}^2+\Vert v_2\Vert _{H^{s+\frac{a}{2}}}^2). \end{aligned}$$
(3.7)

It follows from (3.4)–(3.7) that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d} t}\int \left( (J^s\eta )^2+\eta \big ((J^sv_1)^2+(J^sv_2)^2\big )+\big ((\sqrt{\mathcal {K}}J^sv_1)^2+(\sqrt{\mathcal {K}}J^sv_2)^2\big )\right) \, \mathrm {d}x\\&\quad \lesssim (\Vert \eta \Vert _{H^s}+\Vert \varvec{v}\Vert _{H^s}+\Vert \eta \Vert _{H^s}\Vert \varvec{v}\Vert _{H^s})^3+\Vert \varvec{v}\Vert _{H^s}(\Vert v_1\Vert _{H^{s+\frac{a}{2}}}^2+\Vert v_1\Vert _{H^{s+\frac{a}{2}}}^2). \end{aligned} \end{aligned}$$

We instead consider the functional

$$\begin{aligned} H(\eta ,\varvec{v})(t)&=\int \bigg ((J^s\eta )^2+\eta \big ((J^sv_1)^2+(J^sv_2)^2\big )+(J^{s+\frac{a}{2}}v_1)^2\\&\quad +\big ((\sqrt{\mathcal {K}}J^sv_1)^2+(\sqrt{\mathcal {K}}J^sv_2)^2\big )\bigg )\, \mathrm {d}x, \end{aligned}$$

and assume the solution \(\eta \) satisfies the non-cavitation assumption: there exists \(h_1\in (0,1)\) such that

$$\begin{aligned} 1+\eta (t,x) \ge h_1, \quad (x,t)\in {\mathbb {R}}^2\times [0,T], \end{aligned}$$
(3.8)

and the upper bound assumption: there exists \(h_2>0\) such that

$$\begin{aligned} \eta (t,x) \le h_2, \quad (x,t)\in {\mathbb {R}}^2\times [0,T], \end{aligned}$$
(3.9)

where the time \(T\) will be determined by the energy estimate below. Then, as Sect. 2.1 by using (3.8) and (3.9), one obtains

$$\begin{aligned} \begin{aligned} H(\eta ,\varvec{v})(t)\lesssim _{_{a,h_1,h_2}} H(\eta ,u)(0)+\int _0^t \big (H(\eta ,\varvec{v})(\tau )+H(\eta ,\varvec{v})^3(\tau )\big )\, \mathrm {d}\tau , \end{aligned} \end{aligned}$$

which together with the Grönwall’s inequality implies that there exists some appropriately small positive number \(T\) such that

$$\begin{aligned} H(\eta ,\varvec{v})(t)\le C(T,a,h_1,h_2)H(\eta ,\varvec{v})(0),\quad \forall t\in [0,T]. \end{aligned}$$
(3.10)

3.2 Estimates for the Differences of Two Solutions

Assume \(i=1,2\). Let \((\eta _i,\varvec{v}_i)\) be the solutions of (1.2) with the initial data \(({\eta _i}_0,{\varvec{v}_i}_0)\) and satisfy the non-cavitation assumption (3.8) and the upper bound assumption (3.9). Consider the evolutionary equations of the difference \(\mathcal {U}_1-\mathcal {U}_2\) and proceed as Sect. 2.2, we then can show the estimate

$$\begin{aligned} \tilde{H}(t)\le C\tilde{H}(0),\quad \forall t\in [0,T], \end{aligned}$$

where the functional \(\tilde{H}\) is given by

$$\begin{aligned} \tilde{H}(t)&=\int \bigg ((\eta _1-\eta _2)^2+\eta _1 \big (({v_1}_1-{v_1}_2)^2+({v_2}_1-{v_2}_2)^2\big )\\&\quad +\big (J^{\frac{a}{2}}({v_1}_1-{v_1}_2)\big )^2+\big (J^{\frac{a}{2}}({v_2}_1-{v_2}_2)\big )^2\bigg )\, \mathrm {d}x, \end{aligned}$$

and \(T\) is determined in (3.10) and \(C=C\big (T,a,h_1,h_2, H(\eta _1,\varvec{v}_1)(0),H(\eta _2,\varvec{v}_2)(0)\big )\).