Abstract
This work concerns the local well-posedness to the Cauchy problem of a fully dispersive Boussinesq system which models fully dispersive water waves in two and three spatial dimensions. Our purpose is to understand the modified energy approach (Kalisch and Pilod in Proc Am Math Soc 147:2545–2559, 2019) in a different point view by utilizing the symmetrization of hyperbolic systems which produces an equivalent modified energy.
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1 Introduction and Main Results
The one-dimensional fully dispersive Boussinesq system is governed by
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:
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
for some smooth even function \(m:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\) with the properties
and
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
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
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
The integration in the LHS of (1.6) is exactly the modified energy that we need. Notice that
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
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
and a unique solution
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
and a unique solution
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
We priorly assume the solution \(\eta \) satisfies the non-cavitation assumption: there exists \(h_1\in (0,1)\) such that
and the upper bound assumption: there exists \(h_2>0\) such that
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
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
Multiplying (1.1) by the matrix \(A\) yields the following system
Applying \(J^s\) to (2.5) and multiplying it by \(J^sU\) give
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
It remains to treat the terms \(I_1\) and \(I_5\). In view of (1.1)\(_1\), we estimate
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
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:
The term \(I_7\), by integration by parts, can be easily controlled by
We next estimate the term \(I_8\). By the Plancherel’s theorem, we rewrite
Via the mean value theorem, there exists some \(\theta \in (0,1)\) such that
with \(\zeta =\theta (\xi -\eta )+(1-\theta )\eta \). Then the above equality and the assumptions (1.3) and (1.4) imply
Notice that
for any \(b>1/2\). Substituting (2.12) into (2.11) gives
which leads to
It follows from (2.6)–(2.10) and (2.13) that
By the assumption (1.3), we have
We then claim that
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
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
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
and aim to show
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
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
Multiplying (2.18) by \(\tilde{U}\) yields
Since \(II_2,II_3\) and \(II_4\) are usual hyperbolic terms, standard estimates give
In light of the Eqs. (1.1)\(_1\) and the assumption (1.3), we have
We now handle the term \(II_5\) as follows:
where we have inserted the Eq. (1.1)\(_2\) into the second equality. The term \(II_6\) can be dominated by
It follows from (2.19)–(2.23) that
Since \(\eta _1\) satisfies the non-cavitation assumption (2.2), like (2.16), the following equivalence holds:
We then see from (2.24) that
This estimate completes the proof of (2.17).
2.3 Conclusion
To verify the non-cavitation assumption (2.2), one first rewrites
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
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.,
we then use (3.1) to rewrite the system (1.2) into the following form
Now the system (3.2) allows us to perform the idea of symmetrization as the system (1.1). Let
Multiplying (3.2) by the matrix \(\mathcal {A}\) yields the following system
Applying \(J^s\) to (3.3) and multiplying it by \(J^s\mathcal {U}\) show
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
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
To estimate the term \(III_8\), we commute \(v_1\) out as:
Integration by parts yields
Using the assumptions (1.3) and (1.4), in a similar fashion to (2.13), we obtain
One can similarly treat \(III_{9},III_{10}\) and \(III_{11}\) and finally conclude that
It follows from (3.4)–(3.7) that
We instead consider the functional
and assume the solution \(\eta \) satisfies the non-cavitation assumption: there exists \(h_1\in (0,1)\) such that
and the upper bound assumption: there exists \(h_2>0\) such that
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
which together with the Grönwall’s inequality implies that there exists some appropriately small positive number \(T\) such that
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
where the functional \(\tilde{H}\) is given by
and \(T\) is determined in (3.10) and \(C=C\big (T,a,h_1,h_2, H(\eta _1,\varvec{v}_1)(0),H(\eta _2,\varvec{v}_2)(0)\big )\).
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Open access funding provided by NTNU Norwegian University of Science and Technology (incl St. Olavs Hospital - Trondheim University Hospital). The author is grateful to the referees for their comments and suggestions that helped improve the exposition of the manuscript.
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Y.W. acknowledges the support by Grants Nos. 231668 and 250070 from the Research Council of Norway.
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Wang, Y. Well-Posedness to the Cauchy Problem of a Fully Dispersive Boussinesq System. J Dyn Diff Equat 33, 805–816 (2021). https://doi.org/10.1007/s10884-020-09831-w
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DOI: https://doi.org/10.1007/s10884-020-09831-w