Well-Posedness to the Cauchy Problem of a Fully Dispersive Boussinesq System

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 dimen-sions. 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.


Introduction and Main Results
The one-dimensional fully dispersive Boussinesq system is governed by here t, x ∈ R, and the surface elevation η 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 ∈ R, x = (x 1 , x 2 ) ∈ R 2 and the unknowns η ∈ R, v = (v 1 , v 2 ) ∈ R 2 . The operator K is a Fourier multiplier with symbol m ∈ S a ∞ (R d ), d = 1 or 2 for some a ∈ R\{0}, for some smooth even function m : R d → R d with the properties and |m (ξ )| (1 + |ξ |) a−1 . (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 with m(ξ ) = tanh(ξ )/ξ , 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(ξ ) = tanh(ξ )/ξ [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 η 0 ≥ 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(ξ ) = (tanh(|ξ |)/|ξ |)(1 + |ξ | 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 v 0 . Due to the lack of symmetry of the nonlinearity, the usual energy estimates only give (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 η(J s ∂ x u) 2 dx (which has no fixed sign) to the LHS of (1.5). Another important observation is that the non-cavitation assumption on η (which means that the surface elevation of the wave cannot touch the bottom of the fluid) allows 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 (1.6) 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: (1.7) We then have similar results to [10] under the non-cavitation assumption: satisfy the non-cavitation condition (1.7). Then there exist a positive number ) and a unique solution In addition, the flow function that maps initial data to solutions is continuous.
and a unique solution 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 x l = (1 + |x| 2 ) l 2 and J l f (ξ ) = ξ lf (ξ ), and denote by H l the L 2 based Sobolev space with the norm · H l = J l · 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, . . .).
Proof of Theorem 1.2

Energy Estimates
Denote We priorly assume the solution η satisfies the non-cavitation assumption: there exists h 1 ∈ (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(η, 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 s U give (2.6) The terms I 2 , I 3 and I 4 come from the hyperbolic parts [do not involve the Fourier multiplier 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

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 (2.10) We next estimate the term I 8 . By the Plancherel's theorem, we rewrite Via the mean value theorem, there exists some θ ∈ (0, 1) such that Then the above equality and the assumptions (1.

3) and (1.4) imply
(2.12) Notice that for any b > 1/2. Substituting (2.12) into (2.11) gives which leads to (2.13) It follows from (2.6)-(2.10) and (2.13) that (2.14) By the assumption (1.3), we have (2.15) 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 ξ 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).

Estimates for the Differences of Two Solutions
Assume i = 1, 2. Let (η i , u i ) be the solutions of (1.1) with the initial data (η 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 energỹ and aim to showẼ where T is defined in (2.4) and C = C T , a, h 1 , h 2 , E(η 1 , u 1 )(0), E(η 2 , u 2 )(0) . We let and take U = U 1 and U = U 2 in (2.5), and then find thatŨ = U 1 − U 2 solves the following system Multiplying (2.18) byŨ yields (2.19) Since I I 2 , I I 3 and I I 4 are usual hyperbolic terms, standard estimates give (2.20) In light of the Eqs. (1.1) 1 and the assumption (1.3), we have We now handle the term I I 5 as follows: (2.24) Since η 1 satisfies the non-cavitation assumption (2.2), like (2.16), the following equivalence holds:Ẽ We then see from (2.24) that This estimate completes the proof of (2.17).

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 ∂ t η(τ, ·) L ∞ x , 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.

Energy Estimates
Taking curl on the equation of v in (1.2) we see that curlv = curlv 0 = 0, i.e., (3.1) we then use (3.1) to rewrite the system (1.2) into the following form (3.2) 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 A yields the following system