Unconditional well-posedness for the periodic Boussinesq and Kawahara equations

: In this article, we obtain new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilin-ear estimates

The first partial differential equation is called the "good" Boussinesq equation and it is known to describe electromagnetic waves in nonlinear dielectrics [1].When the quadratic nonlinearity is replaced by 4u 3 − 6u 5 , the resulting equation was, in fact, derived in the context of shape-memory alloys [2].The "good" Boussinesq equation can be seen as an improved frequency dispersion version of the original water wave equation derived by Boussinesq [3], where g is the gravitational acceleration, h is undisturbed depth of the channel where the water waves are observed, and u = u(t, x) is the elevation of the water surface.This equation admits solitary wave solutions of the form where a and c are suitably chosen constants, and many consider this to be the place where the stability theory of solitary wave solutions started.For a detailed review of various mathematical and physical aspects concerning Boussinesq equations and their generalizations, we refer the reader to the excellent review article by Makhankov [4].
The partial differential equation in the second Cauchy problem is referred to as the Kawahara equation and it represents a scalar approximation for the full water wave equations in the shallow water regime, when one also takes into account the surface tension.For the derivation of this type of equations, one uses three non-dimensional parameters: δ and ϵ which quantize the dispersive and nonlinear effects, and µ, also known as the Bond number.Shallow water regime corresponds to δ ≪ 1 and, if ϵ = δ 2 and µ 1/3, then one obtains which is the well-known KdV equation.On the other hand, if ϵ = δ 4 and µ = 1/3 + νϵ 1/2 , then one arrives at the Kawahara equation This equation was first derived by Kakutani and Ono [5] to describe nonlinear hydromagnetic waves in plasmas and by Hasimoto [6] precisely in the context of shallow water waves with surface tension and Bond number close to 1/3.For an extensive mathematical and physical perspective on the Kawahara equation, we ask the interested reader to consult the impressive monograph [7] by Lannes.
When studying the WP of ( The subject of UWP for nonlinear dispersive equations has produced an impressive amount of research for the last 30 years, arguably starting with Kato's work [16] on nonlinear Schrödinger equations.In the same context, we mention the recent seminal papers of Kwon et al. [17] and Kishimoto [18] addressing fairly generic dispersive problems.Specifically for periodic problems, a selection of notable results consists of the ones obtained by Babin et al. [19], Kwon and Oh [20], Guo et al. [21], Kishimoto [22], and Kato and Tsugawa [23].For comprehensive discussions on UWP, as well as access to extensive bibliographies on the subject, we refer the reader to [17,18]. Related to our results, we first recall that UWP in the non-periodic case was previously obtained by Farah [24] for the "good" Boussinesq equation and by Geba and Lin [25] for the Kawahara equation, both for data in L 2 (R).To our knowledge, prior to this paper, there are no UWP arguments in the literature for either (1.1) or (1.2).For both of the sharp locally WP results mentioned before (i.e., Kishimoto [11] for (1.1) and Kato [15] for (1.2)), uniqueness of solutions is derived only in proper subsets of their corresponding Hadamard space.Moreover, it is important to note that the question of what the optimal Sobolev index needs to be in order to guarantee UWP is a fairly delicate issue, as discussed in [17,18].Current methodologies and previous results on related dispersive equations suggest that s = 0 should be the right value for both (1.1) and (1.2).
In proving Theorems 1.1 and 1.2, we apply an abstract framework developed by Kishimoto [18] in the context of nonlinear dispersive equations.This has been successfully used in obtaining UWP for periodic Cauchy problems associated to nonlinear Schrödinger equations [26], the Benjamin-Ono equation [22], and the modified Benjamin-Ono equation [27].At the heart of this method lies a critical set of multilinear bounds, which are used to derive even more complex multilinear estimates.The latter represent the key elements in an infinite iteration scheme of normal form reductions proving UWP for the Cauchy problem under consideration.
In concluding this section, we want to emphasize that one of the goals of this paper is to advocate for the simplicity and the robustness of Kishimoto's method.These two features allowed us to keep the argument quite concise and, in our opinion, very transparent.Of course, one could argue that, at least for (1.2), our result is likely not optimal, since the Cauchy problem for the KdV equation (which enjoys weaker dispersion when compared to the Kawahara equation) is UWP in L 2 (T), as proven in [19] through finitely many normal form reductions. Again, our aim is to present a streamlined approach to UWP questions, which caters to a large audience.

Basic notational conventions and terminology
First, we write A ≲ B to stand for A ≤ CB, where C > 0 is a constant varying from line to line and depending on various fixed parameters.Next, we ask that A ∼ B denotes that both A ≲ B and B ≲ A are valid.We also let A ≪ B signify that A ≤ ϵB for some small absolute constant ϵ > 0.
Secondly, for a function f = f (x) defined on T, its Fourier series is given by In similar fashion, for v = v(t, x) defined on R × T, we write Finally, for s ∈ R, we will operate with the space where ⟨k⟩ = (1

Adapted Kishimoto's methodology
Here, we present a version of Kishimoto's framework specifically suited to be applied to our two Cauchy problems, (1.1) and (1.2).
In the case of (1.2), we consider the generic Cauchy problem and we work with the following version of Theorem 1.1 in [18]: ) ) hold true.
For (1.1), we look at the generic Cauchy problem (2.7) In Section 2.5 of [18], Kishimoto discusses how his scheme can be adapted to abstract systems like the one above, and we take as a working version the following result: Theorem 2.2.The Cauchy problem (2.7) has at most one solution ) and, for some ) ) hold true for all 1 ≤ i ≤ 2 and 1 ≤ j ≤ 4.

Proof of Theorem 1.1
We begin this section by reformulating the Boussinesq Cauchy problem (1.1) in such a way that we can implement the methodology described before.The first step consists in rewriting it as a Cauchy problem for a Schrödinger equation.As in Kishimoto and Tsugawa [28], if we take where Conversely, if v and v 0 satisfy this Cauchy problem, then, by letting it is easy to check that u, u 0 , and u 1 are all real-valued and solve (1.1).It is equally important to notice that, for an arbitrary T > 0, are both Lipschitz continuous.Thus, UWP for (1.1) with data in H s × H s−2 becomes equivalent to UWP for (3.1) in H s .Next, by introducing the Fourier series coefficients for v 0 and v according to (2.1) and (2.2), respectively, it follows that (3.1) can be turned into the infinite coupled system of ordinary differential equations where then the previous system comes to be . Finally, we rely on the notation and to arrive at the following working version of the original Cauchy problem (1.1): It is for this system that we verify the validity of Theorem 2.2 when s > 0, thus proving Theorem 1.1.Proof.We start the argument by recognizing that, in our setting, We deduce directly from (3.4) that which immediately implies (2.8) and (2.9).
From (3.2), we derive Using this fact, the Cauchy-Schwarz inequality, and s ≥ 0, we can easily prove (2.12) as follows: Next, we turn to the argument for (2.10), which can be reduced, with the help of the Cauchy-Schwarz inequality, to proving By relying on (3.6), we can further simplify the previous estimate down to If we are in the second scenario, then, using again the information about the size of |φ i j | and s ≥ 0, we obtain sup This completes the proof of (3.7) and, hence, of (2.10).Finally, we come to the argument for (2.11).First, we use (3.6), the Cauchy-Schwarz inequality, and s ≥ 0 to infer This effectively reduces the proof of (2.11) to the one for Here, we reason in identical fashion with the argument for (3.7), involving the notation (3.8), the information about the size of |φ i j |, and the complementary scenarios (3.10).The case k min = 0 can be easily dispensed with.Otherwise, if 1 again due to δ < 1/2.This concludes the argument for (2.11) and the whole proof.Remark 3.2.It is important to recognize that we used s > 0 only in the argument for (2.10), while the proofs for (2.8), (2.9), (2.11), and (2.12) required only that s ≥ 0.

Proof of Theorem 1.2
For the Kawahara Cauchy problem (1.2), we proceed in similar fashion to the previous section and we reformulate it in a form amenable to the framework we want to implement.First, we notice that its smooth solutions satisfy If we take (v, v 0 ) := (u − µ, u 0 − µ), and now, due to (4.1), one has This will turn out to be important moving forward.Next, by turning to the Fourier series coefficients (2.1) and (2.2) for v 0 and v, respectively, we obtain the equivalent form of (4.2) as an infinite coupled system of ordinary differential equations: In the end, we let w k (t) := e it(−βk 3 −k 5 +2µk) v k (t) to rewrite the previous system as When k = k 1 + k 2 , we have the following important factorization: At this moment, we observe that (4.3) implies v 0 (t) ≡ 0 and, hence, w 0 (t) ≡ 0. This allows us to remove from the system the equation corresponding to k = 0.Moreover, when k 0, we see that (4.4) implies ϕ = 0 ⇐⇒ k 1 = 0 or k 2 = 0.
Thus, a working version for the Cauchy problem (1.2) is represented by for which we argue that Theorem 2.1 holds true when s > 1/4.This clearly proves Theorem 1.2.Proof.All three estimates in Theorem 2.1 (i.e., (2.4)-(2.6)) to be verified share the generic profile Using duality, this bound can be seen to be the consequence of It is important to note that, due to (4.4), β ∈ {−1, 0, 1}, and (4.5), we can further assume above that, in our context, none of k, k 1 , and k 2 are equal to 0.
Here, as in the argument for Proposition 3.1, we rely on the notation (3.8), the bound (3.9), and the complementary scenarios (3.10).We start by proving (2.4), for which Since s ≥ 0, the estimate (3.9) implies Next, we turn to the argument for (2.5), which corresponds to (4.6) with Like above, on the basis of s ≥ 0 and (3.9), we derive  Remark 4.3.It seems that the restriction s > 1/4 is essential in the sense that, by switching from s − 2 to s − σ for some other σ > 0, one encounters it again in the same scenario.As suggested by one of the referees, this is likely one of the shortcomings of this methodology, as (2.6) is particularly not responsive to the presence of dispersion in the original equation.

Use of AI tools declaration
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Remark 4 . 2 .
It is important to note that in the previous proposition we used s > 1/4 only in the proof of (2.6), when 1 ≤ |k| ≲ |k max | ∼ |k min |.All the other arguments simply required that s ≥ 0.
Yet, in the previously listed WP references, uniqueness of solutions to either (1.1) or (1.2) is only attained in a proper subset of X.This has the format X ∩ Y, where Y is an additional functional space.It is then natural to ask under what conditions these solutions become unique in their full Hadamard spaces.This is what is called the studying of unconditional uniqueness or, by extension, unconditional well-posedness (UWP) for these Cauchy problems.In this direction, our article provides the following results: 1.1) and (1.2), the natural solution spaces, also called Hadamard spaces, are X = C(I; H s (T)) ∩ C 1 (I; H s−2 (T)) and X = C(I; H s (T))respectively, where I ⊆ R is a time interval with 0 ∈ I. .4) ≤ |k| ≲ |k max | ∼ |k min |, then we deduce from (4.4) that |ϕ| ∼ |k||k max | 4 and, hence, ≤ |k min | ≪ |k| ∼ |k max |, then (4.4) implies |ϕ| ∼ |k min ||k| 4 and, thus,