Well-posedness of the two-component Fornberg-Whitham system in Besov spaces

The present paper establishes well-posedness for the two-component Fornberg-Whitham system in Besov spaces. First the existence and uniqueness of its solution is proved, then it is shown that the corresponding data-to-solution map is continuous, provided the initial data belong to Besov spaces.


Introduction
In this paper, we consider the following two-component Fornberg-Whitham (FW) system for a fluid where u = u(x, t) is the horizontal velocity of the fluid and η = η(x, t) is the height of the fluid surface above a horizontal bottom.This system was first proposed in [6], where the authors studied the bifurcations of its travelling wave solutions.The Fornberg-Whitham (FW) equation was derived in [8] as a model to study waves on shallow water surfaces.To obtain this, Fornberg and Whitham considered the integro-differential equation The FW equation belongs to the family of nonlinear wave equations described by which has been studied extensively in existing literature.For L(u) = − 1 6 ∂ 3 x u, (1.4) becomes the KdV equation, while for L(u) = −(I − ∂ 2 x ) −1 ∂ x (u 2 + 1 2 u 2 x ), it is the Camassa-Holm (CH) equation.The two-component system (1.1) captures several features of surface waves in an incompressible fluid, e.g.nonlocality and wave breaking, which the KdV equation does not.Therefore it is useful to look into properties of the FW system (equation) as an alternative to the KdV equation for water waves.
Establishing well-posedness of strong solutions for the system (1.1) in various spaces is a challenging problem.In [11], the authors proved local well-posedness of the FW system with initial data in H s × H s−1 for s > 3  2 and presented a blowup criterion by a local-in-time a priori estimate.They also imposed sufficient conditions on the initial data that may lead to wave breaking and analytically demonstrated the existence of periodic traveling waves using a local bifurcation theorem.
Our objective is to establish local well-posedness of the two component FW system in Besov spaces B s p,r × B s−1 p,r for s > max{2 + 1 p , 5 2 }, where s is a Sobolev exponent, p is an L p space exponent and r is related to a Hölder exponent.Besov spaces are an interesting class of functions of growing relevance in the study of nonlinear partial differential equations as they generalize Sobolev spaces and are more effective at measuring regularity properties of functions.To examine the FW system in Besov spaces, we define ρ = η − 1 and study the following system: For our problem to be well-posed in the sense of Hadamard, we must show existence and uniqueness of the solution to the system (1.5) and also continuity of the data-to-solution map when initial data belongs to the aforementioned Besov spaces.To prove existence, we consider a sequence of approximate solutions to (1.5) and construct a system of linear transport equations.First we show that the solutions to this system are uniformly bounded on a common lifespan.Then using a compactness argument, we extract a subsequence that converges to a solution of the original system (1.5).Uniqueness and continuous dependence on initial data follow using an approximation argument similar to that in [10] which dealt with the single FW equation, with appropriate modifications required upon adding ρ to the model.This paper is organized as follows.In Section 2, we state important definitions and properties related to Besov spaces and linear transport equations.Section 3 begins with the main existence result Theorem 3.1 for the FW system (1.5).
The proof involves finding the minimum lifespan, which is presented in Lemma 3.2.Uniqueness of the solution is verified in Proposition 3.3.Finally we prove continuity of the data-to-solution map when the initial data belong to B s p,r × B s−1 p,r for s > max{2 + 1 p , 5 2 }, thus establishing local well-posedness for the twocomponent FW system.

Preliminaries
This section is a review of relevant definitions and results on Besov spaces and linear transport equations, from [1,3,4], that will be used throughout the rest of this paper.Additionally, it describes some analytical tools used in Section 3.

Besov spaces
Let S(R) denote the Schwartz space of smooth functions on R whose derivatives of all orders decay at infinity.Then the set S ′ (R) of temperate distributions is the dual set of S(R) for the usual pairing.
Lemma 2.1 (Littlewood-Paley decomposition) There exists a pair of smooth radial functions (χ, ϕ) taking values in [0, 1] such that χ is supported in the ball B = {ξ ∈ R |ξ| ≤ 4  3 } and ϕ is supported in the ring Then for u ∈ S ′ (R), the nonhomogeneous dyadic intervals are defined as follows:

Definition 2.2
The low frequency cut-off S q is defined by Then we have and by Young's inequality it also follows that for all p ∈ [1, ∞], where M is a positive constant independent of q.
Using the Littlewood-Paley decomposition we define Besov spaces as follows.
Following are some important properties proved in [1, Section 2.8] and [4, Section 1.3] that facilitate the study of nonlinear partial differential equations in Besov spaces.
Lemma 2.4 Let s, s j ∈ R and 1 ≤ p, r, p j , r j ≤ ∞, for j = 1, 2. Then the following hold: (1) Topological property: B s p,r is a Banach space which is continuously embedded in S ′ (R).
(2) Density: C ∞ c is dense in B s p,r if and only if p and r are finite.
(3) Embedding: , if p 1 ≤ p 2 and r 1 ≤ r 2 , and (4) Algebraic property: For all s > 0, , where M is a constant that depends on s, p and r.
(6) Fatou property: If the sequence {f n } n∈N is uniformly bounded in B s p,r and converges weakly in where κ is a constant that depends on s, p and r.

Dutta
Definition 2.5 A smooth function f : R → R is said to be an S m -multiplier if for all multi-index α, there exists a constant C α such that for all ξ ∈ R Proposition 2.6 Let m ∈ R and f be an S m -multiplier.Then for all s ∈ R, 1 ≤ p and r ≤ ∞, the operator f (D) defined for all u ∈ S ′ (R) as

Linear transport equation
Given a linear transport equation, Proposition A.1 in [3] proves the following estimate for its solution size in Besov spaces.
Proposition 2.7 Consider the linear transport equation where Then for some constant C which depends on s and p, and we have ) for all s ′ < s.

Miscellaneous
The following notions have been utilized in the proof of Theorem 3.1.
II. Operator Λ: For our problem, we define Λ = 1 − ∂ 2 x .Then the system (1.5) becomes (2.4) Λ −1 is a well-defined operator whose Fourier transform is for any test function f .Now, Λ −1 is an S 2 -multiplier.Therefore by Proposition 2.6, for some constant θ depending on s, p and r it holds that 3 Local Well-posedness In this section, we prove existence and uniqueness of the solution to the FW system (1.5), and establish continuous dependence of its data-to-solution map in B s p,r × B s−1 p,r .Proof: Consider the sequences of smooth functions {u n } n≥0 and {ρ n } n≥0 with u 0 = 0 and ρ 0 = 0 that solve the following system of linear transport equations

Existence and Lifespan
where J n+1 is a Friedrichs mollifier as defined in (2.3) and Λ = 1 − ∂ 2 x .First we need to show that solutions to the system (3.1) are uniformly bounded on a common lifespan.Applying Proposition 2.7, for some constants K 1 and K 2 that depend on s, p and r, we have where From (2.5), it follows that for some constant M 1 , we have and by ( 6) in Lemma 2.4, for some constant M 2 , it holds that Using (3.5) in (3.2), (3.6) in (3.3) and setting and dτ . (3.8) Taking C := 2 max{L 1 , L 2 } and using we combine (3.7) and (3.8) to write Now we state and prove a lemma that provides the minimum lifespan.
Lemma 3.2 There exists a minimum lifespan T as stated in Theorem 3.1, such that for all n ∈ N and for all t ∈ [0, T ], Then for every t, τ ∈ [0, T ], .
Next we show that {(u n , ρ n )} n≥0 converges to a solution (u, ρ) of the system (1.5).We begin by using Arzela-Ascoli's theorem to find limit points for {u n } n≥0 and {ρ n } n≥0 respectively.

Uniqueness
The following proposition establishes uniqueness of the solution to (1.5) by showing how a change in initial data affects the solution.