Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane

This paper focuses on the two-dimensional Benjamin-Bona-Mahony and Benjamin-Bona-Mahony-Burgers equations with a general flux function. The aim is at the global (in time) well-posedness of the initial-and boundary-value problem for these equations defined in the upper half-plane. Under suitable growth conditions on the flux function, we are able to establish the global well-posedness in a Sobolev class. When the initial- and boundary-data become more regular, the corresponding solutions are shown to be classical. In addition, the continuous dependence on the data is also obtained.


Introduction
This paper is concerned with the two-dimensional (2D) Benjamin-Bona-Mahony-Burgers equation of the form u t + div (φ(u)) = ν 1 ∆u + ν 2 ∆u t in Ω × (0, T ) , In addition, (1.1a) has also been derived to model the two-phase fluid flow in a porous medium, as in the oil recovery. In fact, (1.1a) is a special case of the well-known Buckley-Leverett equation u t + div(φ(u)) = −div{H(u)∇(J(u) − τ u t )}, (1.4) where u denotes the saturation of water, the functions φ, H and J are related to the capillary pressure and the permeability of water and oil [14]. (1.1a) follows from (1.4) by linearizing the static capillary pressure J(u) and H around a constant state.
Attention here will be focused on the case when Ω = R 2 + , the upper half-plane. The aim is at the global well-posedness of (1.1) with inhomogeneous boundary data, namely h ≡ 0. One motivation behind this study is to rigorously validate the laboratory experiments involving water waves generated by a wavemaker mounted at the end of a water channel. We are able to prove the global existence and uniqueness of the mild and classical solutions to (1.1). In addition, a continuous dependence result is also obtained. Our main theorems can be stated as follows.
It is worth remarking that Theorems 1.1 and 1.2 hold with either ν 1 = 0 or with ν 1 > 0 and do not rely on the regularization due to the Burgers dissipation. We briefly review related well-posedness results and then explain the main difficulties in proving Theorems 1.1 and 1.2. There is a very large literature on the global well-posedness and asymptotic behavior of solutions for the 1D BBM equation on the whole line (see, e.g. [1,4,5,21]). Extensive results have also been obtained on the global well-posedness for the initial-and boundary-value problem of the BBM equation posed on the halfline (see, e.g. [3,6,7,8,9,10,11,12,13,19,22]). In particular, in the well-known articles [3,4], the existence of classical solutions and their continuous dependence on the specified data were investigated. While current analytic results for the multidimensional BBM or BBM-Burgers equations only dealt with the existence of mild solutions on either the whole space or bounded domains with homogeneous boundary data (see, e.g. [2,15,17,20]). The results presented here allow inhomogeneous boundary data, which correspond to the setup of a wavemaker mounted at the end of a channel in laboratory experiments. We emphasize that, our methods are also suitable for the corresponding initial value problem. Therefore, Theorems 1.1 and 1.2 are the complete extensions of the results in [3,4] to the multi-dimensional case.
The main difficulty in proving Theorem 1.1 is from two sources. First, the Green function for operator I−∆ in 2D is much more singular than the 1D case; and second, the inhomogeneous boundary data prevents us from obtaining a time-independent H 1 upper bound, which very much simplifies the process of global-in-time estimates. To overcome the difficulties, we introduce a new function that assumes the homogeneous boundary data and rewrite the equation in an integral form through the Green function of the elliptic operator. In addition, we use the bootstrapping technique to obtain the classical solution of (1.1) instead of looking for the solution in classical spaces directly.
The rest of this paper is divided into six sections. The first five sections deal with the case when ν 1 = 0 while the last section explains why the results for ν 1 = 0 can be extended to the case when ν 1 > 0. Section 2 introduces a new function that assumes homogenous boundary data and converts (1. and in Hölder spaces. Section 6 contains the continuous dependence results. The continuous dependence of the solution on the initial data and the boundary data is proven in two functional settings and the proof is lengthy. As aforementioned, Section 7 is devoted to the case when ν 1 > 0.

An alternative formulation
In this section we set ν 1 = 0. The case when ν 1 > 0 is handled in Section 7. This section provides an integral formulation of (1.1).
In order to apply the standard elliptic theory in the functional framework of Sobolev spaces, we shall rewrite equation (1.1) with homogeneous boundary data. This is For short, we rewrite (2.4) as the form v = Av = g + Bh + Cv, (2.5) where, for x ∈ R 2 + and t ≥ 0,

Preliminary results
This section specifies the functional spaces and provides two preliminary estimates on the solutions to the elliptic equation (2.3). In the rest of this paper, we write .
The spaces with the particular indices k = 0, 1 and ℓ = 1, 2 will be frequently used.
For the simplicity of notation, when k = 0, we omit the super-index 0, that is, x . We will also need the space u(t) L p (R 2 + ) . Similar notation is used to define the space of the boundary data which is only defined on the real line R. We introduce To study the classical solutions, we let C k,α (Ω) denote the space of k-times classically differentiable functions whose k-th derivatives are Hölder continuous with exponent α. The norm on C k,α (Ω) is given by where D j u denotes the j-th classical derivative of u.
To deal with the integral representation (2.4), we need some crucial estimates on the operator (I − ∆) −1 . In particular, the bounds in the following propositions will be employed in the subsequent sections.
If f is instead in a Hölder space, then we have the following Hölder's estimates for the solution of (2.3).

Local-in-time existence
This section proves that (1.1) has a unique local (in time) classical solution. We make use of the integral representation (2.5). Due to the difficulty of applying the contraction mapping principle in the setting of Hölder spaces, the proof is divided into two steps. The first step applies the contraction mapping principle to (2.5) in the setting of Sobolev spaces to obtain a unique local solution. The second step obtains the desired regularity of the local solution through a bootstrapping procedure.
, and φ satisfy the condition (1.5). Then there is S with 0 < S ≤ T , depending only on g and h, such that (2.5) has a unique solution v ∈ C([0, S]; H 2 (R 2 + )).
Proof. This local existence and uniqueness result is proven through the contraction mapping principle. More precisely, we show that A defined in (2.5) is a contraction map from B(0, R) ⊂ C([0, S]; H 2 (R 2 + )) to itself, where B(0, R) denotes the closed ball centered at 0 with radius R in C([0, S]; H 2 (R 2 + )). S and R will be specified later in the proof. It follows from (1.5), (2.5), Proposition 3.1 and the mean value theorem that, for v, w ∈ B(0, R), and Av − Aw 2 where v lies between the line segment joining v and w, C 1 is a constant depending on g H 2 and h CtH 2 , and C 2 is a constant depending on h CtH 2 Hence, by (4.2), A is a contraction mapping of this ball if C 2 S(1 + R) < 1. These conditions will be met if we take R = 2C 1 and find a positive value S > 0 small enough such that The contraction mapping principle gives that the sequence v n (x, t) converges in If the initial data g and the boundary data h are also Hölder, then the corresponding solution can also be shown to be Hölder. This is achieved through the Sobolev embeddings and a bootstrapping procedure.

Global-in-time existence
This section shows that the local (in time) solution obtained in the previous section can be extended into a global one. This is achieved by establishing a global bound for v(t) H 2 under the condition that the flux φ obeys suitable growth condition. We start with a global H 1 -bound.
, and φ satisfy the condition (1.5). Then the solution v of (2.5) obtained in Lemma 4.1 satisfies the estimates where C > 0 is a constant depending only on φ.
Proof. Multiplying (2.1a) by v and integrating over R 2 + , we get 1 2 Hence ). (5.6) Applying the mean value theorem and (1.5) again, we have As a consequence, From (5.3)-(5.7), we can conclude that where C depends only on φ. Gronwall's inequality gives ) . Now we derive the H 2 -estimates based on the H 1 -estimates we just obtained. Proof. Multiplying (2.1a) by ∆u and then integrating on R 2 + , we have By the mean value theorem and (1.5), Thus, Hölder's inequality gives By the Sobolev embedding H 1 (R 2 + ) ֒→ L 4 (R 2 + ) and Young's inequality, where C > 0 depends only on φ; that is, where C depends only on g, h, and φ. Therefore, by Gronwall's inequality, v H 2 ≤ C(1 + S) 1/2 exp CS(1 + S) 1/2 e CS which concludes the proof of the lemma.

Continuous dependence of the solution on data
This section is devoted to proving Theorem 1.2. That is, we establish the desired continuous dependence. For the sake of clarity, we will divide the rest of this section into two subsections. The first subsection proves the continuous dependence in the regularity setting of H 2 while the second subsection focuses on the continuous dependence in the intersection space of H 2 and a Hölder class. The precise statements can be found in the lemmas below.
6.1. Continuous dependence in H 2 . Let L m denote the mapping that takes the data g and h to the corresponding solutions of (1.1). By Theorem 1.1 we have Since H 2 (R 2 + ) and C 1 ([0, T ]; H 2 (R)) are Banach spaces, the space X m equipped with the usual product topology is also a Banach space. Lemma 6.1. Suppose that φ ∈ C 2 (R, R 2 ) satisfies the condition (1.5). Then L m is continuous.
Proof. Let (g i , h i ) ∈ X m and u i = L m (g i , h i ) be the mild solution of (1.1) corresponding to the initial data g i and the boundary data Then v i satisfies the following initial-boundary value problem: Then w satisfies: In addition, we derive that w satisfies the following integral equation: Given ε > 0. Suppose that the distance between (g 1 , h 1 ) and (g 2 , h 2 ) in X m is small enough such that Taking H 2 norm on both sides of (6.2) and using Proposition 3.1, we derive the mean value theorem and condition (1.5) yield Applying (6.5) and Hölder's inequality to (6.3), we obtain By Sobolev's inequality, where C depends only on v 1 , v 2 , h 1 , h 2 , and φ. Then Gronwall's inequality gives Note that Therefore, by (6.7), }e CT ≤ e CT ε.

Continuous dependence in the intersection of H 2 and a Hölder space.
This subsection proves the continuous dependence in the setting of the intersection of H 2 and a Höler space. First, we introduce the metrics on the spaces C 2,α loc (Ω) and be an increasing sequence of compact subsets of Ω satisfy For a function f ∈ C 1 ([0, T ]; C 2,α loc (Ω)), we define Then {ρ i } forms a family of seminorms on C 2,α loc (Ω). For f 1 , f 2 ∈ C 1 ([0, T ]; C 2,α loc (Ω)), we define Then d is a metric on C 1 ([0, T ]; C 2,α loc (Ω)). It is clear that . A metric on the space C 2,α loc (Ω) can be defined similarly if we replace the seminorm above by for f ∈ C 2,α loc (Ω) and i ∈ N. Now we fix a sequence {Ω i } ∞ i=1 of compact convex sets in R 2 + such that the conditions (i) and (ii) hold. Let I i denote the projection of Ω i to x 1 -axis. Then forms a sequence of compact sets in R satisfying (i) and (ii). As stated above, the se- T ]; C 2,α loc (R)), and d 3 on C 1 ([0, T ]; C 2,α loc (R 2 + )) respectively. In Theorem 1.1, we get that for a given pair of initial and a boundary data, then (1.1) admits a unique classical solution If we let L c denote the mapping that takes the pair (g, h) into the corresponding classical solution u, then L c : X c → Y. Proof. By the discussions before the statement of this lemma, it suffices to prove are both continuous. Comparing the metrics of the spaces X m and X c , we can easily get the continuity of the mapping L c : X c → C([0, T ]; H 2 (R 2 + )) from Lemma 6.1. In this proof, we focus on showing that the second mapping of (6.9) is sequentially continuous.
k=1 be a sequence of X c that converges to (g 0 , h 0 ) in X c . Suppose that u k = L c (g k , h k ), k ∈ N ∪ {0} be the corresponding classical solutions of (1.1) with respect to the initial data g k and the boundary data h k . Set, for k ∈ N∪{0}, v k = u k − h k e −x 2 , g k = g k − h k e −x 2 .

Results for the GBBM-Burgers equation
The purpose of this section is to generalize the above results to the 2D GBBM-Burgers equation, that is, equation (1.1) with ν 1 = 1. The results established in previous sections also hold for the GBBM-Burgers equation.
The proofs of Theorems 1.1 and 1.2 for the case when ν 1 = 1 are essentially the same as those for the case when ν 1 = 0. In fact, as in the case when ν 1 = 0, we rewrite equation (1.1) as where g is again given by (2.2) and h is defined by