The entropy weak solution to a generalized Fornberg–Whitham equation

Recently, Holmes and Thompson [3] proved the well-posedness of Eq. (3) in the Besov space in the periodic and nonperiodic cases and established a Cauchy–Kowalevski-type theorem for Eq. (3) to show the existence and uniqueness of analytic solutions. The blowup criterion for the solutions is given in [3]. Using several estimates derived from the Fornberg–Whitham equation itself and the conclusions in [4], Haziot [5] found sufficient conditions on the initial data to guarantee the wave breaking of solutions of Eq. (3). Gao et al. [6] proved the L1 local stability of strong solutions of Eq. (3).


Introduction
Consider the nonlinear partial differential equation where m > 0 and k are constants. Assume that V 0 (x) = V (0, x) is an initial value to Eq. (1). We establish the inequality where c 1 > 0 and c 2 > 0 are constants independent of t.
If k = -1 and m = 3 2 , then Eq. (1) becomes the Fornberg-Whitham equation [1,2] Recently, Holmes and Thompson [3] proved the well-posedness of Eq. (3) in the Besov space in the periodic and nonperiodic cases and established a Cauchy-Kowalevski-type theorem for Eq. (3) to show the existence and uniqueness of analytic solutions. The blowup criterion for the solutions is given in [3]. Using several estimates derived from the Fornberg-Whitham equation itself and the conclusions in [4], Haziot [5] found sufficient conditions on the initial data to guarantee the wave breaking of solutions of Eq. (3). Gao et al. [6] proved the L 1 local stability of strong solutions of Eq. (3).
We write the Cauchy problem for Eq. (1): which is equivalent to where m > 0 is a constant, . Motivated by the desire to further investigate the Fornberg-Whitham equation (3), the objective of this work is to establish the existence and uniqueness of entropy solutions for Eq. (1). Using the viscous approximation techniques and assuming that the initial value V 0 (x) belongs to the space L 1 (R) ∩ L ∞ (R), we prove the well-posedness of the entropy solutions. The novelty is that we derive a new L 2 (R) conservation law for Eq. (1). The ideas for obtaining our main result come from those in [22].
The structure of this paper is as follows. In Sect. 2, we establish several estimates for the viscous approximations of problem (5), and in Sect. 3, we present our main results and their proofs.

Estimates of viscous approximations
For conciseness in this paper, we let c denote an arbitrary positive constant, which is independent of parameter ε and time t.
For a smooth function V 0,ε and s ≥ 0, we have For problem (4), we will discuss the limiting behavior of a sequence of smooth functions {V ε } ε>0 , where each function V ε satisfies the viscous problem or, in the equivalent form, where Proof Using Theorem 2.3 in [23], we directly get the result of this lemma. Now we give the following lemma, which plays a key role in our investigation of Eq. (1). (7), V 0 ∈ L 2 (R), and t > 0. Then

Lemma 2.2 Suppose that V ε is a solution of problem
where c 1 , c 2 , and c 3 are positive constants independent of ε and t. Proof Multiplying the first equation of problem (7) by g ε -∂ 2 xx g ε and integrating over R yields We have For the right-hand side of (13), integrating by parts and using (11) result in From (12), (13), and (14) we conclude that Using the smoothness of the function V 0,ε , we have It follows from (11) that Using (15) and (16), we derive that there exist constants c 1 and c 2 such that and The proof of Lemma 2.2 follows from (17) and (18).
Letting ε = 0 in the proof of Lemma 2.2, for Eq. (1), we obtain inequality (2). Using Lemma 2.2, we give the following conclusion for the term Q ε (t, x).
Proof We have and Using the Schwarz inequality leads to Utilizing the Tonelli theorem and (23), we get and From (21)- (25) and Lemma 2.2 we derive that (19) and (20) hold. The proof is finished.
Applying Lemma 2.4 and the methods presented in [22], we obtain the following result.
Lemma 2.5 (Oleinik-type estimate) Let V 0 ∈ L 1 (R) ∩ L ∞ (R) and T > 0. Then where the constant C T depends on T .
We omit the proof of this lemma since it is similar to that of Lemma 2.11 in [22]. We state the concepts of weak solution and entropy weak solution (see [22,24]).

Definition 2.6 (Weak solution) A function
Remark 2.8 As stated by Coclite and Karsen [22], by a standard argument we get that the Kruzkov entropies/entropy fluxes where k 1 is an arbitrary constant, satisfy (33).

Main results
We state the following L 1 (R) stability result of entropy weak solutions for Eq. (1).
The proof of Theorem 3.1 is the standard argument presented in Gao et al. [6]. We omit its proof.
We employ the compensated compactness method in [25,26] to discuss the strong convergence of a subsequence of the viscosity approximations.

Lemma 3.2 Let {V ε } ε>0 be a family of functions defined on
where the constant C T > 0 depends on T , and the family is compact in H -1 loc ((0, ∞) × R) for any convex η ∈ C 2 (R), where q(V ) = aV η (V ) with constant a > 0. Then there exist a sequence {ε n } n∈N , ε n → 0, and a function V ∈ L ∞ ((0, T) × R), T > 0, such that V ε n → V a.e. and in L p Lemma 3.2 can be found in [25] or [26].

Lemma 3.4 Let
. Then there exists a subsequence V ε n , n ∈ 1, 2, 3, . . . , of {V ε } ε>0 and a limit function Proof Suppose that η : R → R is an arbitrary convex C 2 entropy function that is compactly supported, and q : R → R is the corresponding entropy flux defined by q (V ) = 3 4 η (V )V . We set where We claim that Using Lemmas 2.2-2.5 yields Therefore we know that (40) holds. Using Lemmas 3.2 and 3.3, we confirm that there exists a subsequence {V ε n } and a limit function V satisfying (36) such that, as n → ∞, and V ε n → V a.e. in R + × R.
where the sequence ε n , and the function V are constructed in Lemma 3.4.
We omit the proof of Lemma 3.5 since it is similar to that of Lemma 4.4 in [22]. Theorem 3.6 Let V 0 ∈ L 1 (R) ∩ L ∞ (R). Then there exists at least one entropy weak solution to problem (5).
Proof If f ∈ C ∞ c (R + × R), then from (31) we get Using Lemmas 3.4, we make sure that the function V presented in Lemma 3.4 is a weak solution of problem (5) in the sense of Definition 2.6. We have to verify that V satisfies the entropy inequalities in Definition 2.7. Let η ∈ C 2 (R) be a convex entropy with flux q defined by q (V ) = 3 4 V η (V ). Using the convexity of η and problem (7) results in Thus by Lemmas 3.4 and 3.5 it follows that the entropy inequality holds. The proof is finished.
From Theorems 3.1 and 3.6 we have the following: