Abstract
We investigate a nonlinear generalized Fornberg–Whitham equation. The key element is that we derive an L^{2}(mathbb{R}) conservation law for solutions of the equation. We establish several estimates by utilizing the L^{2}(mathbb{R}) conservation law. These estimates lead to the proof of the existence and uniqueness of entropy weak solution of the equation in the space L^{1}(mathbb{R})cap L^{infty}(mathbb{R}).
Highlights
Consider the nonlinear partial differential equationVt – Vtxx + kVx + mVVx = 2 VxVxx + 2 VVxxx, (t, x) ∈ R+ × R, (1)where m > 0 and k are constants
Using the viscous approximation techniques and assuming that the initial value V0(x) belongs to the space L1(R) ∩ L∞(R), we prove the well-posedness of the entropy solutions
Using (15) and (16), we derive that there exist constants c1 and c2 such that c1 V0 L2(R) ≤ Vε L2(R) ≤ c2 V0 L2(R)
Summary
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. Using the viscous approximation techniques and assuming that the initial value V0(x) belongs to the space L1(R) ∩ L∞(R), we prove the well-posedness of the entropy solutions. 2, we establish several estimates for the viscous approximations of problem (5), and, we present our main results and their proofs. Lemma 2.1 If V0 ∈ L2(R), for any fixed ε > 0, there exists a unique global smooth solution Vε = Vε(t, x) to the Cauchy problem (6) belonging to C([0, ∞); Hs(R)) with s ≥ 0. Where c1, c2, and c3 are positive constants independent of ε and t
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