Wave-breaking phenomena and global existence for the generalized Fornberg-Whitham equation

Abstract

We consider the generalized Fornberg-Whitham equation. We study sufficient conditions for blow-up of solutions and show the global existence with small initial data. Also we give some relations to the Burgers equation.

Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation

“…Note that the guaranteed time of existence according to [12, Theorem 1] is inverse proportional to u 0 H 2 . The paper mentioned above, [16] discussing a whole class of Fornberg-Whitham-type equations, contains also detailed information on the existence time.…”

“…We split the further analysis into two cases depending on the value of β: Case 2β + 1 ≤ 0: Equation ( 15) implies that Z ′ ≥ 0. By (13), this leaves only the option that Z ′ = 0, hence 0 ≤ (Y + 1) 2 = 2β + 1 ≤ 0, so that Y would have to be constant (equal to −1), which is a contradiction to (16).…”

Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

“…Many mathematical properties of the Fornberg-Whitham equation have been discussed, this equation was first proposed for studying the qualitative behavior of wave breaking in [21]. Some investigation of wave breaking conditions can be found in [14,12].…”