Well-posedness of the Fornberg–Whitham equation on the circle

Holmes, John

doi:10.1016/j.jde.2016.02.030

Cited by 36 publications

(37 citation statements)

References 24 publications

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“…This was motivated by the works of Himonas [20], Thompson [29] and Grayshan [13] on CH-type equations. Furthermore, Holmes investigated this result for the FW equation in Sobolev spaces in [21].…”

Section: )mentioning

confidence: 98%

Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces

Holmes

Thompson

2017

Journal of Differential Equations

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Abstract. In this paper, we prove well-posedness of the Fornberg-Whitham equation in Besov spaces B s 2,r in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-tosolution map provided that the initial data belongs to B s 2,r . We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of B s 2,r to C([−T, T ]; B s 2,r ). Furthermore, we prove a Cauchy-Kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow-up criterion for solutions.

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Section: )mentioning

confidence: 98%

Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces

Holmes

Thompson

2017

Journal of Differential Equations

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“…It may exist forever, or it may have a shock after quite a long, numerically undetectable time has passed. 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.…”

Section: Remarks On Shock Conditions and Asymptotic Propertiesmentioning

confidence: 99%

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

Hörmann¹,

Okamoto²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…Let s > 3/2, u 0 ∈ H s , u be the corresponding unique solution to (3)(4), and denote by T its maximal life span. We begin by drawing a simple immediate consequence from (7).…”

Section: Wave Breaking In Case Of Finite Maximal Life Spanmentioning

confidence: 99%

“…Well-posedness results for (3)(4) with spatial regularity according to Sobolev or Besov scales have been obtained in [7,8]. Here we will make use of the following simpler consequence: If s > 3/2 and u 0 ∈ H s (T), then there exists T 0 > 0 such that (3-4) possesses a…”

Section: Introductionmentioning

confidence: 99%

Wave breaking of periodic solutions to the Fornberg-Whitham equation

Hörmann¹

2018

Discrete &Amp; Continuous Dynamical Systems - A

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Based on recent well-posedness results in Sobolev (or Besov spaces) for periodic solutions to the Fornberg-Whitham equations we investigate here the questions of wave breaking and blow-up for these solutions. We show first that finite maximal life time of a solution necessarily leads to wave breaking. Second, we prove that for a certain class of initial wave profiles the corresponding solutions do indeed blow-up in finite time.

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Well-posedness of the Fornberg–Whitham equation on the circle

Cited by 36 publications

References 24 publications

Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces

Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces

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

Wave breaking of periodic solutions to the Fornberg-Whitham equation

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