Non-uniform Dependence for the Novikov Equation in Besov Spaces

Li, Jinlu; Li, Min; Zhu, Weipeng

doi:10.1007/s00021-020-00511-9

Cited by 19 publications

(20 citation statements)

References 43 publications

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“…Utilizing the well‐posedness result above along with techniques developed in [29,30], we are able to demonstrate that the continuity of the data‐to‐solution map to the R‐b‐family () is sharp, as summarized in the following theorem. Theorem Assume that

s, p, r, μ, b, A, σ

satisfies

s > \max \{1 + \frac{1}{p}, 2 - \frac{1}{p}\}, false(p, r false) \in false[1, \infty false] \times false[1, \infty false), and μ false(b + 1 false) = A σ .

Then the data‐to‐solution map corresponding to the R‐b‐family system () is not uniformly continuous from

B_{p, r}^{s} \times B_{p, r}^{s - 1}

into

C (false[0, T false]; B_{p, r}^{s} \times B_{p, r}^{s - 1})

.…”

Section: Introductionmentioning

confidence: 90%

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Nonuniform dependence of the R‐b‐family system in Besov spaces

2021

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In this paper, we consider a generalized rotation b‐family system (R‐b‐family system) which models the evolution of equatorial water waves. Based on local well‐posedness results and lifespan estimates, we establish sharpness of continuity on the data‐to‐solution map by showing that it is not uniformly continuous from Bp,rs×Bp,rs−1 to C(false[0,Tfalse];Bp,rs×Bp,rs−1). The proof of nonuniform dependence is based upon approximate solutions and Littlewood‐Paley decomposition theory.

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s, p, r, μ, b, A, σ

satisfies

s > \max \{1 + \frac{1}{p}, 2 - \frac{1}{p}\}, false(p, r false) \in false[1, \infty false] \times false[1, \infty false), and μ false(b + 1 false) = A σ .

Then the data‐to‐solution map corresponding to the R‐b‐family system () is not uniformly continuous from

B_{p, r}^{s} \times B_{p, r}^{s - 1}

into

C (false[0, T false]; B_{p, r}^{s} \times B_{p, r}^{s - 1})

.…”

Section: Introductionmentioning

confidence: 90%

“…[24]. Under Li et al., these regularity results were extended to the broader range of Besov spaces

B_{p, r}^{s}

for the CH [30] and Novikov [29] equations where

(p, r) \in [1, \infty] \times [1, \infty)

and

s > \max {1 + \frac{1}{p}, \frac{3}{2}}

.…”

Section: Introductionmentioning

confidence: 99%

Nonuniform dependence of the R‐b‐family system in Besov spaces

2021

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“…Also, it admits exact peaked soliton solutions (peakons) of the form ce −|x−ct| with c > 0, which are orbitally stable [13] and models wave breaking (i.e., the solution remains bounded, while its slope becomes unbounded in finite time [5,9,10]). When k = 2, (1.1) becomes the famous Novikov equation [18,19,[23][24][25][26][27][28]30]…”

Section: Introductionmentioning

confidence: 99%

Sharp ill-posedness for the generalized Camassa-Holm equation in Besov spaces

Li¹,

Yu²,

Zhu³

2021

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In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation that includes the Camassa-Holm as well as the Novikov equation on the line. We present a new and unified method to prove the sharp ill-posedness for the generalized Camassa-Holm equation in B s p,∞ with s > max{1 + 1/p, 3/2} and 1 ≤ p ≤ ∞ in the sense that the solution map to this equation starting from u 0 is discontinuous at t = 0 in the metric of B s p,∞ . Our results cover and improve the previous work given in [22], solving an open problem left in [22].

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“…Another important integrable equation similar to the CH equation (1.1) is the Novikov equation (see, e.g., [17,18,24,25])…”

Section: Introductionmentioning

confidence: 99%

“…The local well-posedness of the Novikov equation with initial data in Sobolev spaces and Besov spaces was studied in [17,31,34,35,36]. Li et al [24,25] showed that the solution map of the Novikov equation are not uniformly continuous from…”

Section: Introductionmentioning

confidence: 99%