AN EXTENSION OF SELIGERS WAVE BREAKING CONDITION FOR THE NONLOCAL WHITHAM TYPE EQUATION YONGKI LEE

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AN EXTENSION OF SELIGER’S WAVE BREAKING CONDITION

FOR THE NONLOCAL WHITHAM TYPE EQUATION

YONGKI LEE

Abstract. We extend the wave breaking condition in Seliger’s work [Proc. R. Soc.

Lond. Ser. A., 303 (1968)], which has been used widely to prove wave breaking phenom-

ena for nonlinear nonlocal shallow water equations.

1. Introduction and statement of main result

In this paper, we are concerned with the wave breaking phenomena - bounded solutions

with unbounded derivatives - for the nonlocal Whitham type equation [10]:

(1.1) ∂tu+u∂xu+RRK(x−ξ)uξ(t, ξ)dξ = 0, t > 0, x ∈R,

u(0, x) = u0(x), x ∈R,

where K(x) = 1

2πRRc(κ)eiκx dκ, is the Fourier transform of the desired phase velocity

c(κ). The function u(t, x) models the deﬂection of the ﬂuid surface from the rest position

and the equation was proposed by Whitham as an alternative to the Korteweg-de Vries

(KdV) equation for the description of wave motion at the surface of a perfect ﬂuid.

Whitham emphasized that the breaking phenomena is one of the most intriguing long-

standing problems of water wave theory, and since the KdV equation can’t describe break-

ing, he suggested (1.1) with the singular kernel

(1.2) K0(x) = ZRtanh ξ

ξ1/2

eiξx dξ,

as a model equation combining full linear dispersion with long wave nonlinearity, and

conjectured wave breaking in (1.1)-(1.2).

The formal approach to prove wave breaking for Whitham type equation originated

from Seliger’s ingenious argument [9], i.e., tracing the dynamics of

(1.3) m1(t) := inf

x∈R[ux(t, x)] and m2(t) := sup

x∈R

[ux(t, x)],

attained at x=ξ1(t) and x=ξ2(t), respectively, provided that Kbe bounded and

integrable, among other hypotheses. The mapping t→ξi(t), however, may be multi-

valued so the curves in general are not necessarily well-deﬁned. In addition to this, to

carry out Seliger’s formal analysis, one needs to assume that the curves ξ1(t) and ξ2(t)

are smooth. These additional strong assumptions were shown unnecessary later by the

rigorous analytical proof of Constantin and Escher [3].

Following the argument in [9,3], in this paper K(x) is assumed to be regular (smooth

and integrable over R), symmetric and monotonically decreasing on x∈[0,∞). For

2020 Mathematics Subject Classiﬁcation. Primary, 35L05; Secondary, 35B30.

Key words and phrases. Wave breaking, Whitham equation, Shallow water equations.

arXiv:2210.13405v1 [math.AP] 24 Oct 2022

2 YONGKI LEE

non-integrable K(x) case, we refer to [5] and references therein. Diﬀerentiating the ﬁrst

equation in (1.1) with respect to xand evaluating the resulting equations at x=ξ1(t)

and x=ξ2(t), two coupled diﬀerential inequalities are deduced in [9]:

(1.4a) dm1

dt ≤ −m2

1(t) + K(0)(m2(t)−m1(t)) a.e.,

(1.4b) dm1

dt ≤ −m2

2(t) + K(0)(m2(t)−m1(t)) a.e,

where K(0) >0.

The wave breaking condition of the Whitham type equation in [9] is

(1.5) inf

x∈R[u0

0(x)] + sup

x∈R

[u0

0(x)] ≤ −2K(0),

indeed, represents that a suﬃciently asymmetric initial proﬁle yields wave breaking in

ﬁnite time. The aforementioned arguments and the rigorous analytical proof in [3] have

been considered as the cornerstone work for proving wave breaking for nonlinear nonlocal

shallow water equations. Further, the condition (1.5) has been used widely in many

studies, including very recent works in [4,6]. This is because it preserves a useful structure

for the proof: if m1(0) + m2(0) ≤ −2K(0), then this relation remains so for all time.

The main contribution of this study is extending (1.5) into a larger set, thereby ob-

taining a lower threshold for the wave breaking and extending the works in several afore-

mentioned papers. Also, we provide an upper bound of wave breaking time. Our proof is

base on simple phase plane analysis equipped with delicate time estimates, e.g [7].

To state our main result, for the sake of simplicity, we let K(0) := 1 then (1.4) is

reduced to

(1.6a) m0

1(t)≤ −m2

1(t) + m2(t)−m1(t)a.e.,

and

(1.6b) m0

2(t)≤ −m2

2(t) + m2(t)−m1(t)a.e.

From (1.3), we necessarily have m1(t)≤0≤m2(t), as long as they exist.

We now let

Ω := {(m1, m2)|m1<−2 and 0 ≤m2< m2

1+m1},

as shown in Figure 1, and state the main theorem.

Theorem 1.1. Consider (1.1). If u0∈H∞satisﬁes

(inf

x∈R[u0

0(x)],sup

x∈R

[u0

0(x)]) ∈Ω,

then the solution must develop wave breaking before T∗, with

T∗=1

2log m1(0)

2 + m1(0)+ max 0,m1(0) + m2(0)

2m1(0)(2 + m1(0)).

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摘要：

ANEXTENSIONOFSELIGER'SWAVEBREAKINGCONDITIONFORTHENONLOCALWHITHAMTYPEEQUATIONYONGKILEEAbstract.WeextendthewavebreakingconditioninSeliger'swork[Proc.R.Soc.Lond.Ser.A.,303(1968)],whichhasbeenusedwidelytoprovewavebreakingphenom-enafornonlinearnonlocalshallowwaterequations.1.Introductionandstatementofmai...

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AN EXTENSION OF SELIGERS WAVE BREAKING CONDITION FOR THE NONLOCAL WHITHAM TYPE EQUATION YONGKI LEE

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