Ill-posedness for a family of nonlinear andnonlocal evolution equations

Alex Himonas

University of Notre Dame & Unversity of Chicago

International ConferencePDE, Complex Analysis, and Related Topics

atFlorida International University

January 4-7, 2016

Abstract

Title: Ill-posedness for a family of nonlinear and nonlocal evolutionequations.

Abstract: We shall discuss the well-posedness of a family ofnonlinear and nonlocal evolution equations depending on a parameterb, which has been introduced by Holm and Staley and which isexpressing a balance between evolution, convection and stretching.This family of equations possess peakon and multipeakon travelingwave solutions for all values of the parameter b. Furthermore, it iswell-posed for initial data in Sobolev spaces Hs for all s > 3/2.However, it has been shown recently that for s < 3/2 it is ill-posed.This work establishes 3/2 as the critical index of well-posedness inSobolev spaces. The talk is based on work with Katelyn Grayshan,Curtis Holliman, Carlos Kenig and Gerard Misiolek.

Historical remarks

• 1973: G. B. Whitham, “Linear and Nonlinear waves, p. 476”

13.14 Breaking and Peaking

“It was remarked earlier that the nonlinear shallow waterequations which neglect dispersion altogether lead to breaking ofthe typical hyperbolic kind, with the development of a verticalslope and a multivalued profile. It seems clear that the thirdderivative term in the Korteweg-deVries equation will prevent thisever happening in its solutions.

. . .

Although both breaking and peaking, as well as criteria for theoccurrence of each, are without doubt contained in the equationsof the exact potential theory, it is intriguing to know whatkind of simpler mathematical equation could include allthese phenomena.”

Historical remarks (cont.)Witham in this book continues by proposing a KdV-type equationwith a weaker dispersion for modeling (hopefully) “breaking andpeaking” shallow water waves. This equation is of the form:

ut +[N(u) + L(u)

]x

= 0 (1)

where

· N = nonlinearity· L = linear “weak” dispersion

KdV(order of dispersion =3): ChoosingN(u) = 1

2u2, L(u) = 1 + 1

6∂2x =⇒

ut + uux + ux +1

6∂3xu = 0. (KdV ) (2)

Whitham Equation(order of dispersion =1/2): Choosing

N(u) =1

2u2, Lf(ξ) =

( tanh ξ

ξ

)1/2f(ξ) ≈ |ξ|−1/2f(ξ) =⇒

ut + uux + ∂xLu = 0. (WE) (3)

u(x, t) =c

2sech2

(√c

2(x− ct)

), u(x, t) = ce−|x−ct|

Figure: KdV soliton & peaked soliton

KdV story

• 1834: Scott Russell reported his “large wave of translation”,nowadays known as soliton, that he observed in the Edinburghchannel while following a barge on horseback.

• 1877: Boussinesq, in his attempt to explain the soliton, derivedapproximations to the Navier-Stokes equations in severaldifferent regimes, among which the so-called KdV equation

ut + ux + uux + uxxx = 0.

• 1895: Korteweg and de Vries rederived the KdV.

Do Camassa & Holm answer Whitham’s question?• 1993: Camassa & Holm derived the “0-order dispersion” equation

ut + uux + ∂x(1− ∂2x)−1[u2 +

1

2u2x

]= 0. (CH) (4)

starting from Euler equations and showed that it admits peakontravelling waves that interact like solitons, thus providing analternative model to Whitham’s equation.

• 1981: Fokas and Fuchssteiner obtained CH earlier based onhereditary symmetries of bi-hamiltonian systems.

• 1755: Euler derived the system of equations

∂u

∂t+ (u · ∇)u = −∇p, u = (u, v, w),

known as Euler equations, modeling the flow of an inviscid fluid.

• 1822 – 1845: Navier (1822), Cauchy (1828), Poisson (1829) andfinally Stokes (1845) derived the Navier-Stokes equations

∂u

∂t+ (u · ∇)u = −∇p+ ν∆u,

modeling the flow of a viscous fluid.

The b-family of equations

In this talk we consider the initial value problem for the b-family ofequations (b-equation), which in its nonlocal form reads as follows

ut + uux + ∂x(1− ∂2x)−1[b

2u2 +

3− b2

(∂xu)2]

= 0. (5)

The b-equation was introduced by Holm and Staley [HS2003] in thefollowing local form

ut︸︷︷︸Evolution

+ umx︸︷︷︸Convection

+ buxm︸ ︷︷ ︸Stretching

= 0, where m = (1− ∂2x)u, (6)

expressing a balance between evolution, convection and stretching.

The b-equation (cont.)

In fact, in deriving the b-equation they begun with an asymptoticexpansion in the small parameters ε1 = a/h and ε2 = h2/`2, where a,h and ` denote the wave amplitude, mean water depth andwavelength, respectively. At the linear order of accuracy thisexpansion gives the Korteweg-de Vries equation while at thequadratic order of accuracy gives the family

ut︸︷︷︸Evolution

+ c0ux +3ε220

uxxx︸ ︷︷ ︸Dispersion

+ ε1(umx + buxm)︸ ︷︷ ︸Nonlinearity

= 0, (7)

where the real constant b depends on the parameters in the Kodamatransformation used in these derivations. The dispersion provides thebalance against the steepening caused by the nonlinearity andtogether with it forms the primary mechanism for the propagation ofsolitary water waves.

Comparing the balance in KdV and b-equationThe balance is the celebrated KdV equation

ut + uux + uxxx = 0. (8)

is provided by the dispersion uxxx. In the absence of dispersion, thebalance in the b-equation is expressed by the nonlocal/nonlinear term

∂x(1− ∂2x)−1[ b

2u2 +

3− b2

(∂xu)2]

(9)

involving the parameter b. This different kind of balance is seenclearly by thinking of both the KdV and the b-equation as twodifferent “dispersive” perturbations of the Burgers equationut + uux = 0. However, the dispersion uxxx in KdV is too strong andin terms of well-posedness this results in the fact that every local intime solution of KdV is also global. In physical terms, Whitham[W1974] expresses this as follows when he writes about breaking andpeaking: “It seems clear that the third derivative term in theKorteweg-deVries equation will prevent this ever happening in itssolutions.” And, he continues: “Although both breaking and peaking,as well as criteria for the occurrence of each, are without doubtcontained in the equations of the exact potential theory, it isintriguing to know what kind of simpler mathematical equation couldinclude all these phenomena.”

The b-equation has peakon traveling wavesThis kind of balance in the b-equation allows for a rich variety oftraveling wave solutions and for interesting phenomena concerning theexistence of global in time solutions. First, for all values of b itpossess peakon traveling wave solutions of the form

u(x, t) = ce−|x−ct|.

In fact, this equation possesses n-peakons (Degasperis, Holm andHone 2002). These are solutions of the form

u(x, t) =

n∑j=1

pj(t)e−|x−qj(t)|, (10)

where the positions qj and the momenta pj satisfy the system ofdifferential equations:

dqkdt

=n∑

j=1

pje−|qk−qj |,

dpkdt

= (b− 1)pk

n∑j=1

pjsgn(qk − qj)e−|qk−qj |. (11)

Special two-peakon solutions (peakon-antipeakon) are used here toprove that the b-equation are not well-posed in Sobolev spaces Hs

when s < 3/2.

Integrable equations

The b-equation contains only two integrable equations (Mikhailov andNovikov 2002):

∗ b = 2 gives the well known Camassa-Holm (CH) equation.

∗ b = 3 gives the Degasperis-Procesi (DP) equation.

Integrable equations possess many special properties including:

• an infinite hierarchy of higher symmetries,

• infinitely many conserved quantities,

• a Lax pair,

• a bi-Hamiltonian formulation,

• and they can be solved by the Inverse Scattering Method.

Conserved quantities are useful for proving global in time solutions. Abi-Hamiltonian formulation is used for finding conserved quantities. ALax pair is used for decoupling the equation into two equations, onethat describes the spatial structure of the equation and helps to solveit at the initial time and another that helps to compute the timeevolution. This is implemented by the Inverse Scattering Method.

Well-posedness in Sobolev spaces

Hadamard well-posedness

There exists a solution, it is unique and depends continuously on theinitial data, i.e. the data-to-solution map u(0) 7→ u(t) is continuous.

Well-posedness of Cauchy problem for CH type equations

(i) For any initial data u(0) ∈ Hs there exists T = Tu(0) > 0 and asolution u ∈ C([0, T ];Hs) to the CH Cauchy problem.

(ii) This solution u is unique in the space u ∈ C([0, T ];Hs).

(iii) The data-to-solution map u(0) 7→ u(t) is continuous. Moreprecisely, if un(0) is a sequence of initial data converging tou∞(0) in Hs and if un(t) ∈ C([0, Tn];Hs) is the solution to theCauchy problem with initial data un(0), then there is T ∈ (0, T∞)such that the solutions un(t) can be extended to the interval[0, T ] for all sufficiently large n and

limn→∞

sup0≤t≤T

‖un(t)− u∞(t)‖Hs = 0. (12)

Jacques HadamardJacques Hadamard (French mathematician) published Lectures onCauchy’s Problem in Linear Partial Differential Equations in 1923.The text was based on a series of lectures he had given at YaleUniversity. In this book, Hadamard put forth the notion of awell-posed problem.

Hadamard’s example (p. 33: Lectures on the Cauchy’sproblem in linear partial differential equations)

. . .I have often maintained, against different geometers, the importanceof this distinction. Some of them indeed argued that you may alwaysconsider any functions as analytic, as, in the contrary case, they couldbe approximated with any required precision by analytic ones. But, inmy opinion, this objection would not apply, the question not beingwhether such an approximation would alter the data very little, butwhether it would alter the solution very little. It is easy to see that, inthe case we are dealing with, the two are not at all equivalent. Let ustake the classic equation of two-dimensional potentials

∂2u

∂x2+∂2u

∂y2= 0

with the following data of Cauchy’s

u(0, y) = 0,∂u

∂x(0, y) = An sin(ny),

n being a very large number, but An a function of n assumed to bevery small as n grows very large (for instance An = 1/np).. . .

These data differ from zero as little as can be wished. Nevertheless,such a Cauchy problem has for its solution

u =Ann

sin(ny)Sh(nx),

which, if An = 1n , 1

np , e−√n, is very large for any determinate value of

x different from zero on account of the mode of growth of enx andconsequently Sh(nx).In this case, the presence of the factor sin ny produces a “fluting” ofthe surface, and we see that this fluting, however imperceptible in theimmediate neighbourhood of the y-axis, becomes enormous at anygiven distance of it however small, provided the fluting be takensufficiently thin by taking n sufficiently great.

Hadamard well-posedness for the b-equation

Theorem (Well-posedness with optimal solution stability)

(1) [Hadamard well-posedness] If s > 3/2 and u0 ∈ Hs then thereexists T > 0 and a unique solution u ∈ C([0, T ];Hs) of the initialvalue problem for b-equation which depends continuously on theinitial data u0. Furthermore, we have the estimate

‖u(t)‖Hs ≤ 2‖u0‖Hs , for 0 ≤ t ≤ T ≤ 1

2cs‖u0‖Hs, (13)

where cs > 0 is a constant depending on s.

(2) [Optimal stability] Also, the data-to-solution map is notuniformly continuous from any bounded subset in Hs intoC([0, T ];Hs).

Nonuniform continuity was proved:

- CH (R: H-Kenig DIE 2009 & T: H-Kenig-Misiolek, CPDE 2010)

- DP (H-Holliman, DCDS 2011, both line R and circle T)

The Proof of well-posedness

Writing b-equation in the following nonlocal form

∂tu+ u∂xu+ F (u) = 0, (14)

where

F (u) = ∂x(1− ∂2x)−1[ b

2u2 +

3− b2

u2x

](15)

and noticing that F maps Hs into Hs, we see that mollifying thelocal term u∂xu we obtain the following ivp

∂tu+ Jε

[(Jεu)Jεux

]+ F (u) = 0, u(x, 0) = u0(x), (16)

which we solve by applying the fundamental ode theorem in aninfinite dimensional Banach space..... . .

Proof of non-uniform continuity for CH (b = 2)

The Proof of non-uniform continuity for CH on T

(Here, we follow work with Kenig and Misio lek)

We shall prove that there exist two sequences of CH solutions un(t)and vn(t) in C([0, T ];Hs(T)) such that:

• supn‖un(t)‖Hs + sup

n‖vn(t)‖Hs . 1,

• limn→∞

‖un(0)− vn(0)‖Hs = 0

• lim infn‖un(t)− vn(t)‖Hs & sin t, 0 ≤ t < T ≤ 1.

Approximate solutions of CH on T

We consider approximate solutions of the form

uω,n(x, t) = ωn−1 + n−s cos(nx− ωt), (17)

where ω = ±1 and n ∈ Z+. Substituting (28) into the CH equationgives the error

F.= ∂tu

ω,n + uω,n∂xuω,n + ∂xD

−2[(uω,n)2 +1

2(∂xu

ω,n)2]

=(((((((((ωn−s cos(nx− ωt)−(((((((((

ωn−s cos(nx− ωt)

− 1

2n−2s+1 sin 2(nx− ωt)

− n−2s+1D−2[

sin(2nx− 2ωt)]

− 2ωn−sD−2[

sin(nx− ωt)]

+1

2n−2s+3D−2

[sin(2nx− 2ωt) 6= 0.

Goal. Hs-norm of error term F is small!

Sobolev norm of error F

Since for σ ∈ R and n� 1

‖ cos(nx− α)‖Hσ(T) ≈ nσ, and (18)

‖ sin(nx− α)‖Hσ(T) ≈ nσ, α ∈ R, (19)

we obtain the following estimate for the error F .

LemmaIf 1/2 < σ < min{1, s− 1}, then the Hσ norm of the error F can beestimated by

‖F‖Hσ . n−rs , where rs =

{2s− σ − 1, s ≤ 3,

s− σ + 2, s ≥ 3.(20)

Remark. The cancelled term ωn−s cos(nx− ωt) in the error is badsince

‖ωn−s cos(nx− ωt)‖Hs(T) ≈ 1.

u±,n(t) satisfy the 3 nonuniform continuity conditionsChoosing ω = ±1 we get the two sequences solutions of approximate

u1,n(t) = n−1+n−s cos(nx−t), and u−1,n(t) = −n−1+n−s cos(nx+t),

having difference

u1,n(t)− u−1,n(t) = 2n−1 + 2n−s sin(nx) sin t.

Then we have

• supn‖u1,n(t)‖Hs + sup

n‖u−1,n(t)‖Hs . 1,

• limn→∞

‖u1,n(0)− u−1,n(0)‖Hs = limn→∞

‖2n−n‖Hs = 0

• Furthermore, we have

‖u1,n(t)− u−1,n(t)‖Hs(T) ≥ 2n−s‖ sin(nx)‖Hs(T)| sin t| − 2n−1‖1‖Hs(T)

Now, letting n↗∞ gives

lim infn→∞

‖u1,n(t)− u−1,n(t)‖Hs(T) & | sin t|. (21)

Constructing actual solutions u±,n(t) close to u±,n(t)

To prove nonuniform continuity, it suffices to construct actual CHsolutions uω,n(t) close to the approximate ones uω,n(t), that is

‖uω,n(t)− uω,n(t)‖Hs ≤ n−ε, for some ε > 0.

In fact, by adding and subtracting and then applying the triangleinequality we have

‖u1,n(t)− u−1,n(t)‖Hs ≥ ‖u1,n(t)− u−1,n(t)‖Hs− ‖u1,n(t)− u1,n(t)‖Hs− ‖u−1,n(t)− u−1,n(t)‖Hs

Therefore, we will have

lim infn‖u1,n(t)− u−1,n(t)‖Hs ≥ lim inf

n‖u1,n(t)− u−1,n(t)‖Hs

& | sin t|.

Actual CH solutions

We construct the actual CH solutions by solving its ivp with initialdata the value of the approximate solutions at t = 0, that is uω,n(x, t)are defined by

∂tuω,n + uω,n∂xuω,n +D−2∂x

[u2ω,n +

1

2(∂xuω,n)2

]= 0, (22)

uω,n(x, 0) = uω,n0 (x) = ωn−1 + n−s cos(nx). (23)

Note that uω,n(x, 0) belong in H∞ and

‖uω,n(t)‖Hs(R) ≈ 1, (24)

Therefore, applying the well-posednes results stated earlier weconclude that there is a T > 0 such that for any ω in a bounded setand n� 1 this i.v.p. has a unique solution uω,n(t) inC([−T, T ];Hs(T)).

Difference between approximate and actual solutionsThe difference between approximate and actual solutions

v.= uω,n − uω,n (25)

satisfies the Cauchy problem

∂tv = F − 1

2∂x[(uω,n + uω,n)v]

−D−2∂x[(uω,n + uω,n)v +1

2∂x(uω,n + uω,n)∂xv]

(26)

v(x, 0) = 0, x ∈ T, t ∈ R, (27)

where F satisfies the Hσ-estimate (20).

PropositionIf s > 3/2 and 1/2 < σ < min{1, s− 1}, then

‖v(t)‖Hσ . n−rs . (28)

Proof. It is based on energy estimates and the error estimate

‖F‖Hσ . n−rs .

Hs estimate for the difference v

Also, using the well-posedness estimates (13) we have that

‖v(t)‖Hs+1 . ‖uω,n(0)‖Hs+1 . n, t ∈ [0, T ]. (29)

Now, interpolating between σ and s+ 1 and using estimates (28) and(29) we get

‖v(t)‖Hs ≤ ‖v(t)‖1/(s+1−σ)Hσ ‖v(t)‖(s−σ)/(s+1−σ)

Hs+1 . n−1

s+1−σ (rs−s+σ).

Finally, from this inequality and the definition of rs (31) we obtain

‖v(t)‖Hs(T) . n−ρs , t ∈ [0, T ], (30)

where

ρs =

{(s− 1)/(s+ 1− σ), if s ≤ 3

2/(s+ 1− σ), if s ≥ 3,(31)

which shows that the the Hs norm of the difference between actualand approximate solutions is small!

Approximate solutions for more equations• The b-equation has the same approximate solutions

uω,n(x, t) = ωn−1 + n−s cos(nx− ωt), ω = ±1.

• The Euler equations on T2 have approximate solutions

uω,n(t, x)=(ωn−1+n−scos(nx2−ωt), ωn−1+n−scos(nx1−ωt)

), ω = ±1,

which are solutions! (H.–Misio lek, CMP 2010)

• Novikov and FORQ equations have approximate solutions

uω,n = ωn−1/2 + n−s cos(nx− ωt), ω = 0, 1.

• The Benjamin-Ono (BO) equation ut + uux +Huxx = 0 where

Hilbert transform H is defined by Hf(ξ) = −isgn(ξ)f(ξ) hasapproximate solutions

uω,n(x, t) = ωn−1 + n−s cos(−n2t+ nx− ωt), ω = ±1.

The nonperiodic version of these were introduced by Koch andTzvetkov, IMRN 2005.

Approximate solutions of b-equation in nonperiodic case

They are of the form uω,n = u` + uh, with uh the high frequency

uh = uh,ω,n(x, t) = n−δ2−sϕ(

x

nδ) cos(nx− ωt), ω = ±1,

where ϕ is in C∞ and such that

ϕ(x) =

{1, if |x| < 1,

0, if |x| ≥ 2.

u` = u`,ω,n(x, t) is the solution of the Cauchy problem for the CHequation with the low frequency initial data ωn−1ϕ( x

nδ)

∂tu` + u∂xu` + F (u`) = 0,

u`(x, 0) = ωn−1ϕ(x

nδ), x ∈ R, t ∈ R,

where ϕ is a C∞0 (R) function such that

ϕ(x) = 1, if x ∈ supp ϕ.

I``-posedness for the b-equation

Theorem (H-Grayshan-Holliman, 2015)

For s < 3/2 the Cauchy problem for the b-equation, b > 1, on both theline and the circle, is not well-posed in Hs in the sense of Hadamard.

• The case b = 3 (DP) was done by H-Grayshan-Holliman, CPDE2014 and the proof is based on peakon-antipeakon traveling wavesolutions and the conservation of a twisted L2 norm.

• The case b = 2 (CH) was done by Byers and the proof is based onpeakon-antipeakon traveling wave solutions and the conservation ofH1 norm.

Open Problem: Is b-equation well-posed at the critical Sobolevexponent s = 3/2?

Proof of Ill-posedness for b-equation

Nonlocal form of b-equation

The Cauchy problem for the b-equation, written in its nonlocal form,is given by

ut + uux + ∂x(1− ∂2x)−1[ b

2u2 +

3− b2

u2x

]= 0, (32)

u(x, 0) = u0(x). (33)

The regions of Ill-posedness

To describe precisely the reason for which well-posedness fails wepartition the region under consideration in the (b, s)-plane

A= {(b, s) ∈ R : 1 < b <∞ and s < 3/2} ,

into the following four subregions:

A1= {(b, s) ∈ A : b > 3; (b = 3, s ≥ 1/2); or (b < 3, s > 2− b/2)}A2= {(b, 2− b/2) ∈ A : b < 3}A3= {(b, s) ∈ A : b < 3, s < 2− b/2}A4= {(3, s) ∈ A : s < 1/2} .

b

s

3/2

1/2

31

A1A2

A3 A4

0

Behavior of Hs-norm and inflationThe first ingredient in our ill-posedness results is the construction ofpeakon-antipeakon solutions with arbitrarily small lifespan whichinitially have an arbitrarily small Hs norm and later have interestingproperties, including norm inflation.

Theorem (Norm behavior and Inflation)Let s < 3/2 and b > 1. Then, for any given ε > 0 there existsT = T (ε) > 0 such that the b-equation Cauchy problem (32)–(33) hasa solution u ∈ C([0, T );Hs) satisfying the following three properties:

1. Lifespan T < ε,

2. ‖u0‖Hs < ε,

3. limt↗T− ‖u(t)‖Hs '

∞, (b, s) ∈ A1, (norm inflation)

1, (b, s) ∈ A2 ∪A4,

0, (b, s) ∈ A3.

Remark. The notation ‖u(t)‖Hs ' Q(t) means that there is constantcs > 0 such that c−1s Q(t) ≤ ‖u(t)‖Hs ≤ csQ(t) for all t ∈ [0, T ).

Peakon-antipeakonWe make a unified presentation of our results. For this, on R we let

E(x) = e−|x|, (34)

and on T we let

E(x) = cosh(x− 2π

[ x2π

]− π

). (35)

Using this notation we have the following result.

Lemma (Peakon-antipeakon solutions and their DE)On both the line and the circle, the peakon-antipeakon traveling wave

u(x, t) = p(t)E(x+ q(t))− p(t)E(x− q(t)) (36)

is a solution to the b-equation if and only if the momentum p = p(t)and the position q = q(t) are solutions to the following system ofdifferential equations

p′ = (1− b)p2E′(2q) and q′ = p (E(2q)− E(0)) . (37)

b = 3 (DP equation): Peakon-antipeakon collision

xBefore collision

xAfter collision

Figure 5: Degasperis–Pro cesi peakon-antipeakon collision in the symmetric caseλ1 + λ2 = 0, computed from the exact solution formulas (3.8) and (3.9). Solidcurves show u(x, t) at evenly sampled times, with some additional samples closeto the collision shown by dashed curves. As the arrows indicate, the peaksapproach each other with constant speed and height, and form a stationaryshockpeakon whose shock strength decays like 1/t as t � +∞ .

Proof of Peakon-antipeakon

Computing E′ and E′′ as distributional derivatives we obtain theformulas

E′(x) =

{−sgn(x)e−|x|, on Rsinh

(x− 2π

[x2π

]− π

), on T

,

E′′(x) =

{e−|x| − 2δx, on Rcosh

(x− 2π

[x2π

]− π

)− 2 sinh(π)δx, on T

,

and the relation

(1− ∂2x)E(x) = 2γδx, (38)

where γ = 1 on R and γ = sinh(π) on T. Substituting u(x, t) given by(36) into the b-equation written in its m-form (6), and using theabove formulas we find that the peakon-antipeakon (36) is a solutionto b-family if and only if the p(t) and q(t) satisfy the system ofdifferential equations (37). We note, that on R this system can alsobe obtained from the 2-peakon solution (10) in [DHH] by choosingp = p1 = −p2 and q = −q1 = q2.

Properties of Peakon-antipeakonLemma (Solving the Peakon-antipeakon DEs)Given ε > 0 we choose initial data for the peakon-antipeakon so that

p(0) = p0 >2

(b− 1)εand 0 < q(0) = q0 <

ln 2

8. (39)

Then, there is a unique solution (p(t), q(t)) to the initial value problem forthe system of differential equations (37) such that p is given by the formula

p(t) = p0

(E(0)− E(2q0)

E(0)− E(2q(t))

) b−12

, (40)

and q(t) solves the differential equation

q′(t) = −p0 (E(0)− E(2q0))b−12 (E(0)− E(2q))

3−b2 . (41)

Furthermore, the lifespan T satisfies the estimate

0 < T ≤ 2

(b− 1)p0< ε. (42)

Finally, p(t) is increasing, q(t) is decreasing, and

limt↗T−

p(t) =∞, and limt↗T−

q(t) = 0. (43)

The case of DP (b = 3) and CH (b = 2)When b = 3 (DP equation) then the differential equation for q is

q′(t) = −p0(1− e−2q0

)(44)

Solving (44) with the initial condition q(0) = q0 we get the formula

q(t) = −p0(1− e−2q0

)t + q0. (45)

Solving q(t) = 0 for t we get an exact formula for the DP lifespan

T =q0

p0 (1− e−2q0). (46)

When b = 2 (CH equation) then the differential equation for q is

q′(t) = −p0(1− e−2q0

) 12

(1− e−2q(t)

) 12. (47)

Solving (47) we get the formula

q(t) = ln (cosh [q1t− q2]) , (48)

where q1 = p0(1− e−2q0

) 12 , and q2 = ln

(eq0 +

√e2q0 − 1

). Now, looking at

formula (48) we see that the CH lifespan is given by

T =q2q1

=ln(eq0 +

√e2q0 − 1

)p0√

1− e−2q0. (49)

b = 3 (DP equation): Peakon-antipeakon collision

xBefore collision

xAfter collision

Figure 5: Degasperis–Pro cesi peakon-antipeakon collision in the symmetric caseλ1 + λ2 = 0, computed from the exact solution formulas (3.8) and (3.9). Solidcurves show u(x, t) at evenly sampled times, with some additional samples closeto the collision shown by dashed curves. As the arrows indicate, the peaksapproach each other with constant speed and height, and form a stationaryshockpeakon whose shock strength decays like 1/t as t � +∞ .

Hs-norm sizeTo study the limiting value of the Hs-norm of peakon-antipeakonconstructed earlier we need the following result.

Proposition (Size of peakon-antipeakon at all times)Let u(t) be the peakon-antipeakon solution constructed in Lemma 5.Then, for all t ∈ [0, T ), we have the estimates

‖u(t)‖Hs '

p(t)q(t)3/2−s, 1/2 < s < 3/2

p(t)q(t)√

ln(1/q(t)), s = 1/2

p(t)q(t), s < 1/2.

(50)

Proof (H.-Holliman-Grayshan, CPDE-2014). Defining

fy(x).= e−|x| − e−|x−y|, (51)

we see that by taking y = 2q(t), the Hs-norm of u can be expressed as

‖u(t)‖Hs = p(t)‖f2q(t)(x+ q(t))‖Hs = p(t)‖fy(x)‖Hs . (52)

Then, using the definition of the Hs-norm we have

‖fy‖2Hs(R) = 16

∫R(1 + ξ2)s−2 sin2

(yξ/2

)dξ. (53)

. . .

b-equation ill-posedness: (b, s) in inflation region A1

For (b, s) in region A1, where norm-inflation occurs: Denote byun(t) the peakon-antipeakon solution of b-family corresponding to thechoice of ε = 1/n, and let u∞(t) = 0. Then, by property (2) ofTheorem 3, we have ‖un(0)‖Hs < 1/n. Therefore, un(0) converges tou∞(0) = 0 in Hs. Furthermore, by property (1), the lifespan Tn ofeach solution un(t) satisfies the inequality Tn < 1/n, whereas thelifespan T∞ of u∞(t) is equal to ∞. By property (3),limt↗∞ ‖u(t)‖Hs =∞, so there is no T > 0 such that the solutionsun(t) can be extended to the interval [0, T ] for all sufficiently large n.Hence, we see that condition (iii) of well-posedness fails. Thus, theb-equation is ill-posed in Hs for (b, s) ∈ A1.

Ill-posedness: (b, s) in the zero-limit region A3

For (b, s) in region A3, where the limit is equal to zero: Recallthat

A3= {(b, s) ∈ A : b < 3, s < 2− b/2}For a given ε > 0, by Theorem 3, there a peakon-antipeakon b-familysolution u(t) with lifespan 0 < T < ε and limt→T ‖u(t)‖Hs = 0, whichgives ‖u(T )‖Hs = 0, or u(T ) = 0. Since v(t) = 0 is another b-equationsolution with initial data v(T ) = 0, it follows that uniqueness ofsolution (condition (ii)) fails to hold. Thus, b-equation is ill-posed inHs for (b, s) ∈ A3.

Remark. The case A2 is similar. The case A4, which corresponds toDP can be found in Himonas-Grayshan-Holliman, CPDE 2014.

Analytic theory

Analytic spaces Gδ,s

For δ > 0 and s ≥ 0, in the periodic case we define

Gδ,s(T) = {ϕ ∈ L2(T) : ||ϕ||2Gδ,s=∑k∈Z

(1+k2)se2δ|k||ϕ(k)|2 <∞}, (54)

while in the nonperiodic case we define

Gδ,s(R) = {ϕ ∈ L2(R) : ||ϕ||2Gδ,s=∫R

(1 + ξ2)se2δ|ξ||ϕ(ξ)|2dξ <∞}.(55)

Remark. If ϕ ∈ Gδ,s(T) then ϕ has an analytic extension to asymmetric strip around the real axis with width δ. This δ is called theradius of analyticity of ϕ.

Well-posedness of CH equations in analytic spaces

Theorem (Barostichi-H.-Petronilho, JFA 2016)Let s > 1

2 . If u0 ∈ G1,s+2 on the circle or the line, then there exists apositive time T , which depends on the initial data u0 and s, such thatfor every δ ∈ (0, 1), the Cauchy problem for the b-equation has aunique solution u which is a holomorphic function in the discD(0, T (1− δ)) valued in Gδ,s+2. Furthermore, the analytic lifespan Tsatisfies the estimate

T ≈ 1

||u0||G1,s

. (56)

Theorem (Barostichi-H.-Petronilho, JFA 2016)If s > 1

2 , then the data-to-solution map u(0) 7→ u(t) of the Cauchyproblem for the b-equation is continuous from Gδ,s+2 into thesolutions space.

Remark. The above theorems have been proved in Barostichi,Himonas and Petronilho, JFA 2016, for several other CH typeequations and systems.

Thanks!