5.1 Quadratic dispersion relations
5.2.5 Proof of new regularity results for the modified Korteweg-de
Since there are four factors with frequency higher or equal toK2, there is enough smoothing from theL6t,x-estimate to sum the expression even for negative regulari-ties choosingδsufficiently small.
Subsubcase BIIb:K2∼K3.
IfK7∼K6, then|Ω(1)(n1, n5, n6, n7)|&K53K23and the argument from Subcase BI applies.
Similarly, ifK2K7K5we find
|Ω(1)(n1, n5, n6, n7)| ∼K52K7K23&|Ω(1)(n1, n2, n3, n4)|
Thus, we can suppose thatK7 .K2. In this case the argument from Subsubcase BIIa applies because there are at least two frequencies comparable to K2 and at most two frequencies, namelyK5andK6, much higher thanK2.
The proof is complete.
Remark 5.2.24. We point out from the proofs of Propositions 5.2.20, 5.2.22 and 5.2.23 and Lemma 5.2.21 that there is some slack in the regularity. In fact, we can lower the regularity on the right hand-side depending ons(after making ε=ε(s) smaller if necessary). This observation becomes important in the construction of the data-to-solution mapping.
To conclude the proof of the energy estimate, one derives a bound for the thresh-olds of the frequency localized energy (cmp. Lemma 5.1.16). For details we refer to Section 5.1.3.
5.2.5 Proof of new regularity results for the modified
To carry out the continuity argument, recall the continuity and limit properties of T 7→ kukEs(T),T 7→ kukNs(T).
We are ready to prove a priori estimates for smooth solutions.
Proof of Lemma 5.2.25. Assuming that u0 is a smooth and real-valued initial da-tum, we find from the classical well-posedness theory the global existence of a smooth and real-valued solutionu∈C(R, H∞) (see e.g. [Bou93a]) which satisfies the set of estimates (5.98).
We defineX(T) =kukEs(T)+kN(u)kNs,1(T) and find the bound
X(T)2≤C1ku0k2Hs+C2((TθMc(s)+M−d(s))X(T)2+TθX(T)4)X(T)2 by eliminatingkukFs,α(T) in the above system of estimates.
SetR=C11/2ku0kHs and chooseM =M(R) large enough so that C2M−d(s)(2R)2<1/4.
Next, chooseT0=T0(R)≤1 small enough so that C2T0θ(Mc(s)(2R)2+ (2R)4)<1/4.
Together with the limiting properties for T → 0 a continuity argument yields X(T)≤2RforT ≤T0.
Iterating the argument gives supt∈[0,T
0]ku(t)kHs(T).ku0kHs forT0=T0(ku0kHs).
The proof is complete.
We establish the existence of the solution mapping. For u0 ∈ Hs(T), we set u0,n=P≤nu0forn∈N. Obviously,u0,n∈H∞(T) and hence, the initial data give rise to smooth global solutionsun∈C(R, H∞(T)). According to Lemma 5.2.25 we already have a priori estimates on a time interval [0, T0] where T0 =T0(ku0kHs) independent ofn. Moreover, we have the following compactness lemma (cf. Lemma 5.1.18), which is proven by the same means like above:
Lemma 5.2.26. Let u0 ∈ Hs(T) for some s > 0. Let un be the smooth global solutions to (5.52) withun(0) =u0,n like above.
Then,(un)n∈N is precompact inC([−T, T], Hs(T))forT ≤T0=T0(ku0kHs).
Key ingredient like above is the uniform tail estimate, i.e., there isn0∈Nsuch that for anyn∈N
kP≥n0unkCTHs < ε. (5.99) We are ready to prove the main result:
Proof of Theorem 5.2.2. For u0 ∈ Hs(T) let (un)n∈N denote the smooth global solutions generated from the initial data P≤nu0 as described above. By Lemma 5.2.26 we find a convergent subsequence (unk), which converges to a functionu∈ C([−T, T], Hs). We observe that due to (5.99) the sequence also converges inEs(T).
WithkN(un−u)kNs,1(T).Tθku0k2Hskun−ukFs,1(T), we find for T =T(ku0kHs) the estimate
kun−ukFs,1(T).kun−ukEs(T)
to hold. The convergence in Fs,1(T) already gives the a priori estimate for the limit. Moreover, we deduce from the multilinear estimates in Proposition 5.2.14 that {N(un)}converges toN(u) inNs,1(T),→ D0. We conclude thatusatisfies (5.52) in the distributional sense with the claimed properties and the proof is complete.
For the proof of non-existence of solutions to the unrenormalized mKdV equa-tion, we compare smooth solutions to (5.2) and (5.52) via a gauge transform. The argument parallels [GO18].
We sketch the argument for the sake of self-containedness and for details refer to [GO18].
Proof of Theorem 5.2.3. Existence and a priori estimates of solutions to (5.52) for negative regularity conditional upon conjectured Strichartz estimates are proved like above. Here, corresponding estimates to (5.98) are utilized.
For the proof of non-existence of solutions to (5.2), we argue by contradiction.
Fixs <0 from the hypothesis of Theorem 5.2.3 and u0∈Hs(T)\L2(T). Suppose that there existsT >0 and a solution u∈C([−T, T], Hs(T)) to (5.2) in the sense of definition 5.2.1.
By defining
vn(t) =e−2itRT|u0,n|2dxun(t), we find a sequence of smooth solutions to (5.52).
Further, by assumption
vn(t= 0) =un(0)→u0 inHs(T).
By a variant of Lemma 5.2.26, there is a subsequence (vnk)k converging to v in C([−T, T], Hs) with T = T(ku0kHs). The convergence of un implies the con-vergence of vn to 0 in the sense of distributions: Let φ ∈ Cc∞([−T, T], C∞(T)).
Then,
hun(t), φ(t)iL2
x →F(t) :=hu(t), φ(t)iL2 x
by convergence ofun(t) inC([−T, T], Hs). Further,F ∈Cc(R).
It follows that
| Z Z
vnφdxdt|=| Z
e−2itRT|u0,n|2dxhun(t), φ(t)iL2xdt|
≤ | Z
e−2itRT|u0,n|2dxF(t)dt|+ Z
|hu(t)−un(t), φ(t)iL2
x|dt→0.
The first term vanishes according to the Riemann-Lebesgue lemma and the second term due toun → u in CTHs. This means that vn converges to 0 in the distri-butional sense, and sincevnk →v in CTHs, this implies v ≡0. This contradicts u0=v(0)6= 0.
Chapter 6
Local and global
well-posedness for dispersion generalized Benjamin-Ono equations on the circle
6.1 Introduction to dispersion generalized Benja-min-Ono equations
In this chapter we prove new regularity results for the one-dimensional fractional Benjamin-Ono equation in the periodic case
∂tu+∂xDaxu =u∂xu, (t, x)∈R×T,
u(0) =u0, (6.1)
where 1< a <2 is considered in the following.
Previous works on dispersion generalized Benjamin-Ono equations include [HIKK10, Guo12] in the real line case and [MV15] in the periodic case. In [MV15]
global well-posedness was proved in Hs(T) for s ≥ 1−a/2, where 1 ≤ a ≤ 2.
For details on these works, we refer to the remarks on the well-posedness theory of the Benjamin-Ono equation in Chapter 1. The following results are proven via short-time analysis:
Theorem 6.1.1. For 1 < a ≤3/2, (6.1) is locally well-posed in Hs(T) provided thats >3/2−a.
For3/2< a <2,(6.1)is globally well-posed in L2(T).
Remark 6.1.2. Recall that Molinet pointed out in [Mol08] that in the Benjamin-Ono case the periodic data-to-solution mapping is C∞ on hyperplanes of initial data with fixed mean. From this, one might suspect that this is also true in the dispersion generalized case. However, Herr proved in [Her08] that (6.1) can not be solved via Picard iteration for 1≤a <2 explaining our use of short-time analysis.
The analysis extends the short-time Strichartz analysis from Chapter 3, which is further improved by considering modified energies. By this we mean correction
terms for the frequency localized energy corresponding to normal form transforma-tions like in Chapter 5, but without symmetrization.
The improved symmetrized expression does not yield new information when an-alyzing differences of solutions because of reduced symmetry. Still, normal form transformations allow us to improve the energy estimates.
An early application of modified energies was given by Kwon in [Kwo08]. In this work, modified energies were combined with short-time Strichartz estimates (cf. [KT03]) in order to improve the local well-posedness theory for the fifth-order KdV equation. This was refined by Kenig-Pilod in [KP15] using short-time Fourier restriction spaces to prove global well-posedness in the energy space. In the inde-pendent work by Guo-Kwak-Kwon [GKK13] a modulation weight was used to prove the same result.
An application of modified energies in the context of short-time Fourier restriction spaces for periodic solutions was given by Kwak in [Kwa16]. In this work, the global well-posedness of the fifth-order KdV equation on the circle was proved inH2.
On the real line, short-time analysis for dispersion generalized Benjamin-Ono equations was already carried out in [Guo12]. In [Guo12] no normal form transfor-mations were used, which gave local well-posedness fors≥2−a, where 1≤a≤2.
The gain from introducing modified energies is most significant for large dispersion coefficients allowing us to prove well-posedness inL2(T). Further, it appears as if some of the arguments can be applied in the low-dispersion case 0< a < 1. For these equations on the circle, which are also of physical interest, are currently no well-posedness results beyond the energy method available.
On the real line, there is the recent work by Molinet-Pilod-Vento [MPV18] refin-ing the analysis from [MV15] by normal form transformations. Since this analysis makes use of smoothing effects unavailable on the circle, it is not clear how to extend the analysis from [MPV18] to the circle.
The local well-posedness result from Theorem 6.1.1 for 1 < a < 2, which is globalized fora >3/2 due to conservation of mass onTis currently the best. This improves global well-posedness fors≥1−a/2, where 1< a <2, proved in [MV15].
As argued in the previous chapters, the analysis can be transferred to the real line.
This yields a possible simplification of the analysis from [HIKK10]. On the real line, the multilinear estimates relying on linear and bilinear Strichartz estimates are improved due to dispersive effects. However, the introduction of a modified energy would require additional care because the resonance
Ω(ξ1, ξ2, ξ3) =ξ1|ξ1|a+ξ2|ξ2|a+ξ3|ξ3|a ξi∈R, ξ1+ξ2+ξ3= 0
might become arbitrarily small in modulus for non-vanishingξi∈R. To avoid this, we confine ourselves to initial data with vanishing mean. As this is a conserved quantity, there is no loss of generality in assuming
Z
T
u(x)dx= 0.
When we localize time, we do not work in Euclidean windows, but rather base the analysis on the time localization T = T(N) = Na−2: This is explained by interpolating between Euclidean windows in the Benjamin-Ono case and the Fourier restriction norm analysis for a = 2, where frequency dependent time localization is no longer required. For the large data theory, it turns out to be convenient to consider the slightly shorter timesNa−2−δ giving an additional factor ofTθin the
nonlinear estimates (cf. Lemma 2.5.3).
The following set of estimates will be established for the proof of Theorem 6.1.1 for a smooth solutionu to (6.1) with vanishing mean. For 1< a < 2,T ∈(0,1], M ∈2N0 ands0≥s≥max(3/2−a,0), there areδ(a, s)>0,c(a, s)>0,d(a, s)>0 andθ(a, s)>0 such that
kukFas0,δ(T) .kukEs0(T)+ku∂xuk
Nas0,δ(T)
ku∂xukNs0,δ
a (T) .TθkukFs0,δ
a (T)kukFs,δ a (T)
kuk2
Es0(T) .ku(0)k2
Hs0(T)
+McTkuk2
Fas0,δ(T)kukFs,δ a (T)
+M−dkuk2
Fas0,δ(T)kukFs,δ
a (T)+Tθkuk2
Fas0,δ(T)kuk2
Fas,δ(T). By the usual bootstrap arguments (cf. Section 5.2.5), the above display gives a priori estimates. In this chapter we omit the bootstrap arguments to avoid repetition.
For differences of solutions v =u1−u2, where ui denote smooth solutions to (6.1) with vanishing mean, we have the following set of estimates fors >3/2−a in case 1< a≤3/2 and s= 0 in case 3/2< a <2 and the remaining parameters like in the previous display:
kvkF−1/2,δ
a (T) .kvkE−1/2(T)+k∂x(v(u1+u2))kN−1/2,δ a (T)
k∂x((u1+u2)v)kN−1/2,δ
a (T) .TθkvkF−1/2,δ
a (T)(ku1kFs,δ
a (T)+ku2kFs,δ a (T)) kvk2E−1/2(T) .kv(0)k2H−1/2
+McTkvk2
Fa−1/2,δ(T)(ku1kFs,δ
a (T)+ku2kFs,δ a (T)) +M−dkvk2
Fa−1/2,δ(T)(ku1kFs,δ
a (T)+ku2kFs,δ a (T)) +Tθkvk2
Fa−1/2,δ(T)(ku1k2
Fas,δ(T)+ku2k2
Fas,δ(T)), which yields Lipschitz-continuity inH−1/2 for initial data inHs.
The related set of estimates with parameters like in the previous display
kvkFs,δ
a (T) .kvkEs(T)+k∂x(v(u1+u2))kNs,δ a (T)
k∂x(v(u1+u2))kNs,δ
a (T) .TθkvkFs,δ a (T)
ku1kFs,δ
a (T)+ku2kFs,δ a (T)
kvk2Es(T) .kv(0)k2Hs
+McTkvk2
Fas,δ(T)(ku2kFs,δ
a (T)+kvkFs,δ a (T)) +M−dkvk2
Fas,δ(T)(ku2kFs,δ
a (T)+kvkFs,δ a (T)) +Tθ(kvk2
Fas,δ(T)(ku2k2
Fas,δ(T)+kvk2
Fas,δ(T)) +kvkF−1/2,δ
a (T)kvkFs,δ
a (T)ku2kFr,δ
a (T)ku2kFs,δ a (T)), wherer= (2−a) +s, yields continuous dependence by a variant of the Bona-Smith approximation (cf. Section 3.5).
This chapter is structured as follows: After introducing function spaces in Sec-tion 6.2, we consider linear and bilinear estimates of funcSec-tions localized in frequency and modulation in Section 6.3. In Section 6.4 the short-time bilinear estimate is carried out, and in Section 6.5 the energy estimates are proved.