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'On the Wellposedness of the Kdv Equation on the Space of Pseudomeasures' Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2018 On the wellposedness of the KdV equation on the space of pseudomeasures Kappeler, Thomas ; Molnar, Jan Abstract: In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudomeasures, also referred to as the Fourier Lebesgue space F฀∞(T,R), where F฀∞(T,R) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space F฀s,∞(T,R) with −1/2<s฀0 and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution q in the Sobolev space H−1(T,R) to be in F฀∞(T,R) in terms of asymptotic behavior of spectral quantities of the Hill operator −฀2x+q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved. DOI: https://doi.org/10.1007/s00029-017-0347-1 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-149834 Journal Article Accepted Version Originally published at: Kappeler, Thomas; Molnar, Jan (2018). On the wellposedness of the KdV equation on the space of pseudomeasures. Selecta Mathematica, 24(2):1479-1526. DOI: https://doi.org/10.1007/s00029-017-0347-1 On the wellposedness of the KdV equation on the space of pseudomeasures Thomas Kappeler,∗ Jan Molnar† October 4, 2016 Abstract In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudo-measures, also referred to as the Fourier Lebesgue space Fℓ∞(T, R), where Fℓ∞(T, R) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space Fℓs,∞(T, R) with −1/2 < s 6 0 and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution q in the Sobolev space 1 H − (T, R) to be in Fℓ∞(T, R) in terms of asymptotic behavior of spectral 2 quantities of the Hill operator −∂x +q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved. Keywords. KdV equation, well-posedness, Birkhoff coordinates 2000 AMS Subject Classification. 37K10 (primary) 35Q53, 35D05 (secondary) 1 Introduction In this paper we consider the initial value problem for the Korteweg-de Vries equation on the circle T = R/Z, arXiv:1610.00278v1 [math.AP] 2 Oct 2016 ∂ u = ∂3u + 6u∂ u, x T, t R. (1) t − x x ∈ ∈ Our goal is to improve the result of Bourgain [2] on global wellposedness for F ∞ solutions evolving in the Fourier Lebesgue space ℓ0 with small Borel measures as initial data. The space Fℓ∞ consists of 1-periodic distributions q S′(T, R) 0 ∈ ∗Partially supported by the Swiss National Science Foundation †Partially supported by the Swiss National Science Foundation 1 whose Fourier coefficients q = q, eikπx , k Z, satisfy (q ) Z ℓ∞(Z, R) and k h i ∈ k k∈ ∈ q0 = 0. Here and in the sequel we view for convenience 1-periodic distributions as 2-periodic ones and denote by f, g the L2-inner product 1 2 f(x)g(x) dx h i 2 0 extended by duality to S′(R/2Z, C) C∞(R/2Z, C). We point out that q = × R 2k+1 0 for any k Z since q is a 1-periodic distribution. We succeed in dropping ∈ the smallness condition on the initial data and can allow for arbitrary initial data q Fℓ∞. In fact, our wellposedness results hold true for any of the spaces ∈ 0 Fℓs,∞ with 1/2 <s 6 0 where 0 − s,∞ ′ Fℓ = q S (T, R) : q = 0 and (q ) Z < , 0 { ∈ 0 k k k∈ ks,∞ ∞} and s (qk)k∈Z s,∞ := sup k qk , α :=1+ α . k k k∈Zh i | | h i | | Informally stated, our result says that for any 1/2 <s 6 0, the KdV equation − 0 F s,∞ is globally C -wellposed on ℓ0 . To state it more precisely, we first need to recall the wellposedness results established in [17] on the Sobolev space −1 −1 T R H0 H0 ( , ). Let 6 a < b 6 be given. A continuous curve ≡ −1 −∞ −1 ∞ γ : (a, b) H0 with γ(0) = q H0 is called a solution of (1) with initial → ∈ ∞ (m) data q if and only if for any sequence of C -potentials (q )m>1 converging −1 (m) to q in H , the corresponding sequence ( (t, q )) > of solutions of (1) 0 S m 1 with initial data q(m) converges to γ(t) in H−1 for any t (a, b). In[17] it was 0 ∈ proved that the KdV equation is globally in time C0-wellposed meaning that for any q H−1 (1) admits a solution γ : R H−1 with initial data in the ∈ 0 → 0 above sense and for any T > 0 the solution map : H−1 C([ T, T ], H−1) is S 0 → − 0 continuous. Note that for any 1/2 < s 6 0, Fℓs,∞ continuously embeds into − 0 −1 F s,∞ F s,∞ F −s,1 H0 . On ℓ0 we denote by τw∗ the weak* topology σ( ℓ0 , ℓ0 ). We refer to Appendix B for a discussion. F s,∞ Theorem 1.1 For any q ℓ0 with 1/2 < s 6 0, the solution curve s,∈∞ − t (t, q) evolves in Fℓ . It is bounded, sup R (t, q) < and 7→ S 0 t∈ kS ks,∞ ∞ continuous with respect to the weak* topology τw∗. ⋊ Remark 1.2. It is easy to see that for generic initial data, the solution curve 3 t Airy(t, q) of the Airy equation, ∂tu = ∂xu, is not continuous with respect 7→ S F s,∞ − to the norm topology of ℓ0 . A similar result holds true for the KdV equation at least for small initial data – see Section 4. ⊸ We say that a subset V Fℓs,∞ is KdV-invariant if for any q Fℓs,∞, ⊂ 0 ∈ 0 (t, q) V for any t R. S ∈ ∈ Theorem 1.3 Let V Fℓs,∞ with 1/2 <s 6 0 be a KdV-invariant - ⊂ 0 − k·ks,∞ norm bounded subset, that is R (t, q) V, t , q V ; sup q s,∞ < . S ∈ ∀ ∈ ∈ q∈V k k ∞ October 4, 2016 2 Then for any T > 0, the restriction of the solution map to V is weak* S continuous, : (V, τ ) C([ T, T ], (V, τ )). ⋊ S w∗ → − w∗ By the same methods we also prove that the KdV equation is globally Cω- F 0,1 wellposed on the Wiener algebra ℓ0 – see Section 5 where such a result is N,1 proved for the weighted Fourier Lebesgue space Fℓ , N Z> . 0 ∈ 0 Method of proof. Theorem 1.1 and Theorem 1.3 are proved by the method of normal forms. We show that the restriction of the Birkhoff map Φ: H−1 0 → −1/2,2 F s,∞ ℓ0 q z(q)=(zn(q))n∈Z, constructed in [10], to ℓ0 is a map with 7→s+1/2,∞ Z C values in ℓ0 ( , ), having the following properties: Theorem 1.4 For any 1/2 <s 6 0, Φ: Fℓs,∞ ℓs,∞ is a bijective, bounded, − 0 → 0 real analytic map between the two Banach spaces. Near the origin, Φ is a local diffeomorphism. When restricted to any -norm bounded subset k·ks,∞ F s,∞ V ℓ0 , Φ: V Φ(V ) is a homeomorphism when V and Φ(V ) are en- ⊂ → F s,∞ F −s,1 s+1/2,∞ −(s+1/2),1 dowed with the weak* topologies σ( ℓ0 , ℓ0 ) and σ(ℓ0 ,ℓ0 ), respectively. Furthermore for any q Fℓs,∞, the set Iso(q) of elements q˜ H−1 ∈ 0 ∈ 0 so that ∂2 + q and ∂2 +q ˜ have the same periodic spectrum is a -norm − x − x k·ks,∞ bounded subset of Fℓs,∞ and hence Φ: Iso(q) Φ(Iso(q)) is a homeomorphism 0 → when Iso(q) and Φ(Iso(q)) are endowed with the weak* topologies. ⋊ Remark 1.5. Note that by [10] for any q H−1, Φ(Iso(q)) = where ∈ 0 TΦ(q) −1/2,2 = z˜ = (˜z ) Z ℓ : z˜ = Φ(q) k Z . (2) TΦ(q) { k k∈ ∈ 0 | k| | k| ∀ ∈ } Furthermore, since by Theorem 1.4 for any q Fℓs,∞, 1/2 <s 6 0, Iso(q) is ∈ 0 − F s,∞ bounded in ℓ0 , the weak* topology on Iso(q) coincides with the one induced by the norm for any 1/2 <σ<s, 2 6 p < with (s σ)p > 1 – cf. k·kσ,p − ∞ − Lemma B.1 from Appendix B. ⊸ Key ingredient for studying the restriction of the Birkhoff map to the Ba- F s,∞ nach spaces ℓ0 are pertinent asymptotic estimates of spectral quantities of the Schrödinger operator ∂2 +q, which appear in the estimates of the Birkhoff − x coordinates in [10, 7] – see Section 2. The proofs of Theorem 1.1 and Theo- rem 1.3 then are obtained by studying the restriction of the solution map , S −1 F s,∞ defined in [17] on H0 , to ℓ0 . To this end, the KdV equation is expressed in Birkhoff coordinates z =(zn)n∈Z. It takes the form ∂ z = iω z , ∂ z =iω z , n > 1, t n − n n t −n n −n 1 where ωn, n > 1, are the KdV frequencies. For q H0 , these frequencies are ∈1 1 2 3 defined in terms of the KdV Hamiltonian H(q)= 0 2 (∂xq) + q dx.
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