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Global wellposedness of KdV in \(H^{-1}(\mathbb T,\mathbb R)\). (English) Zbl 1106.35081
In the two following theorems the global in time well-posedness in Sobolev space \(H^{\beta}(\mathbb{T},\mathbb{R}),\;\; \beta \geq -1\) (\(\mathbb{T}\) is the circle \(\mathbb{R}/\mathbb{Z}\) of the length \(1\)) of the Cauchy problem for the KdV equation \[ \partial_tv(x,t)=-\partial_x^3v(x,t)+6v(x,t)\partial_x v(x,t), v(0,x)=q \in H^{\beta}(\mathbb{T}) \] is proved.
(1) KdV is globally \(C^0\)-wellposed in \(H^{\beta}(\mathbb{T})\) for any \(\beta\geq -1.\) In particular, the solution map \(\mathcal{J}:H^{\beta}(\mathbb{T})\to C(\mathbb{R},H^{\beta}(\mathbb{T}))\) has the group property \(\mathcal{J}(t+s,q)=\mathcal{J}(t,\mathcal{J}(s,q)),\;\forall t,s \in \mathbb{R}\) and for any \(t\in \mathbb{R}\;\; \mathcal{J}_t:H^{\beta}(\mathbb{T})\to H^{\beta}(\mathbb{T}),\; q\longmapsto \mathcal{J}(t,q)\) is a homomorphism.
(2) For any \(q\in H^{\beta}(\mathbb{T})\) with \(\beta\geq -1\) the solution \(t\longmapsto \mathcal{J}(t,q)\) has the following properties:
(i) \(\{\mathcal{J}(t,q)|t\in \mathbb{R}\}\)is relatively compact in \(H^{\beta}(\mathbb{T})\);
(ii) the solution \(t\longmapsto \mathcal{J}(t,q)\) is almost periodic;
(iii) \(\mathcal{J}(t,q)\in I_{s0}(L_q)=\{p\in H^{-1}(\mathbb{T})|\text{spec}(L_p)=\text{spec}(L_q)\},\) with \(L_q=-\frac{d^2}{dx^2}+q\) being the Hill operator and spec\((L_q)\) denoting the periodic spectrum of \(L_q\).

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35D05 Existence of generalized solutions of PDE (MSC2000)
35G25 Initial value problems for nonlinear higher-order PDEs
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[1] D. BäTtig, T. Kappeler, and B. Mityagin, On the Korteweg –.de Vries equation: Frequencies and initial value problem , Pacific J. Math 181 (1997), 1–55. · Zbl 0899.35096 · doi:10.2140/pjm.1997.181.1 · nyjm.albany.edu:8000
[2] H. Bohr, Fastperiodische Funktionen , reprint, Springer, Berlin, 1974.
[3] J. L. Bona and R. Smith, The initial-value problem for the Korteweg –.de Vries equation , Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555–601. JSTOR: · Zbl 0306.35027 · doi:10.1098/rsta.1975.0035 · links.jstor.org
[4] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV-equation , Geom. Funct. Anal. 3 (1993), 209–262. · Zbl 0787.35098 · doi:10.1007/BF01895688 · eudml:58123
[5] -, “On the Cauchy problem for periodic KDV-type equations” in Conference in Honor of Jean-Pierre Kahan (Orsay, France, 1993) , J. Fourier Anal. Appl. 1995 , special issue, 17–86.
[6] -, Periodic Korteweg de Vries equation with measures as initial data , Selecta Math. (N.S.) 3 (1997), 115–159. · Zbl 0891.35138 · doi:10.1007/s000290050008
[7] -, Global Solutions of Nonlinear Schrödinger Equations , Amer. Math. Soc. Colloq. Publ., Amer. Math. Soc., Providence, 1999. · Zbl 0933.35178
[8] M. Christ, J. Colliander, and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations , Amer. J. Math. 125 (2003), 1235–1293. · Zbl 1048.35101 · doi:10.1353/ajm.2003.0040
[9] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Sharp global well-posedness for KdV and modified KdV on \(\R\) and \(\T\) , J. Amer. Math. Soc. 16 (2003), 705–749. · Zbl 1025.35025 · doi:10.1090/S0894-0347-03-00421-1
[10] -, Symplectic nonsqueezing of the KdV flow , · arxiv.org
[11] -, Local and global well-posedness for non-linear dispersive and wave equations , Web site, http://www.math.ucla.edu/\(\sim\)tao/Dispersive B. A. Dubrovin, A periodic problem for the Korteweg –.de Vries equation in a class of short-range potentials (in Russian), Funkcional. Anal. i Priložen 9 , no. 3 (1975), 41–51.; English translation in Functional Anal. Appl. 9 (1975), 215–223.
[12] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable Systems” in Dynamical Systems, IV , Encyclopaedia Math. Sci. 4 , Springer, Berlin, 2001, 177–332.
[13] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Nonlinear equations of Korteweg –.de Vries type, finite-band linear operators and Abelian varieties (in Russian), Uspehi Mat. Nauk 31 , no. 1 (1976), 55–136. · Zbl 0326.35011
[14] H. Flaschka and D. W. Mclaughlin, Canonically conjugate variables for the Korteweg –.de Vries equation and the Toda lattice with periodic boundary conditions , Progr. Theoret. Phys. 55 (1976), 438–456. · Zbl 1109.35374 · doi:10.1143/PTP.55.438
[15] A. R. Its and V. B. Matveev, “A class of solutions of the Korteweg –.de Vries equation” (in Russian) in Problems in Mathematical Physics, No. 8 , Izdat. Leningrad Univ., Leningrad, 1976, 70–92.
[16] T. Kappeler and M. Makarov, On Birkhoff coordinates for KdV , Ann. Henri Poincaré 2 (2001), 807–856. · Zbl 1017.76015 · doi:10.1007/s00023-001-8595-0
[17] T. Kappeler and C. MöHr, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator with singular potentials , J. Funct. Anal. 186 (2001), 62–91. · Zbl 1004.34077 · doi:10.1006/jfan.2001.3779
[18] T. Kappeler, C. MöHr, and P. Topalov, Birkhoff coordinates for KdV on phase spaces of distributions , Selecta Math. (N.S.) 11 (2005), 37–98. · Zbl 1089.37042 · doi:10.1007/s00029-005-0009-6
[19] T. Kappeler and J. PöSchel, KdV & KAM , Ergeb. Math. Grenzgeb. 3 , Springer, Berlin, 2003.
[20] T. Kappeler and P. Topalov, Riccati representation for elements in \(H^-1(\T^1)\) and its applications , Pliska Stud. Math. Bulgar. 15 (2003), 171–188.
[21] -, Global well-posedness of mKdV in \(L^2(\T,\R)\) , Comm. Partial Differential Equations 30 (2005), 435–449. · Zbl 1080.35119 · doi:10.1081/PDE-200050089
[22] -, Riccati map on \(L^ 2_ 0(\mathbb T)\) and its applications , J. Math. Anal. Appl. 309 (2005), 544–566. · Zbl 1085.34067 · doi:10.1016/j.jmaa.2004.09.061
[23] C. E. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation , J. Amer. Math. Soc. 9 (1996), 573–603. JSTOR: · Zbl 0848.35114 · doi:10.1090/S0894-0347-96-00200-7 · links.jstor.org
[24] -, On the ill-posedness of some canonical dispersive equations , Duke Math. J. 106 (2001), 617–633. · Zbl 1034.35145 · doi:10.1215/S0012-7094-01-10638-8
[25] E. Korotyaev, Characterization of the spectrum for Schrödinger operators with periodic distributions , Int. Math. Res. Not. 2003 , no. 37, 2019–2031. · Zbl 1104.34059 · doi:10.1155/S1073792803209107
[26] S. B. Kuksin, Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDEs , Comm. Math. Phys. 167 (1995), 531–552. · Zbl 0827.35121 · doi:10.1007/BF02101534
[27] P. D. Lax, Periodic solutions of the KdV equation , Comm. Pure Appl. Math. 28 (1975), 141–188. JSTOR: · Zbl 0295.35004 · links.jstor.org
[28] H. P. Mckean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points , Comm. Pure Appl. Math. 29 (1976), 143–226. · Zbl 0339.34024 · doi:10.1002/cpa.3160290203
[29] H. P. Mckean and P. Van Moerbeke, The spectrum of Hill’s equation , Invent. Math. 30 (1975), 217–274. · Zbl 0319.34024 · doi:10.1007/BF01425567 · eudml:142350
[30] H. P. Mckean and K. L. Vaninsky, Action-angle variables for the cubic Schrödinger equation , Comm. Pure Appl. Math. 50 (1997), 489–562. · Zbl 0990.35047 · doi:10.1002/(SICI)1097-0312(199706)50:6<489::AID-CPA1>3.0.CO;2-4
[31] C. MöHr, Schrödinger operators with singular potentials on the circle: Spectral analysis and applications, Ph.D. dissertation, University of Zurich, Zurich, 2001.
[32] J. PöSchel and E. Trubowitz, Inverse Spectral Theory , Pure Appl. Math. 130 , Academic Press, Boston, 1987. · Zbl 0623.34001
[33] J. C. Saut and R. Temam, Remarks on the Korteweg –.de Vries equation , Israel J. Math. 24 (1976), 78–87. · Zbl 0334.35062 · doi:10.1007/BF02761431
[34] R. Temam, Sur un problème non linéaire , J. Math. Pures Appl. (9) 48 (1969), 159–172. · Zbl 0187.03902
[35] Y. Zhou, Uniqueness of weak solution of the KdV equation , Internat. Math. Res. Notices 1997 , no. 6, 271–283. \endthebibliography · Zbl 0883.35105 · doi:10.1155/S1073792897000202
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