Download e-book for iPad: Invariant Manifolds and Dispersive Hamiltonian Evolution by Kenji Nakanishi, Wilhelm Schlag

By Kenji Nakanishi, Wilhelm Schlag

ISBN-10: 3037190957

ISBN-13: 9783037190951

The inspiration of an invariant manifold arises certainly within the asymptotic balance research of desk bound or status wave strategies of volatile dispersive Hamiltonian evolution equations reminiscent of the focusing semilinear Klein Gordon and Schr?¶dinger equations. this can be on the grounds that the linearized operators approximately such particular options normally express adverse eigenvalues (a unmarried one for the floor state), which bring about exponential instability of the linearized circulate and enables rules from hyperbolic dynamics to go into. one of many major effects proved right here for power subcritical equations is that the center-stable manifold linked to the floor country seems to be as a hyper-surface which separates a area of finite-time blowup in ahead time from one that indicates international life and scattering to 0 in ahead time. Our whole research occurs within the strength topology, and the conserved strength can exceed the floor country strength merely via a small volume. This monograph is predicated on fresh examine by means of the authors and the proofs depend upon an interaction among the variational constitution of the floor states at the one hand, and the nonlinear hyperbolic dynamics close to those states nonetheless. A key aspect within the evidence is a virial-type argument aside from nearly homoclinic orbits originating close to the floor states, and returning to them, most likely after an extended day trip. those lectures are compatible for graduate scholars and researchers in partial differential equations and mathematical physics. For the cubic Klein Gordon equation in 3 dimensions all information are supplied, together with the derivation of Strichartz estimates for the unfastened equation and the concentration-compactness argument resulting in scattering as a result of Kenig and Merle.

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Example text

62) is seen to be false, concluding the proof of compactness of KC . 4 Scattering in P SC Finally, we show that KC precompact leads to a contradiction via the following virial argument; this is the “rigidity” step from [84], cf. Part (iv) above. First, let be a compactly supported cutoff function so that D 1 on jxj Ä 1. 47). The O. /-term here is uniformly small by the precompactness of the forward trajectory of u . R/ (by energy conservation and the fact that we are in the region fK 0g which ensures that the free energy of u is uniformly bounded in time), whereas if the entire right-hand side is < ı < 0 for some fixed ı > 0, then a contradiction follows by taking t0 large.

0/ 0. 54) Note that the left-hand side does not depend on time since nk is a free wave. 4 Scattering in P SC as n ! 0/ > ı0 > 0 for large k and n. UE j / < E for each j . By the minimality of E each U j is a global solution and scatters with kU j kL3 L6x < 1. 36). v/ D . t C tnj / 3 : j

74) as a convolution with a fixed Schwartz function and passing to the limit n ! 75) with an absolute constant C0 . 0/, and define n2 WD un v 1 . C tn1 ; C xn1 /. By construction, En2 . tn1 ; xn1 / * 0. P 2 k1 2 n /. tn2 ; xn2 / 2 ˇ tn2 ; xn2 /ˇ > 2 2 2 for large n. 0/ defined by En2 . 0/. / in H. Suppose jtn1 tn2 j C jxn1 xn2 j remains bounded as n ! 1. Then we may assume that tn1 tn2 ! and xn1 xn2 ! whence En2 . tn1 tn2 / En2 . tn1 ; xn1 C xn1 xn2 / * 0; n ! 1: Thus, jtn1 tn2 j C jxn1 xn2 j !

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Invariant Manifolds and Dispersive Hamiltonian Evolution Equations by Kenji Nakanishi, Wilhelm Schlag

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