Fix s > 1. Colliander, Keel, Staffilani, Tao and Takaoka proved in [CKS + 10] the existence of solutions of the cubic defocusing nonlinear Schrödinger equation in the two torus with s-Sobolev norm growing in time. In this paper we generalize their result to the cubic defocusing nonlinear Schrödinger equation with a convolution potential. Moreover, we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear Schrödinger equation in [GK12]. The results we obtain can deal with any potential in H s0 (T 2 ), s 0 > 0.Are there solutions of (5) with periodic boundary conditions in dimension 2 or higher with unbounded growth of H s -norm for s > 1?Moreover, he conjectured, that in case this is true, the upper bound that he had obtained in [Bou96] was not optimal and that the growth should be subpolynomial in time, that is, u(t) H s ≪ t ε u(0) H s for t → ∞, for all ε > 0.
We prove there are finitely many isometry classes of planar central configurations (also called relative equilibria) in the Newtonian 5-body problem, except perhaps if the 5-tuple of positive masses belongs to a given codimension 2 subvariety of the mass space.
We consider the image of a fractal set X in a Banach space under typical linear and nonlinear projections π into R N . We prove that when N exceeds twice the box-counting dimension of X, then almost every (in the sense of prevalence) such π is one-to-one on X, and we give an explicit bound on the Hölder exponent of the inverse of the restriction of π to X. The same quantity also bounds the factor by which the Hausdorff dimension of X can decrease under these projections. Such a bound is motivated by our discovery that the Hausdorff dimension of X need not be preserved by typical projections, in contrast to classical results on the preservation of a Hausdorff dimension by projections between finite-dimensional spaces. We give an example for any positive number d of a set X with box-counting and Hausdorff dimension d in the real Hilbert space 2 such that for all projections π into R N , no matter how large N is, the Hausdorff dimension of π(X) is less than d (and in fact, is less than two, no matter how large d is).AMS classification scheme numbers: 28A20, 58F11, 31A15, 94A17, 49Q15, 60B05 0951-7715/99/051263+13$30.00
We prove a form of Arnold diffusion in the a priori stable case. Letbe a nearly integrable system of arbitrary degrees of freedom n 2 with a strictly convex H0. We show that for a "generic" εH1, there exists an orbit (θ, p)(t) satisfyingwhere l(H1) is independent of ε. The diffusion orbit travels along a co-dimension one resonance, and the only obstruction to our construction is a finite set of additional resonances.For the proof we use a combination geometric and variational methods, and manage to adapt tools which have recently been developed in the a priori unstable case.
We introduce a new potential-theoretic definition of the dimension spectrum D q of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if 1 < q 2 and µ is a Borel probability measure with compact support in R n , then under almost every linear transformation from R n to R m , the q-dimension of the image of µ is min(m, D q (µ)); in particular, the q-dimension of µ is preserved provided m D q (µ). We also present results on the preservation of information dimension D 1 and pointwise dimension. Finally, for 0 q < 1 and q > 2 we give examples for which D q is not preserved by any linear transformation into R m. All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.
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