240 research outputs found
On Singularity formation for the L^2-critical Boson star equation
We prove a general, non-perturbative result about finite-time blowup
solutions for the -critical boson star equation in 3 space dimensions. Under
the sole assumption that the solution blows up in at finite time, we
show that has a unique weak limit in and that has a
unique weak limit in the sense of measures. Moreover, we prove that the
limiting measure exhibits minimal mass concentration. A central ingredient used
in the proof is a "finite speed of propagation" property, which puts a strong
rigidity on the blowup behavior of .
As the second main result, we prove that any radial finite-time blowup
solution converges strongly in away from the origin. For radial
solutions, this result establishes a large data blowup conjecture for the
-critical boson star equation, similar to a conjecture which was
originally formulated by F. Merle and P. Raphael for the -critical
nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704].
We also discuss some extensions of our results to other -critical
theories of gravitational collapse, in particular to critical Hartree-type
equations.Comment: 24 pages. Accepted in Nonlinearit
Tosio Kato (1917–1999)
Tosio Kato was born August 25, 1917, in Kanuma City, Tochigi-ken, Japan. His early training was in physics. He obtained
a B.S. in 1941 and the degree of Doctor of Science in 1951, both at the University of Tokyo. Between these events he published
papers on a variety of subjects, including pair creation by gamma rays, motion of an object in a fluid, and results
on spectral theory of operators arising in quantum mechanics. His dissertation was entitled “On the convergence of the
perturbation method”.
Kato was appointed assistant professor of physics at the University of Tokyo in 1951 and was promoted to professor of
physics in 1958. During this time he visited the University of California at Berkeley in 1954–55, New York University in 1955,
the National Bureau of Standards in 1955–56, and Berkeley and the California Institute of Technology in 1957–58. He was
appointed professor of mathematics at Berkeley in 1962 and taught there until his retirement in 1988. He supervised
twenty-one Ph.D. students at Berkeley and three at the University of Tokyo.
Kato published over 160 papers and 6 monographs, including his famous book Perturbation Theory for Linear
Operators [K66b]. Recognition for his important work included the Norbert Wiener Prize in Applied Mathematics, awarded
in 1980 by the AMS and the Society for Industrial and Applied Mathematics. He was particularly well known for his work on
Schrödinger equations of nonrelativistic quantum mechanics and his work on the Navier-Stokes and Euler equations of
classical fluid mechanics. His activity in the latter area remained at a high level well past retirement and continued until his
death on October 2, 1999
Soliton dynamics for fractional Schrodinger equations
We investigate the soliton dynamics for the fractional nonlinear Schrodinger
equation by a suitable modulational inequality. In the semiclassical limit, the
solution concentrates along a trajectory determined by a Newtonian equation
depending of the fractional diffusion parameter.Comment: 22 page
On the fractional abstract Schrodinger type evolution equations on the Hilbert space and its applications to the fractional dispersive equations
In this paper we prove the local and global well-posedness of the time
fractional abstract Schr\"odinger type evolution equation on the Hilbert space
and as an application, we prove the local and global well-posedness of the
fractional dispersive equation with static potential under the only assumption
that the symbol of P(D) behaves like a polynomial of highest degree m at
infinity. In appendix, we also give the Holder regularities and the asymptotic
behaviors of the mild solution to the linear time fractional abstract
Schr\"odinger type equation. Because of the lack of the semigroup properties of
the solution operators, we employ a strategy of proof based on the spectral
theorem of the self-adjoint operators and the asymptotic behaviors of the
Mittag-Leffler functions.Comment: 44 pages, 0 figure
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