arXiv subjects

Bradley Isom

Publications and source records attributed to Bradley Isom.

2 recordsLinked to original sources

arXiv2020

Growth Bound and Nonlinear Smoothing for the Periodic Derivative Nonlinear Schrödinger Equation

A polynomial-in-time growth bound is established for global Sobolev $H^s(\mathbb T)$ solutions to the derivative nonlinear Schrödinger equation on the circle with $s>1$. These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.

math.AP↗

arXiv2020

Polynomial Bound and Nonlinear Smoothing for the Benjamin-Ono Equation on the Circle

For initial data in Sobolev spaces $H^s(\mathbb T)$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-\frac 12) + ε}$, $0<ε\ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.

math.AP↗