45 research outputs found
Uniqueness of the 2D Euler equation on rough domains
We consider the 2D incompressible Euler equation on a bounded simply
connected domain . We give sufficient conditions on the domain
so that for all initial vorticity the weak
solutions are unique. Our sufficient condition is slightly more general than
the condition that is a domain for some ,
with its boundary belonging to . As a corollary we prove
uniqueness for domains for and for convex domains
which are also domains for some . Previously
uniqueness for general initial vorticity in was only known
for domains with possibly a finite number of acute angled corners.
The fundamental barrier to proving uniqueness below the regularity is
the fact that for less regular domains, the velocity near the boundary is no
longer log-Lipschitz. We overcome this barrier by defining a new change of
variable which we then use to define a novel energy functional.Comment: 33 pages, comments welcom
Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner
We consider the 2D incompressible Euler equation on a corner domain
with angle with . We prove that if the initial
vorticity and if
is non-negative and supported on one side of the angle bisector of the domain,
then the weak solutions are unique. This is the first result which proves
uniqueness when the velocity is far from Lipschitz and the initial vorticity is
nontrivial around the boundary.Comment: 34 page
Absolute continuity of Brownian bridges under certain gauge transformations
We prove absolute continuity of Gaussian measures associated to complex
Brownian bridges under certain gauge transformations. As an application we
prove that the invariant measure for the periodic derivative nonlinear
Schr\"odinger equation obtained by Nahmod, Oh, Rey-Bellet and Staffilani in
[20], and with respect to which they proved almost surely global
well-posedness, coincides with the weighted Wiener measure constructed by
Thomann and Tzvetkov [24]. Thus, in particular we prove the invariance of the
measure constructed in [24].Comment: 12 pages. Submitte
Scattering and Blow up for the Two Dimensional Focusing Quintic Nonlinear Schr\"odinger Equation
Using the concentration-compactness method and the localized virial type
arguments, we study the behavior of solutions to the focusing quintic NLS
in , namely,
Denoting by and , the mass and energy of a solution
respectively, and the ground state solution to , and
assuming , we characterize the threshold for global versus
finite time existence. Moreover, we show scattering for global existing time
solutions and finite or "weak" blow up for the complement region. This work is
in the spirit of Kenig and Merle and Duyckaerts, Holmer, and Roudenko.Comment: 37 pages, 2 figures and updated reference
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Bilinear Operators with Non-Smooth Symbol, I
This paper proves the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding Lp-boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part II, our subsequent paper [11], using phase-plane analysis
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On Schrodinger Maps
We study the question of well-posedness of the Cauchy problem for Schr¨odinger maps from R 1 ×R 2 to the sphere S 2 or to H2 , the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schr¨odinger system of equations and then study this modified Schr¨odinger map system (MSM). We then prove local well posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well posedness of the Schr¨odinger map itself from it. In proving well posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted L 2 functions
The probabilistic scaling paradigm
In this note we further discuss the probabilistic scaling introduced by the
authors in [21, 22]. In particular we do a case study comparing the stochastic
heat equation, the nonlinear wave equation and the nonlinear Schrodinger
equation.Comment: Expository paper, 14 page