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On global existence and scattering for the wave maps equation
Amer.
J. Math. 123(2001) no. 1, 37-77. This article extends the result in the
above one to dimensions 2 and 3, but is considerably more technical. Thanks
to Kenji Nakanishi, who found the error in the first manuscript.
The function spaces are now built using frequency localized classical solutions
to the inhomogeneous wave equation with respect to characteristic directions.
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Strichartz estimates for operators with nonsmooth coefficients and the
nonlinear wave equation American Journal of Mathematics 122 (2000),
no. 2, 349--376. The aim of this article is threefold. First we use the
FBI transform to set up a calculus for partial differential operators with
non-smooth coefficients. Secondly, we use this calculus to prove Strichartz
type estimates for the wave equation with nonsmooth coefficients. Here
we do this in the case when the coefficients are $C^s$ for $0 \leq s \leq
1$. Finally, we use a version of these Strichartz estimates to improve
the local theory for second order nonlinear hyperbolic equations.
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On the equation $\Box u = |\nabla u|^2$ in $5+1$ dimensions. Math.
Res. Letters 6 (1999), no. 5-6, 469--485. The aim of this article is to
prove sharp local well-posedness results for the the equation in the title.
This problem is more difficult in dimension 5+1 and higher. Then well-posedness
should hold up to the scaling level, and the Strichartz estimates and the
$X^{s,\theta}$ spaces are insufficient. So, if you are curious about more
complicated function spaces related to the wave operator, then this is
the place. Availlable in dvi-letter
(without
pictures) and ps-letter
format.
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Strichartz estimates for second order hyperbolic operators with nonsmooth
coefficients II Amer. J. Math. 123 (2001), no. 3, 385--423. This
is a revised and greatly expanded spin-off of the original article. The
main result is that the full Strichartz estimates hold for operators with
$C^2$ coefficients. Partial estimates are then proved for operators with
$C^s$ coefficients for $1 \leq s \leq 2$.
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Strichartz estimates for second order hyperbolic operators with nonsmooth
coefficients III J. Amer. Math. Soc. 15 (2002),
no. 2, 419--442 This article improves the results in the previous article.
It is proved here that the full Strichartz estimates hold for second order
hyperbolic operators whose coefficients have two derivatives in $L^1(L^\infty)$.
Partial estimates are obtain if the coefficients have less than two derivatives
in the same space. This result also leads to further improvement for the
local theory for nonlinear hyperbolic equations. Availlable in dvi-letter
and
ps-letter
format.
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Strichartz estimates for a Schrodinger operator with nonsmooth coefficients
(joint
with Gigliola Staffilani) Comm. Partial Differential Equations
27 (2002), no. 7-8, 1337--1372. Here we begin the study of
dispersive estimates for variable coefficient Schroedinger operators.
For now we confine ourselves to compactly supported $C^2$ perturbations
of the identity subject to a nontrapping condition. An essential role in
our approach is played by the local smoothing effect, which allows us to
properly localize the problem. Availlable in dvi-letter
and
ps-letter
format.
(March 2001, revised April 2002)
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Null form estimates for second order hyperbolic operators with
rough coefficients Here we prove certain $L^p_t L^q_x$ null form
estimates for second order hyperbolic equations with rough coefficients.
This extends some earlier L^p results of Wolff and Tao for the constant
coefficient case, and also some L^2 null form estimates for operators with
rough coefficients due to Smith and Sogge. Also it proves most of
(the non-endpoint part of) a conjecture of Foschi and Klainerman. Availlable
in dvi-letter
and
ps-letter
format.
(Oct. 2001, revised Feb. 2002)
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Sharp local well-posedness results for the nonlinear wave equation
(joint
with Hart Smith) We prove that the Cauchy problem for generic nonlinear
hyperbolic equations is locally well-posed for initial data in $H^s \times
H^{s-1}$ provided that s> n/2+ 3/4 for n=2 respectively s >
(n+1)/2 for n > 2. This result is sharp in dimensions n=2,3. Availlable
in dvi-letter
and
ps-letter
format.
(Oct. 2001)
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Sharp counterexamples for Strichartz estimates for low regularity metrics.(joint
with Hart Smith) We show that the Strichartz estimates for second order
hperbolic equations with rough coefficients proved in the papers
(II) (III) above are sharp. Availlable in dvi-letter
and
ps-letter
format.
(April 2003)
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Rough solutions for the Wave-Maps equation We consider the
wave maps equation with values into a Riemannian manifold which is
isometrically embedded in $\R^m$. The main result asserts that the Cauchy
problem is globally well-posed for initial data which is small in the critical
Sobolev spaces. This extends and completes recent work of Tao and other
authors. Availlable in pdf
and
ps-letter
format.
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The wave maps equation . These are some notes which I wrote
last winter for the AMS meeting in Baltimore. If you found the previous
paper too long and technical, then maybe this one is more useful. Availlable
in pdf
and
ps-letter
format.
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Dispersive estimates for wave equations with rough
coefficients. (joint with Dan Geba ) In this article we obtain a
multiscale wave packet representation for the fundamental solution of
the wave equation with rough coefficients. This leads to pointwise and
weighted Lp bounds on the fundamental solution and also to a proof of
dispersive estimates for such operators. Availlable in pdf
and ps-letter
format.
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Parametrices and dispersive estimates for Schroedinger
operators with variable coefficients
This is about variable coefficient time dependent
Schr\"odinger evolutions in $\R^n$. Using phase space
methods we construct outgoing parametrices and to prove global in
time Strichartz type estimates. This is done in the context of $C^2$
metrics which satisfy a weak asymptotic flatness condition at
infinity. Availlable in pdf