What's new:
What was new in 2002? 2001?
2000? 1999?
Dec 11, 2003
- Travel:
Due to visa delays, I will now not be returning to the US until January
21. My teaching duties until
then will be covered by Christoph Thiele (for Math 133) and John Garnett
(for Math 245B) respectively.
- Uploaded:
“A Strichartz inequality
for the Schrodinger equation on non-trapping asymptotically conic
manifolds”, with Andrew Hassell and Jared Wunsch. Here we extend the interaction (or
“two-particle”) Morawetz inequality, which was previously derived for
linear and non-linear Schrodinger equations on R^3 to obtain an L^4-type
Strichartz bound, to more general manifolds, thus obtaining a new L^4
Strichartz inequality for all smooth asymptotically conic non-trapping
manifolds (and in particular for asymptotically flat non-trapping
manifolds). The assumption of
no trapped geodesics is crucial, since otherwise there are pseudomode
solutions to the Schrodinger equation which have no local smoothing and in
particular must necessarily obey worse Strichartz estimates than in
Euclidean space. Similar
results have been obtained before by Burq, Gerard, Tzvetkov, Staffilani,
and Tataru; our result is weaker in that it is restricted to L^4 on three
dimensions, but we do not lose any derivatives even in the asymptotically
conic case. Unlike previous
methods, which rely (among other things) on a microlocal parametrix which
obeys good dispersive estimates, we do not use a parametrix here. Instead our argument is primarily
based on the positive commutator method applied to the product solution;
in fact one can phrase the main argument as an integration by parts based
primarily on understanding the convexity properties of the distance
function d(x,y), even when x and y are far apart; thus the paper requires
a certain amount of Riemannian geometry computations. The trouble comes near infinity
and at far distances when the distance function begins to develop
singularities; we must apply cutoffs to avoid these, which in turn
generate further terms which need to be estimated. Most of these can be estimated by
local smoothing estimates (or “one-particle” Morawetz estimates), but
there is an error term which requires a new type of Morawetz estimate (one
with only one angular derivative instead of the usual two), which may also
be of independent interest.
Nov 17, 2003
- Uploaded:
“Instability of the
periodic nonlinear Schrodinger equation”, with Michael Christ and Jim
Colliander. This is third in
our sequence of papers analyzing the instability at low regularities of
model dispersive equations; this time we focus on one-dimensional periodic
non-linear Schrodinger equation, which has a particularly clear separation
into high frequency and low frequency modes due to the periodic Fourier transform. Here we find that while, as
expected, these evolutions become ill-posed and discontinuous once the
data is rougher than L^2 in Sobolev norms, the nature of this
ill-posedness depends on the power of the non-linearity. In the cubic case (which should be
much more stable anyway, being completely integrable), it is possible for
a small change in the initial data in a negative Sobolev norm to lead to a
large change in even the zeroth Fourier mode in an arbitrarily short time;
the problem here is that while the negative Sobolev norm is small, the
mass is large. Thus the
solution map is not continuous below L^2 at the origin. For the quintic and higher
equations this phenomenon occurs, but there is another mechanism for
ill-posedness too... arbitrarily small perturbations in C^infty norm
can lead to a large oscillation in the zeroth Fourier mode in arbitrarily
small time if the solution is small in negative Sobolev spaces but not in
L^2; the quintic nature of the nonlinearity allows small perturbations at
low modes to interact with high modes to return to cause massive
instability in low modes. We
emphasize that in both cases we show ill-posedness but not blowup; indeed
we suspect in the completely integrable cubic case there is no blowup in
negative Sobolev spaces despite the presence of instabilities; this may be
a renormalization issue, and we are continuing to look into this question.
- Added:
23rd Annual Western States Mathematical Physics Meeting
(Caltech, Feb 16-17).
Nov 12, 2003
- Updated
“Recent progress
on the restriction conjecture”.
These Park City lecture notes have been revised; the most
noticeable change is the addition of 9 new figures, and removal of several
typos, especially from the exercises. Special thanks to Julia Garibaldi for her proofreading
while TA’ing the Park City Course!
Nov 5, 2003
- Uploaded:
“On multilinear oscillatory
integrals, nonsingular and singular”, with Michael Christ, Xiaochun
Li, and Christoph Thiele.
Here we consider multilinear operators similar to that of the
bilinear Hilbert transform, but with an additional polynomial phase
oscillation (thus these are the multilinear analogue of the singular oscillatory
integrals studied for instance by Ricci and Stein, and are also related to
the polyomial phase Carleson maximal operators studied for instance by
Lacey). It turns out that one
can fairly quickly decouple the singular and oscillatory parts of this
operator into a singular piece where the oscillation is not significant
(and which can be dealt with by existing multilinear multiplier theorems),
plus an oscillatory part where the singularity of the kernel is
irrelevant. This oscillatory
part is the main focus of the paper, and in many ways we found it more
interesting than the original problem. The question is under what conditions one can obtain
decay estimates for these multilinear oscillatory integrals analogous to
what one can obtain using Van der Corput’s lemma for oscillatory integrals
of the first kind, or via e.g. the analysis of Phong and Stein for
oscillatory integrals for the second kind. There is a natural conjecture, which is that decay
occurs whenever the polynomial phase is non-degenerate (i.e. it cannot be
factored away into the various component functions). We prove this conjecture when the
number of functions is less than twice the number of integration
variables, which co-incidentally is the same range that is needed to
handle the singular part.
When there are very few functions one can use a Cauchy-Schwarz (or
Van der Corput) trick to repeatedly eliminate functions until one is left
with a plain oscillatory integral; however this technique does not always
work once there are too many functions, and instead we rely on some
polynomial Fourier analysis (inspired by the recent work of Gowers on
arithmetic progressions), in particular dividing into the cases when a
function has a large polynomial Fourier coefficient (which can be treated
by an inductive hypothesis), and when all functions are “polynomially
uniform” (in which case we can descend to a simple two-dimensional
problem). The cases of higher
functions is still pretty much open however. In analogy with the work of Christ, Carbery, and
Wright, we also consider level set analogues of this oscillatory problem,
for which we have a more satisfactory range of results.
- Uploaded:
“Ill-posedness for
nonlinear Schrodinger and wave equations”, with Michael Christ and Jim
Colliander. This continues our previous work
on low regularity ill-posedness for the KdV and cubic non-linear
Schrodinger equation, but now extended to nonlinear Schrodinger (NLS) and
wave (NLW) equations of arbitrary power (and in particular not relying on
the specific feature of the 1D cubic NLS that it is on the borderline
between long-range and short-range scattering behavior). We introduce a general technique
for demonstrating ill-posedness of these equations (even in the defocusing
case) below scale-invariant or Gallilean invariant regularities, by
rescaling the problem to a small dispersion limit and showing that for
sufficiently small dispersion, the non-linear part of the equation
dominates and causes phase decoherence. The illposedness we obtain is in the sense of failure
of uniform continuity of the solution map; there seem to be subtle
distinctions between this and actual failure of continuity of the solution
map (or of blowup), which we shall partially address in a forthcoming
paper. Unfortunately the
method does not work so well with the Lorentz-invariant regularity of the
wave equation both because of the second-order nature in time of the wave
equation and because the Lorentz transformation distorts time as well as
space (so in particular does not preserve the concept of initial data), so
this remains the major remaining obstacle towards a complete
well-posedness and ill-posedness for NLW. We do however obtain a satisfactory one-dimensional NLW
theory showing that the obstruction of needing enough regularity for the
solution to be a distribution is a necessary one for uniform
well-posedness. We also relate
our NLW work with the instability work of Lebeau for the supercritical
wave equation; the results are similar but not directly comparable.
Oct 23, 2003
- Uploaded:
“Bi-parameter paraproducts”,
with Camil Muscalu, Jill Pipher, and Christoph Thiele, submitted to Acta
Math. Here we study
paraproducts on R^2 where the multiplier m(xi_1, xi_2) is not of classical
Coifman-Meyer type (in that each derivative in xi_1 or xi_2 gains a power
of |xi_1| + |xi_2|) but is rather of product Coifman-Meyer type, where we
split xi_1 = (xi’_1, xi’’_1) and xi_2 = (xi’_2, xi’’_2), and each
derivative in xi’_1 and xi’_2 gains a power of |xi’_1| + |xi’_2|, and
similarly for the xi’’ terms.
This particular type of “bi-parameter paraproduct” arises if one
wishes to do things like fractional differentiation in the x_1 and x_2
directions simultaneously. We
obtain the expected bilinear L^p x L^q -> L^r estimates for this
operator for p,q > 1; we allow the possibility that r is less than
one. For r > 1 the
argument is straightforward, proceeding by the usual Littlewood-Paley type
decomposition and then using boundedness of “maximal-maximal”,
“maximal-square function”, “squarefunction-maximal” or
“squarefunction-squarefunction” operators (which are concatenations of the
one-dimensional Hardy-Littlewood maximal or Littlewood-Paley square
function operators in either direction) in L^p, p > 1 to conclude the
result. For r <= 1 one
must take a little more care, but one can still proceed without much
difficulty by using the time-frequency analysis of Fefferman and
Lacey-Thiele. Specifically,
we introduce biparameter space-frequency tiles, gather them into “trees”
of various “size” (a phase space localized Hardy-Littlewood maximal function)
and “energy” (a phase space localized Littlewood-Paley square function),
organize the trees in a sort of descending order, and sum them
seperately. Despite using the
product theory, the arguments are simpler than those for say, the bilinear
Hilbert transform, because the nature of the paraproduct is such that the
only frequency singularity is at the origin (or more precisely, the two
co-ordinate axes) so there are far fewer tiles to deal with in the dyadic
decomposition. Indeed, we
also show in the paper that if we consider the simplest biparameter
operator of bilinear Hilbert transform type, namely the double bilinear
Hilbert transform, then this operator is in fact unbounded in any Lebesgue
space (this may have been discovered earlier as “folklore”, but does not
appear explicitly in print).
Sep 26, 2003
- Uploaded:
“On the asymptotic behavior
of large radial data for a focusing non-linear Schr\"odinger equation”,
submitted to Journal of Partial Differential Equations and Dynamical Systems. Here we give some partial results
toward the problem of soliton resolution for non-integrable focusing
nonlinear Schrodinger equations.
There seems to be a fundamental distinction between L^2 subcritical
and L^2 supercritical equations here; I chose the best of both worlds,
namely an L^2 subcritical equation with an a priori assumption of no blow
up (i.e. H^1 norm stays bounded).
Specifically, I chose the three-dimensional cubic focusing NLS
assuming bounded energy. (One
can remove this bounded energy assumption if one is willing to mollify the
nonlinearity smoothly at infinity)
Here we show that the solution splits into a linear part
(radiation), plus a ‘weakly bound’ state which is asymptotically
orthogonal to all linear solutions.
Specializing then to the radial case, we show that this weakly
bound state must either have mass and energy strictly bounded away from
zero, or else decay to zero.
Also, it converges (modulo an error which decays to 0 in homogeneous
H^1 norm) to a smooth, decaying time-dependent object which obeys symbol
bounds like <x>^{-3/2+}, and obeys an asymptotic Pohozaev
identity. This smooth
decaying object is supposed to be the soliton, although I was unable to
get the exponential decay that one expects for solitons, and also I obtained
no control on the long-term dynamics of this object (though in the short
term it obeys an approximate NLS equation, of course). This seems to require some “low
frequency” techniques, e.g. from stability analysis of periodic orbits or
other machinery from dynamical systems or Hamiltonian mechanics; the
argument here is mainly “high frequency” and is focused on showing that
high frequency radiation must eventually leave the origin, thus leaving
the residual bound state smooth.
To obtain the (somewhat unsatisfactory) decay of <x>^{-3/2+},
we need to show that the further one gets from the origin, the lower the
frequencies which can escape to infinity (basically at a distance R from
the origin, any component with frequency R^{-1+} can escape to infinity). There is one novel trick, which is
to distinguish between incoming waves (heading towards the origin) and
outgoing waves (heading away from the origin), and to choose either
Duhamel’s formula backwards in time or forwards in time appropriately
(such a trick will also appear in a forthcoming paper with Igor Rodnianski
concerning decay estimates for the Schrodinger equation on manifolds).
- Added:
IPAM program – Multiscale
geometry and analysis in high dimensions (Sep – Nov 2004, IPAM) and Recent Developments
in Applied Harmonic Analysis: Multiscale Geometric Analysis (Apr 15-17
2004, U. Arkansas Fayetteville).
Sep 22, 2003
- Uploaded:
“Global regularity
for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in
high dimensions”, joint with Igor
Rodnianski. In this paper
we obtain a small data critical regularity result for the
Maxwell-Klein-Gordon equation in the Coulomb Gauge (MKG-CG) in six and
higher dimensions. The
statement of this result is similar to recent work on wave maps, but the
techniques are slightly different.
As with wave maps, the problem is critical with derivatives and so
standard iteration methods do not work unless one works in a Besov space
instead of a Sobolev space.
For wave maps, the passage to the Coulomb gauge (either globally,
or microlocally) ameliorates the derivatives in the nonlinearities to an
extent that one can indeed close the argument by iteration methods;
however for MKG, the Coulomb gauge does not go far enough to eradicate
derivatives (though it is undeniably useful nevertheless, for instance, it
allows all the nonlinear terms in the equation for the connection A to be
treatable, at least in high dimension). In fact to close the argument what must be done is to
obtain global Strichartz estimates for the covariant wave equation
assuming that the connection A is in the Coulomb gauge and is small in a
certain critical norm. The
key idea here is to do this by constructing a global parametrix for the
covariant wave equation which is like the usual Fourier parametrix for the
free wave equation but distorted by what is basically a (microlocal)
Cronstrom gauge potential function for A. This introduces phase terms which look roughly like
exp(i nabla^{-1} A) which would normally be bad (because nabla^{-1} A is
in H^{n/2} which just barely fails to be bounded) however the fact that A
is real makes these phase terms bounded. One then has to carefully check that these parametrices
do make reasonably good Fourier integral operators, and that they solve
the covariant wave equation reasonably accurately, but this can be done by
stationary phase arguments.
One useful technical innovation, which may be useful elsewhere, is
a new decoupling device to get rid of annoying amplitude functions in
oscillatory integrals.
It seems likely that the techniques here extend from MKG to Yang-Mills
(but now the connection evolves according to a covariant wave equation
rather than the free wave equation, which may complicate some of the
Fourier angular decompositions used in our paper). Of course the real prize here is
the small energy implies regularity result for Yang-Mills in four
dimensions, but in order to accomplish that one would need bilinear
estimates for the covariant wave equation and not just Strichartz
estimates. However the recent
theory of bilinear estimates on rough metrics developed for the Einstein
equations by Klainerman and Rodnianski may be useful here.
- Updated
once again: “An
uncertainty principle for cyclic groups of prime order”. More feedback from Gerd
Mockenhaupt and Melvyn Nathanson has been incorporated. A depressingly large number of
typos for a six-page paper have also been detected and eradicated.
- Added:
52nd Midwest PDE
(U. Minnesota, Nov 15-16).
Thanks to Markus Keel for this link.
Sep
19, 2003
Sep
12, 2003
- Uploaded:
The expository note “Viriel,
Morawetz, and interaction Morawetz inequalities”. In this note we gather some
different ways to view the monotonicity formulae listed above for the
non-linear Schrodinger equation; either as a conservation law arising from
the mass and momentum density and current; or as a quantum version of a
classical collision counting inequality; or as positive commutator
estimates on the solution (or of the tensor product of the solution with
itself). We also present
variants of the Morawetz inequality derived by Bourgain, Grillakis, and
Nakanishi, and also speculate on extensions to the Klein-Gordon equation.
Sep
10, 2003
- Updated:
“An uncertainty
principle for cyclic groups of prime order”. This new version incorporates some feedback from Robin
Chapman, Roy Meshulam, Michael Cowling, and Gergely Harcos. In particular, a shorter proof of
the Cauchy-Davenport inequality due to Robin Chapman is included, as well
as mention of Roy Meshulam’s extension to (Z/pZ)^N.
Sep
2, 2003
Aug
29, 2003
- Uploaded:
“An uncertainty
principle for cyclic groups of prime order”, submitted to Math
Research Letters. In a finite
abelian group G, the standard uncertainty principle asserts that the
support supp(f) of a function, and the support supp(\hat{f}) of its
Fourier transform, are related by the formula |supp(f)| |supp(\hat f)|
>= |G|. Here we show that
in the case of a group of prime order, G = Z/pZ, we can improve this to
|supp(f)| + |supp(\hat{f})| >= p+1. Furthermore, this is absolutely sharp; given any two
sets A and B with |A|+|B| >= p+1 we can find an f such that supp(f)=A
and supp(\hat f) = B. The
proof is based on computing minors of the Fourier matrix and uses a number
of tricks from algebra (some basic Galois theory and Vandermonde
determinants, to be precise).
As a rather modest application of this uncertainty principle we can
now give a Fourier-analytic proof of the Cauchy-Davenport inequality. It also seems to have application
to the zeroes of sparse polynomials, although I do not know the literature
in that area well enough to judge the ramifications.
Aug
6, 2003
Jul
23, 2003
- Uploaded:
“Global well-posedness of
the Benjamin-Ono equation in H^1(R)”, submitted to J. Hyperbolic Diff.
Eq. Here we improve the local
well-posedness theory of the Benjamin-Ono equation u_t + Hu_xx = uu_x (a
slightly less dispersive, but still completely integrable, variant of the
KdV equation) from H^{5/4+} to H^1, and thus (by the H^1 conservation law)
obtain global well-posedness.
No new estimates are proven – in fact, one relies on plain old Strichartz
estimates for the 1D Schrodinger equation; the trick here is to apply an
algebraic gauge transformation, similar to the Cole-Hopf transformation,
to eliminate the interaction between very low and very high frequencies.
Jun
13, 2003
- Uploaded: The expository
note "Korner's Besicovitch
set construction". This is a presentation of a recent
argument of Tom Korner, who established that "most" Besicovitch
sets have measure zero, in the sense of the Baire category theorem.
It leads to one of the more intuitive explanations as to why Besicovitch
sets exist, although the quantative bound obtained by this method is
rather poor.
- Uploaded: “A positive proof of
the Littlewood-Richardson rule using the octahedron recurrence”, with Allen Knutson and Chris Woodward. Here we give a combinatorial and self-contained
proof of the Littlewood-Richardson rule for tensor product multiplicities
of GL_n representations.
Indeed we show that any rule which is associative and obeys the
“Pieri rule” on generates must correspond to the puzzle rule (which is
known to be equivalent to the Littlewood-Richardson rule). The key is to show that the puzzle
rule is itself associative, which we do by means of the octahedron
recurrence. An alternate
geometric proof based on “scattering” of honeycombs is also provided.
Jun 9, 2003
- Uploaded: "Fuglede's conjecture is false
in 5 and higher dimensions". We give an explicit example of
a set in R^5 (a finite union of unit cubes, actually), which has an orthogonal
basis of exponentials, but which does not tile R^5 by translations.
This disproves (one direction of) a conjecture of Fuglede in higher
dimensions. The example is based on Hadamard matrices (orthogonal
matrices whose entries are +1 or -1) of order not equal to a power of 2;
actually this only gives a counterexample in R^{11}, to get down to R^5 we
need the corresponding concept for cube roots of unity instead of square
roots of unity. There is still however a chance that the conjecture can
be salvaged in one dimension. Also we do not know the status of the
converse direction (are there sets which tile, but have no orthogonal
basis? The difficulty here is that it is not so obvious how to find
a set which tiles, but only in a non-lattice manner).
- Uploaded: "Park city notes on the Restriction problem".
These are an expanded version of "Some recent progress
on the Restriction conjecture" from the proceedings of the Fourier Analysis and
Convexity workshop. In line with the Park City philosophy, many
homework questions have been added, and also two additional lectures on
the connection with Bochner-Riesz and PDE have also been added.
- Added: Conference
on Classical Analysis in honor of Paul Koosis (Oct 23-26, 2003, CRM
Montreal)
May 30, 2003
- Uploaded: The expository
note "The Fourier
transform on non-abelian finite groups". These are some
notes I made on the Fourier transform on a finite non-abelian group G;
this stuff is of course very basic to representation theorists and finite
group theorists, but does not seem to be as well publicized among
real-variable harmonic analysts as it could be. I restrict to the
finite case for simplicity, similar to how the theory of the finite
Fourier transform is technically much simpler than that of the Fourier
transform on R or Z or T, although algebraically of course they are almost
identical. The notes have a harmonic analysis bias and assume no
prior knowledge of representation theory. Incidentally, this topic
has nothing to do with the nonlinear Fourier transform; that deals with
the Fourier analysis of functions whose range lies in a non-abelian
group, whereas the above notes deal with the Fourier analysis of functions
whose domain is a non-abelian group.
May 21, 2003
May 5, 2003
May 2, 2003
Apr 28, 2003
Apr 9, 2003
- Added: Additive Number
Theory and Applications to Harmonic Analysis (CUNY Graduate Center, NY,
May 15-17)
Apr 1, 2003 (No joke!)
- Uploaded: Tom
Wolff Memorial Lectures on non-linear dispersive equations. These lectures
discuss some selected recent results on non-linear dispersive equations
such as the Korteweg de Vries (KdV) equation, the nonlinear Schrodinger
(NLS) equation, and the wave maps equation. In particular, some results on
global well-posedness for rough data, on scattering for NLS, symplectic
non-squeezing for KdV, and global regularity for wave maps is discussed.
Some of this material has been taken from my earlier talks on this subject
(notably my Chicago
lectures and my Canberra
lectures) but some material is new, particularly the section on
Morawetz inequalities and the application to scattering.
Mar 28, 2003
Mar 21, 2003
Mar 12, 2003
- Uploaded: "Some recent progress
on the Restriction conjecture". This is for the proceedings
of the Fourier Analysis
and Convexity workshop from June 2001. Here I attempt to
summarize the progress on the restriction conjecture, at least for the
model hypersurfaces of the sphere, paraboloid, and cone. In
particular I focus on giving a relatively informal description of some of
the main ideas used in the modern theory, including: (a) using the decay
of the Fourier transform of surface measure to localize a restriction
estimate; (b) working in the bilinear setting instead of the linear one to
eliminate parallel interactions; (c) using the wave packet decomposition
to reduce the issue to that of estimating sums of oscillatory functions on
tubes; (d) use of induction on scale arguments to eliminate "highly
localized" interactions; (e) use of L^4 theory and the additive
geometry of the surface to obtain further orthogonality conditions on the
oscillations.
Mar 10, 2003
Mar 2, 2003
Feb 4, 2003
Jan 31, 2003
- Updated: The link for the Park city program in
harmonic analysis and PDE (Jun 30-18, Park City, UT)
- Added: The Twentyseventh
Summer Symposium in Real Analysis (Jun 23-29, Silesian University, Czech
Republic), Applicable
Harmonic Analysis (Banff, Canada, June 7-12), Wavelet
Theory and Applications: New Directions and Challenges (National U.
Singapore, July 14-18), Analysis and
Geometric Measure Theory (Banff, Canada, July 26-31)
Jan 28, 2003
- Uploaded: "A sum-product estimate
for finite fields, and applications", joint with Jean Bourgain
and Nets Katz.
Here we extend the classical Erdos-Szemeredi estimate |A+A| + |A.A| >=
|A|^{1+epsilon} from finite sets A of integers, to finite subsets A of the
field Z/pZ, with p prime. The method is a variant of the recent
paper of Edgar and Miller used to prove the Erdos ring conjecture,
combined with some standard tools from additive number theory (in
particular, sumset estimates, the Balog-Szemeredi theorem, and a recent
lemma of Katz and myself). As applications we present a non-trivial
incidence bound of Szemeredi-Trotter type in finite fields, some non-trivial
progress on the Erdos distance problem in finite fields, and a new
estimate for the size of Besicovitch sets (or Kakeya sets) in
three-dimensional finite field geometries.
- Added: Analysis and
geometry in Carnot-Caratheodory spaces (Mar 7-8, Fayetteville AR)
Jan 27, 2003
- Uploaded: "Global existence and
scattering for rough solutions of a nonlinear Schroedinger equation on R^3",
joint with Jim Colliander,
Mark Keel, Gigliola
Staffilani, and Hideo Takaoka, submitted to CPAM.
This is the full version of our earlier announcement paper, "Existence globale
et diffusion pour l'équation de Schrödinger nonlinéaire répulsive cubique
sur R^3 en dessous l'espace d'énergie". Here we flesh out
in full detail a new "interaction-Morawetz" inequality for
three-dimensional nonlinear Schrodinger equations, which by using nothing
more than an integration by parts, gives an a priori unweighted
L^4_{x,t} spacetime estimate on solutions to defocusing NLS. From
this estimate one can obtain a new and simpler proof of scattering of the
cubic defocusing NLS in the energy class, and also for the first time
obtain scattering results for regularities below the energy class, and
specifically in H^s for s > 4/5. In particular, we also obtain
global well-posedness in this class (thus superceding our earlier global
well-posedness result, which held for s > 5/6).
Jan 21, 2003
Jan 14, 2003
Jan 6, 2003
- Uploaded: Multiple
Choice Quiz applet. This applet gives simple multiple choice
quizzes. More quizzes will be added later, but currently I have a
satisfactory list of quizzes on set theory, functions, and sequences, and
plan to add more in the future.