7,009 research outputs found
Strong solutions for time-dependent mean field games with non-separable Hamiltonians
We prove existence theorems for strong solutions of time-dependent mean field
games with non-separable Hamiltonian. In a recent announcement, we showed
existence of small, strong solutions for mean field games with local coupling.
We first generalize that prior work to allow for non-separable Hamiltonians.
This proof is inspired by the work of Duchon and Robert on the existence of
small-data vortex sheets in incompressible fluid mechanics. Our next existence
result is in the case of weak coupling of the system; that is, we allow the
data to be of arbitrary size, but instead require that the (still possibly
non-separable) Hamiltonian be small in a certain sense. The proof of this
theorem relies upon an appeal to the implicit function theorem
Existence theory for non-separable mean field games in Sobolev spaces
The mean field games system is a coupled pair of nonlinear partial
differential equations arising in differential game theory, as a limit as the
number of agents tends to infinity. We prove existence and uniqueness of
classical solutions for time-dependent mean field games with Sobolev data. Many
works in the literature assume additive separability of the Hamiltonian, as
well as further structure such as convexity and monotonicity of the resulting
components. Problems arising in practice, however, may not have this separable
structure; we therefore consider the non-separable problem. For our existence
and uniqueness results, we introduce new smallness constraints which
simultaneously consider the size of the time horizon, the size of the data, and
the strength of the coupling in the system.Comment: Added extensions to problems with nonsmoothing payoff function and to
problems with congestion. Made some small correction
Existence theory for a time-dependent mean field games model of household wealth
We study a nonlinear system of partial differential equations arising in
macroeconomics which utilizes a mean field approximation. This system together
with the corresponding data, subject to two moment constraints, is a model for
debt and wealth across a large number of similar households, and was introduced
in a recent paper of Achdou, Buera, Lasry, Lions, and Moll. We introduce a
relaxation of their problem, generalizing one of the moment constraints; any
solution of the original model is a solution of this relaxed problem. We prove
existence and uniqueness of strong solutions to the relaxed problem, under the
assumption that the time horizon is small. Since these solutions are unique and
since solutions of the original problem are also solutions of the relaxed
problem, we conclude that if the original problem does have solutions, then
such solutions must be the solutions we prove to exist. Furthermore, for some
data and for sufficiently small time horizons, we are able to show that
solutions of the relaxed problem are in fact not solutions of the original
problem. In this way we demonstrate nonexistence of solutions for the original
problem in certain cases.Comment: 27 page
Global paths of time-periodic solutions of the Benjamin-Ono equation connecting pairs of traveling waves
We classify all bifurcations from traveling waves to non-trivial
time-periodic solutions of the Benjamin-Ono equation that are predicted by
linearization. We use a spectrally accurate numerical continuation method to
study several paths of non-trivial solutions beyond the realm of linear theory.
These paths are found to either re-connect with a different traveling wave or
to blow up. In the latter case, as the bifurcation parameter approaches a
critical value, the amplitude of the initial condition grows without bound and
the period approaches zero. We then prove a theorem that gives the mapping from
one bifurcation to its counterpart on the other side of the path and exhibits
exact formulas for the time-periodic solutions on this path. The Fourier
coefficients of these solutions are power sums of a finite number of particle
positions whose elementary symmetric functions execute simple orbits (circles
or epicycles) in the unit disk of the complex plane. We also find examples of
interior bifurcations from these paths of already non-trivial solutions, but we
do not attempt to analyze their analytic structure.Comment: 35 pages, 14 figures; changed title slightly, added 7 references,
changed conjecture to a theorem and proved it, moved some material to
appendice
Local Existence Theory for Derivative Nonlinear Schr\"{o}dinger Equations with Non-Integer Power Nonlinearities
We study a derivative nonlinear Schr\"{o}dinger equation, allowing
non-integer powers in the nonlinearity, . Making careful use
of the energy method, we are able to establish short-time existence of
solutions with initial data in the energy space, . For more regular
initial data, we establish not just existence of solutions, but also
well-posedness of the initial value problem. These results hold for real-valued
while prior existence results in the literature require
integer-valued or sufficiently large (), or
use higher-regularity function spaces.Comment: 23 page
Nonexistence of small doubly periodic solutions for dispersive equations
We study the question of existence of time-periodic, spatially periodic
solutions for dispersive evolution equations, and in particular, we introduce a
framework for demonstrating the nonexistence of such solutions. We formulate
the problem so that doubly periodic solutions correspond to fixed points of a
certain operator. We prove that this operator is locally contracting, for
almost every temporal period, if the Duhamel integral associated to the
evolution exhibits a weak smoothing property. This implies the nonexistence of
nontrivial, small-amplitude time-periodic solutions for almost every period if
the smoothing property holds. This can be viewed as a partial analogue of
scattering for dispersive equations on periodic intervals, since scattering in
free space implies the nonexistence of small coherent structures. We use a
normal form to demonstrate the smoothing property on specific examples, so that
it can be seen that there are indeed equations for which the hypotheses of the
general theorem hold. The nonexistence result is thus established through the
novel combination of small divisor estimates and dispersive smoothing
estimates. The examples treated include the Korteweg-de Vries equation and the
Kawahara equation
Magnetic stripe domain pinning and reduction of in plane magnet order due to periodic defects in thin magnetic films
In thin magnetic films with strong perpendicular anisotropy and strong
demagnetizing field two ordered phases are possible. At low temperatures,
perpendicularly oriented magnetic domains form a striped pattern. As
temperature is increased the system can undergo a spin reorientation transition
into a state with in-plane magnetization. Here we present Monte Carlo
simulations of such a magnetic film containing a periodic array of non-magnetic
defects. We find that the presence of defects stabilizes parallel orientation
of stripes against thermal fluctuations at low temperatures. Above the spin
reorientation temperature we find that defects favor perpendicular spin
alignment and disrupt long range ordering of spin components parallel to the
sample. This increases cone angle and reduces in plane correlations, leading to
a reduction in the spontaneous magnetization
Global existence and analyticity for the 2D Kuramoto-Sivashinsky equation
There is little analytical theory for the behavior of solutions of the
Kuramoto-Sivashinsky equation in two spatial dimensions over long times. We
study the case in which the spatial domain is a two-dimensional torus. In this
case, the linearized behavior depends on the size of the torus -- in
particular, for different sizes of the domain, there are different numbers of
linearly growing modes. We prove that small solutions exist for all time if
there are no linearly growing modes, proving also in this case that the radius
of analyticity of solutions grows linearly in time. In the general case (i.e.,
in the presence of a finite number of growing modes), we make estimates for how
the radius of analyticity of solutions changes in time.Comment: 26 page
Well-posedness of fully nonlinear KdV-type evolution equations
We study the well-posedness of the initial value problem for fully nonlinear
evolution equations, where may depend on up to the first
three spatial derivatives of We make three primary assumptions about the
form of a regularity assumption, a dispersivity assumption, and an
assumption related to the strength of backwards diffusion. Because the third
derivative of is present in the right-hand side and we effectively assume
that the equation is dispersive, we say that these fully nonlinear evolution
equations are of KdV-type. We prove the well-posedness of the initial value
problem in the Sobolev space The proof relies on gauged
energy estimates which follow after making two regularizations, a parabolic
regularization and mollification of the initial data
Observation of a charged (DD*bar)- mass peak in e+e- --> pi+ (DD*bar)- at Ecm=4.26 GeV
We report on a study of the process e+e- --> pi+(D D*bar)- at Ecm=4.26 GeV
using a 525 /pb data sample collected with the BESIII detector at the BEPCII
storage ring. A distinct charged structure is observed in the (DD*bar)-
invariant mass distribution. When fitted to a mass-dependent-width Breit-Wigner
lineshape, the pole mass and width are determined to be M_pole=(3883.9 +- 1.5
+- 4.2) MeV and Gamma_pole=(24.8 +- 3.3 +- 11.0) MeV. The mass and width of the
structure, which we refer to as Z_c(3885), are 2sigma and 1sigma, respectively,
below those of the Z_c(3900) --> pi+J/psi peak observed by BESIII and Belle in
pi+pi-J/psi final states produced at the same center-of-mass energy. The
angular distribution of the pi Z_c(3885) system favors a JP=1+ quantum number
assignment for the structure and disfavors 1- or 0-. The Born cross section
times the DD*bar branching fraction of the Z_(3885) is measured to be
sigma(e+e- --> pi+ Z_c(3885)-) x Bf(Z_c(3885)-->DD*bar)=(83.5 +-6.6 +- 22.0)
pb. Assuming the Z_c(3885) --> DD*bar signal reported here and the Z_c(3900)
--> pi J/psi signal are from the same source, the partial width ratio
Gamma(Z_c(3885) --> DD*bar)/Gamma(Z_c(3900) -->pi J/psi)=6.2 +- 1.1 +- 2.7 is
determined.Comment: 7 pages, 3 figures and 3 tables, submitted for publication in
Physical Review Letters, references adde
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