1,166 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
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
Confinement of vorticity for the 2D Euler-alpha equations
In this article we consider weak solutions of the Euler- equations in
the full plane. We take, as initial unfiltered vorticity, an arbitrary
nonnegative, compactly supported, bounded Radon measure. Global well-posedness
for the corresponding initial value problem is due M. Oliver and S. Shkoller.
We show that, for all time, the support of the unfiltered vorticity is
contained in a disk whose radius grows no faster than . This result is an adaptation of the corresponding result for the
incompressible 2D Euler equations with initial vorticity compactly supported,
nonnegative, and -th power integrable, , due to D. Iftimie, T. Sideris
and P. Gamblin and, independently, to Ph. Serfati
The radius of analyticity for solutions to a problem in epitaxial growth on the torus
A certain model for epitaxial film growth has recently attracted attention,
with the existence of small global solutions having being proved in both the
case of the n-dimensional torus and free space. We address a regularity
question for these solutions, showing that in the case of the torus, the
solutions become analytic at any positive time, with the radius of analyticity
growing linearly for all time. As other authors have, we take the Laplacian of
the initial data to be in the Wiener algebra, and we find an explicit smallness
condition on the size of the data. Our particular condition on the torus is
that the Laplacian of the initial data should have norm less than 1/4 in the
Wiener algebra
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