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, $|u|^{2\sigma} u_x$. Making careful use of the energy method, we are able to establish short-time existence of solutions with initial data in the energy space, $H^1$. 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 $\sigma\geq 1,$ while prior existence results in the literature require integer-valued $\sigma$ or $\sigma$ sufficiently large ($\sigma \geq 5/2$), 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, $u_{t}=f[u],$ where $f$ may depend on up to the first three spatial derivatives of $u.$ We make three primary assumptions about the form of $f:$ a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion. Because the third derivative of $u$ 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 $H^{7}(\mathbb{R}).$ 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