Presentation of the Main Results

This chapter collects the main results of the master equation for the convergence of the Nash system. It explains the notation used, specifies the notion of derivatives in the space of measures, and describes the assumptions on the data. One of the striking features of the master equation is that it involves derivatives of the unknown with respect to the measure. This chapter also discusses the link between the two notions of derivatives, which have been used in the mean field game (MFG) theory. The main result states that the master equation has a unique classical solution under the regularity and monotonicity assumptions on H, F, and G. Once the master equation has a solution, this solution can be used to build approximate solutions for the Nash system with N-players.

Uniform time of existence of the smooth solution for 3D Euler-α equations with periodic boundary conditions

After reformulating the incompressible Euler-[Formula: see text] equations in 3D periodic domain, one obtains that there exists a unique classical solution of Euler-[Formula: see text] equations in uniform time interval independent of [Formula: see text]. It is shown that the solutions of the Euler-[Formula: see text] converge to the corresponding solutions of Euler equation in [Formula: see text] in space, uniformly in time. It also follows that the [Formula: see text] [Formula: see text] solutions of Euler-[Formula: see text] equations exist in any fixed sub-interval of the maximum existing interval for the Euler equations provided that initial velocity is regular enough and [Formula: see text] is sufficiently small.

The Existence of a Unique Solution of the Hyperbolic Functional Differential Equation

We consider the Z. Szmydt problem for the hyperbolic functional differential equation. We prove a theorem on existence of a unique classical solution and the Carathéodory solution of the hyperbolic equation.

PRICING EQUATIONS IN JUMP-TO-DEFAULT MODELS

We study pricing equations in jump-to-default models, and we provide conditions under which the option price is the unique classical solution, with a special focus on boundary conditions. In particular, we find precise conditions ensuring that the option price at the default boundary coincides with the recovery payment. We also study spatial convexity of the option price, and we explore the connection between preservation of convexity and parameter monotonicity.

The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.

On the Modified Jump Problem for the Laplace Equation in the Exterior of Cracks in a Plane

The boundary value problem for the Laplace equation outside several cracks in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified on the cracks. The problem has unique classical solution under certain conditions. The new integral representation for the unique solution of this problem is obtained. The problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero. The integral representation and integral equation are essentially simpler than those derived for this problem earlier. The singularities at the ends of the cracks are investigated.

On periodic Stokesian Hele-Shaw flows with surface tension

In this paper we consider a 2π-periodic and two-dimensional Hele-Shaw flow describing the motion of a viscous, incompressible fluid. The free surface is moving under the influence of surface tension and gravity. The motion of the fluid is modelled using a modified version of Darcy's law for Stokesian fluids. The bottom of the cell is assumed to be impermeable. We prove the existence of a unique classical solution for a domain which is a small perturbation of a cylinder. Moreover, we identify the equilibria of the flow and study their stability.

Asymptotic Behavior of The Unique Solution to a Singular Elliptic Problem With Nonlinear Convection Term And Singular Weight

By Karamata regular variation theory, we first derived the exact asymptotic behavior of the local solution to the problem -φʹʹ(s) = g(φ(s)), φ(s) > 0, s ∈ (0, a) and φ(0) = 0. Then, by a perturbation method and constructing comparison functions, we derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem -Δu = b(x)g(u) + λ|▽u|

A remark on the equation of a vibrating plate

SynopsisWe consider the von Karman equations, which describe a vibrating plate either with a clamped boundary or with completely free boundary. In both cases we obtain a unique, classical solution. As the main tool we use a set of integral equations, which we deduce from the well known “variations of constants” formula.