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Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see Lions [6], and also the very helpful notes of Cardialaguet (2013) on Lions’ lectures, the Master Equation has attracted a lot of interest, and various points of view have been expressed, see for example Carmona and Delarue (2014), Bensoussan et al. (2015), Buckdahn et al. [3]. There are several ways to introduce this type of equation; and in those mentioned works, they involve an argument which is a probability measure, while P.L. Lions has recently proposed the alternative idea of working on the Hilbert space of square integrable random variables. Hence writing the equation is an issue; while another issue is its origin. In this article, we discuss all these various aspects. An important reference is the seminar at College de France delivered by P.L. Lions on November 14, 2014.

We study here quasi-linear parabolic equations with quadratic growth in \(\mathbb {R}^{n}\). These parabolic equations are at the core of the theory of PDE; see Friedman (Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964) [6], Ladyzhenskaya et al. (Translations of Mathematical Monographs. AMS, 1968) [4] for details. However, for the applications to physics and mechanics, one deals mostly with boundary value problems. The boundary is often taken to be bounded and the solution is bounded. This brings an important simplification. On the other hand, stochastic control theory leads mostly to problems in \(\mathbb {R}^{n}\). Moreover, the functions are unbounded and the Hamiltonian may have quadratic growth. There may be conflicts which prevent solutions to exist. In stochastic control theory, a very important development deals with BSDE (Backward Stochastic Differential Equations). There is a huge interaction with parabolic PDE in \(\mathbb {R}^{n}\). This is why, although we do not deal with BSDE in this paper, we use many ideas from Briand and Hu (Probab Theory Relat Fields 141(3–4):543–567, 2008) [1], Da Lio and Ley (SIAM J Control Optim 45(1):74–106, 2006) [2], Karoui et al. (Backward stochastic differential equations and applications, Princeton BSDE Lecture Notes, 2009) [3], Kobylanski (Ann Probab 28(2):558–602, 2000) [5]. Our presentation provided here is slightly innovative.

Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007, of Gueant-Lasry-Lions in 2011, of Huang-Caines-Malham in 2007 and many others). There are a lot of applications. In general, the applications concern approximating an infinite number of players with common behavior by a representative agent. This agent has to solve a control problem perturbed by a field equation, representing in some way the behavior of the average infinite number of agents. This approach does not lead easily to the problems of Nash equilibrium for a finite number of players, perturbed by field equations, unless one considers averaging within different groups, which has not been done in the literature, and seems quite challenging. In this paper, the authors approach similar problems with a different motivation which makes sense for control and also for differential games. Thus the systems of nonlinear partial differential equations with mean field terms, which have not been addressed in the literature so far, are considered here.

We consider the loading of an elastic perfectly plastic body governed by the Prandtl-Reuss law. It is shown that the stress velocities of the body have fractional derivatives of order - δ up to the boundary in the direction of the loading parameter, and of order ⅓ - δ in the interior of the body in direction of the space variables.

We consider the Navier–Stokes equations for compressible isothermal flow in the steady two-dimensional case and show the existence of a weak solution for homogeneous Dirichlet boundary conditions.

We consider a class of non-linear partial differential systems like -div$(a(x)\nabla u_{\nu}) +\lambda u_{\nu}=H_{\nu}(x, Du) \, $ with applications for the solution of stochastic differential games with $N$ players, where $N$ is an arbitrary but positive number. The Hamiltonian $H$ of the non-linear system satisfies a quadratic growth condition in $D u$ and contains interactions between the players in the form of non-compact coupling terms $\nabla u_{i} \cdot\nabla u_j$. A $L^{\infty}\cap H^1$-estimate and regularity results are shown, mainly in two-dimensional space. The coupling arises from cyclic non-market interaction of the control variables.

We consider parabolic systems $$u_{t} - {\rm div} \left( a(t, x, u, \nabla u)\right) + a_{0}(t, x, u, \nabla u) = 0$$ in two space dimensions with initial and Dirichlet boundary conditions. The elliptic part including a0 is derived from a potential with quadratic growth in ∇u and is coercive and monotone. The term a0 may grow quadratically in ∇u and satisfies a sign condition a0 · u ≥ −K. We prove the existence of a regular long time solution verifying a regularity criterion of Arkhipova. No smallness is assumed on the data.

We study mathematical properties of the model that has been proposed to explain the phenomenon of hardening due to cyclic loading. The model considers two elastic plastic materials, soft and hard, that co-exist while the soft material can be transformed into the hard material. Regarding elastic responses we remain in a simplified framework of linearized elasticity. Incorporating tools such as variational inequalities, penalty approximations and Sobolev spaces, we prove the existence of weak solution to the corresponding boundary-value problem and investigate its uniqueness and regularity.

Consider a probability space \((\Omega,\mathcal{A},P)\) and a filtration \({\mathcal{F}}^{t}\) generated by a n-dimensional standard Wiener process w(t). The state space is \({\mathbb{R}}^{n}\) with the generic notation x and the control space is \({\mathbb{R}}^{d}\) with generic notation v.

The mean field game and mean field type control problems introduced in Chap. 2 are both control problems for a representative agent, with mean field terms influencing both the evolution and the objective functional of this agent. The terminology game comes from the fact that the optimal feedback of the representative agent can be used as an approximation for a Nash equilibrium of a large community of agents that are identical. In Sect. 8.2 we have shown that the theory extends to a multi-class of representative agents. However, each class still has its individual control problem.

We first assume that the functions f(x, m, v), g(x, m, v), and h(x, m)—as functions of m—can be extended to Dirac measures and the sum of Dirac measures that are probabilities. Since m is no more in L p , the reference topology will be the weak * topology of measures. That will be sufficient for our purpose, but we refer to [14] for metric space topology. At any rate, the vector space property is lost.

Dealing with the analysis of the systems of PDE’s describing unsteady flows of incompressible fluids, we focus on open problems that occur if the equations are supplemented by the homogeneous Dirichlet (no-slip) boundary conditions, but that are successfully solvable in the spatially periodic setting, for example. Within this framework we also present four different approaches to obtain compactness of weakly converging quantities and discuss how these tools can be used in the global existence theory for the power-law fluids and their various generalizations.

In this chapter we assemble many technical results that will be used throughout the book. Function spaces play a central role, and we recall first some notation, fundamental definitions, and properties.

II G(u)ll