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I will review the topic of primordial inflation, covering both the basics of the field and some more advanced topics. The advanced topics will be: initial conditions for inflation, slow-roll eternal inflation and the EFT of inflation.

 I will review the topic of primordial inflation, covering both the basics of the field and some more advanced topics. The advanced topics will be: initial conditions for inflation, slow-roll eternal inflation and the EFT of inflation.

Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-B...

The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.

However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume ...

In this lecture I will cover the developments in the area which followed our work and finally lead Isett to the resolution of the Onsager conjecture. If time allows I will give a glimpse of the challenges that lie ahead if one would like to give a rigorous justification that some dissipative solutions of the Euler equations are indeed limit of classical solutions of Navier-Stokes.

Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-B...

The flow associated to a given smooth vector field $u(t,x)$ is well-defined, it is smooth, and it can be used to solve the classical linear PDEs associated to $u$, namely the transport and the continuity equations. The situation is substantially different when $u$ is not smooth (e.g. when $u$ is not Lipschitz continuous in the space variable, but only Sobolev or BV). Goal of the lecture is to provide an overview on classical and recent results about flow maps and PDEs associated to non-smooth vector fields.

In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.

In this lecture I will explain a second type of iteration, introduced by La'szlo' and myself, which produces continuous solutions of incompressible Euler in a fashion which shares a lot of similarities with the theorem of Nash explained in Lecture 3.

In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.

Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-B...