To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract: 

Convection is buoyancy-driven flow resulting from unstable density stratification in the presence of a gravitational field.  Beyond its central role in myriad engineering heat transfer applications, convection underlies many of nature’s dynamical designs on larger-than-human scales. Indeed, solar heating of Earth’s surface generates buoyancy forces that cause the winds to blow, which in turn drive the oceans’ flow.  Convection in Earth’s mantle on geological timescales makes the continents drift, and thermal and compositional density differences induce buoyancy forces that drive a dynamo in Earth’s liquid metal core—the dynamo that generates the magnetic field protecting us from solar wind that would otherwise extinguish life as we know it on the surface.  The structure of the Sun itself relies on convection in the outer layers to transfer heat from the interior to radiate away from the surface.

The key feature of convection is transport: thermal convection actively transports the heat that generates the density variations that produce the buoyancy forces.  Determining the rate at which “heat rises” in turbulent convection is one of the most important open problems in fluid dynamics.  In this presentation the confounding question of asymptotically high Rayleigh number heat transport in Rayleigh-Bénard convection – the buoyancy-driven flow in a horizontal layer of fluid heated from below modeled by the Boussinesq approximation to the Navier-Stokes equations – is reviewed from viewpoints of theory (models of the model), computation (direct numerical simulations), experiment (laboratory tests), and mathematical analysis (theorems).

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract: The primary purpose of this talk is to elucidate an instability mechanism, which will be referred to as degenerate dispersion, that leads to illposedness of the Cauchy problem in arbitrarily high-regularity Sobolev spaces for a number of nonlinear PDEs of hydrodynamics and magnetohydrodynamics (MHD) that respect conservation of energy. Due to the conservation structure, the instability mechanism is necessarily different from that of, say, the reverse heat equation; rather, it is a mechanism by which energy gets concentrated into small scales at an arbitrarily fast rate due to the degeneration of the dispersion relation. 

In this talk, I will focus on generalized surface quasi-geostrophic (gSQG) models with singular velocities. I will give a heuristic description of the phenomenon via geometric-optical ideas (or classical-quantum correspondence), and then discuss the mathematical tools recently developed to capture this phenomenon in the nonlinear setting. This talk is based on joint works with Dongho Chae and In-Jee Jeong.

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract:  

In this talk, I consider the Muskat problem, modeling the dynamics of two fluids in porous media. I will present  local and global well-posedness of the 2d muskat problem (with viscosity jump and surface tension) in the critical space H^{3/2}. 

This is based on joint works in collaboration with Thomas Alazard.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract: 

In this talk, we illustrate that small-amplitude Stokes waves with infinite depth are nonlinearly unstable under long-wave perturbations. Our approach is based on the modulational approximation of the water wave system and the instability mechanism of the focusing cubic nonlinear Schrödinger equation. This is joint work with Gong Chen.

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract: In this talk, I will present different boundary conditions for the Stokes operator, relevant or not from the physical viewpoint. The general setting will be bounded open connected sets with Lipschitz boundary. The properties of the Stokes semigroup will be described. They will be used to prove existence in critical spaces of mild solutions of the incompressible Navier-Stokes system via a classical fixed point procedure. Uniqueness can be obtained in some cases via the property of maximal regularity.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract:
The question of how roots of polynomials move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the Euler alignment model used in mathematical biology to describe collective behavior and flocking of various species - such as fish, birds or ants. The equation is critical, but due to strongly nonlinear form of its coefficients, proving global regularity for its solutions is harder than for equations such as Burgers, SQG or Euler alignment model. I will discuss joint work with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials. Intriguing connections to free probability and random matrices will also be mentioned.

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

Abstract: The log canonical threshold is an invariant that measures how singular a hypersurface over an algebraically closed field of characteristic zero is. The F-pure threshold is the positive characteristic analog. Hypersurfaces with smaller threshold are more singular.
I will discuss a lower bound for a homogeneous polynomial in characteristic p, relative to its degree, and describe the classification of the hypersurfaces that achieve this bound up to change of coordinates. These results were obtained as part of a project started at the A.W.M. Workshop ``Women in Commutative Algebra” at B.I.R.S.; joint work with Zhibek Kadyrsizova, Jennifer Kenkel, Janet Page, Jyoti Singh, Karen E. Smith and Emily Witt.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Serre's equations have several drawbacks. First, they are inaccurate for relatively  short waves. Second, even for weakly sheared currents, Kelvin-Helmholtz  instabilities  appear. Third, they do not admit physically relevant solutions such as `slug' or `plug' flows. We propose here modified Serre-like equations to address these 

shortcomings. 

The modified Serre equations have improved dispersion properties. In presence of weak  sheared current, no Kelvin--Helmholtz instability appears. With strong sheared current,  Kelvin-Helmholtz instabilities appear at low frequencies, that is consistent with the long wave approximation. For steady waves, the modified Serre equations admit slug-like solutions.  

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract:  Further results are developed for showing the critical nature of the Navier-Stokes equation.  It is proved that global-in-time smooth solutions exist for the Navier-Stokes equations with hyper-dissipation, i. e. the equations \[ \partial_t u+(-\Delta)^\beta u+ u\cdot\nabla u+\nabla p=0 , \quad \textrm{div}\ u=0, \qquad \beta >1 \] with the hypothesis that there is only one singular point at a possible blow-up time.

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

Let R_n be the polynomial ring with n variables over a field K. We consider the natural action of the n-th symmetric group S_n to R_n. In this talk, I will mainly talk about the following problem: Fix monomials u_1,\dots,u_m and consider the ideal I_n of R_n generated by the S_n-orbits of these monomials. How the Betti numbers of I_n change when n increases?
I will explain that there is a simple way to determine non-zero positions of the Betti table of I_n when n is sufficiently large. I also explain that we can determine the Betti numbers of I_n by considering the S_n-module structure of Tor_i(I_n,K).
The above problem is motivated by recent studies of algebraic properties of S_n-invariant ideals and is inspired by studies of Noetherianity up to symmetry. I will explain this motivation and basic combinatorial properties of S_n-invariant ideals in the first part of the talk.
This talk includes a joint work with Claudiu Raicu.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract:

I will consider the 2D incompressible Euler equation in a simply-connected bounded domain. Such domains admit stationary points with the following property: a single point vortex located at that position will not move. In this talk I will discuss the problem of the long time confinement of the vorticity around such a stationary point. This is joint work with Martin Donati.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Tea for members and participants.

To participate, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657