Abstract : Over the last years, substantial breakthroughs have arisen in mathematical fluid mechanics. For example, the smooth blowup of the incompressible, axisymmetric Euler equations via computer assisted proofs, or the smooth self-similar blowup solutions to compressible Euler and Navier-Stokes. The aim of this session is to bring together well-known experts and young researchers to present new developments in partial differential equations describing the dynamics of fluids. Particular emphasis has been put into explaining the aforementioned breakthroughs and exploring new directions from them. Other key topics include corners and cusps solutions in fluids models, and stability results for kinetic equations.
Organizer(s) : Bruno Vergara, Javier Gómez Serrano
01188 (1/3) : 2C @G405 [Chair: Javier Gómez Serrano]
[04609] Recent progress on singularity formation in incompressible fluids
Format : Online Talk on Zoom
Author(s) :
Jiajie Chen (New York University)
Thomas Y Hou (California Institute of Technology)
Abstract : I will talk about recent progress on singularity formation in incompressible fluids with smooth data.
[04708] Gravity Unstable Muskat Bubbles
Format : Online Talk on Zoom
Author(s) :
Neel Patel (University of Maine)
Siddhant Agrawal (ICMAT)
Sijue Wu (University of Michigan)
Abstract : The Muskat problem describes the evolution of the interface between two fluids in porous media. Neglecting surface tension, the well-posedness of this problem depends on the Rayleigh-Taylor condition. For fluids of differing densities, it is required that the denser fluid is below. Otherwise, the system is gravity unstable. We will discuss the stability of a closed curve interface, or a bubble, in which the Rayleigh-Taylor condition cannot hold.
[04950] Whitham’s highest cusped wave
Format : Talk at Waseda University
Author(s) :
Bruno Vergara (Brown University)
Abstract : Whitham’s equation is a nonlinear, nonlocal, very weakly dispersive shallow water wave model in one space dimension. As in the case of the Stokes wave for the Euler equation, non-smooth traveling waves with greatest height between crest and trough have been shown to exist for this model. In this talk I will discuss the existence of a unique cusped, convex and monotone traveling wave solution to the Whitham equation. Our results follow a strategy that combines different ideas from classical analysis and rigorous computer verification methods. Joint work with Alberto Enciso and Javier Gómez Serrano.
[03258] On the (in)stability of smooth self-similar solutions to the compressible Euler equations
Abstract : In this talk, I am going to describe recent progress in smooth self-similar solutions to the compressible Euler equations. I will explain how these solutions, initially found by Merle-Raphael-Rodnianski-Szeftel (2019), arise in the family of Guderley self-similar solutions (1942), how their (in)stability is studied under smooth perturbations, and which are some endpoints of unstable directions. The topic will be introduced making a contrast between its states during the 20th and 21st centuries.
[04242] Small scale creation for the 2D Boussinesq Equation
Format : Talk at Waseda University
Author(s) :
Alexander Kiselev (Duke university)
Yao Yao (National University of Singapore)
Jaemin Park (University of Basel)
Abstract : In this talk, we study long-time behaviors of the two- dimensional incompressible Boussinesq equations without thermal diffusion. While the 2D Boussinesq equations is known to possess global solutions with the presence of viscos- ity, it remains a outstanding open problem whether the inviscid case can exhibit a finite-time blow up. In the viscous case, we established algebraic growth of the Sobolev norms of the solu- tions for all time. For the inviscid case, we obtained the growth of the gradient of the temperature, assuming that the global so- lution exists for all time. The initial data under consideration in this work is not too restrictive. More precisely, we only require certain symmetry and sign conditions. The key ingredient of the proof is to derive a norm-inflation from the decay of an anisotropic Sobolev norm of the temperature, which can be ob- served in the conservation of energy. This work is a joint work with A. Kiselev and Y. Yao.
[05046] On the motion of an internal wave in two-dimensional viscous flow
Format : Talk at Waseda University
Author(s) :
Rafael Granero-Belinchon (Universidad de Cantabria)
Abstract : In this talk we will review some recent results concerning the motion of an internal wave in two-phase viscous flow. In particular we will establish the local and global well-posedness of the free boundary problem associated to this physical situation. Finally, we will also prove an exponential instability result.
These results were obtained in a joint work with Francisco Gancedo and Elena Salguero.
[03025] Smooth imploding solutions for 3D compressible fluids
Format : Online Talk on Zoom
Author(s) :
Gonzalo Cao Labora (Massachusetts Institute of Technology (MIT))
Abstract : We will talk about singularity formation for the 3D isentropic compressible Euler and Navier-Stokes equations for ideal gases. We will construct a new family of self-similar profiles corresponding to larger self-similar exponents than what was previously known. In particular, this will show singularity formation for all adiabatic constants, giving the first known singularity formation result for monoatomic gases. These results are joint work with Tristan Buckmaster and Javier Gomez-Serrano.
[04870] Stability of a point charge for the Vlasov-Poisson system
Format : Online Talk on Zoom
Author(s) :
Benoit Pausader (Brown University)
Abstract : We consider solutions of the Vlasov-Poisson system starting from initial data (i) a dirac mass (the point charge) and (ii) some small density with respect to Liouville measure (the cloud). We show global existence of the solution and describe the asymptotic behavior in terms of modified scattering. This is joint work with K. Widmayer and J. Jiang.
[05033] Global axisymmetric Euler flows with rotation
Format : Online Talk on Zoom
Author(s) :
Klaus Widmayer (University of Vienna & University of Zurich)
Benoit Pausader (Brown University)
Yan Guo (Brown University)
Abstract : We discuss the construction of a class of global, dynamical solutions to the 3d Euler equations near the stationary state given by uniform "rigid body" rotation. These solutions are axisymmetric, of Sobolev regularity and have non-vanishing swirl.
[02875] On the analyticity of the Muskat equation
Format : Online Talk on Zoom
Author(s) :
Jia Shi (MIT)
Abstract : The Muskat equation describes the interface of two liquids in a porous medium. We will show that if a solution to the Muskat problem in the case of same viscosity and different densities is sufficiently smooth, then it must be analytic except at the points where a turnover of the fluids happens. We will also show analyticity in a region that degenerates at the turnover points provided some additional conditions are satisfied.
