Analysis Seminar | |
Student No.： | 40 |
Time： | Tue 8:20-9:50, 2017-9-19~2018-1-5 |
Instructor： | Yu Pin, Alexandru dan Ionescu |
Place： | Conference room 3, Floor 2, Jin Chun Yuan West Building |
Starting Date： | 2017-9-19 |
Ending Date： | 2018-1-5 |
Date: 2017-12-12
Speaker: Luo Chenyun [Vanderbilt University]
Time: Tue 8:20-9:50
Place: Conf. room 3, Floor 2, Jin Chun Yuan West Bldg.
Title: Compressible water wave and related problems
Abstract: I will talk about my recent results on compressible Euler equations with free surface boundary in an unbounded domain, i.e., the compressible water wave equations. It is well known that the motion of an incompressible water wave admits global in time classical solution, but the theory of a compressible water wave is very poorly established. To my knowledge, the only known result is the compressible water wave equations are locally well-posed, due to Yuri Trakhinin, with loss of regularity (Nash-Moser technique is applied).
In my recent paper, I have established a priori estimate for a compressible water wave, and the estimate is uniform as the sound speed of the liquid goes to infinity, which allows us to show the convergence of the solution for compressible water wave equations to that of the incompressible equations in C^2. To achieve this, we first need to put appropriate decay condition on the initial data, which is not trivial since the data has to be chosen to satisfy compatibility condition up to certain order at the same time. This work opens up a new approach to proving long time existence also for compressible water waves with small data, as conjectured by Hans Lindblad.
Finally, I will also cover some recent work on a compressible liquid with surface tension. It does not follow from my previous work when surface tension is not taken into account. In fact, the presence of surface tension introduces severe technical difficulties and new methods are introduced.
Date: 2017-12-5
Speaker: Alexandru dan Ionescu
Time: Tue 8:20-9:50
Place: Conf. room 3, Floor 2, Jin Chun Yuan West Bldg.
Title: On long term regularity of solutions of certain evolution models
Abstract: I will talk about some recent work on long term regularity of several quasilinear physical models, such as water waves, plasma models, and Einstein equations. The focus will be on explaining some new developments that involve the combination of classical vector-field/energy methods with semilinear methods based on the Fourier transform and analysis of resonances.
Date: 2017-11-7
Speaker: Hao Chengchun 郝成春 [中科院CAS]
Time: Tue 8:20-9:50
Place: Conf. room 3, Floor 2, Jin Chun Yuan West Bldg.
Title: On the free boundary problem of compressible Euler equations in physical vacuum with general initial densities
Description:
In this talk, I will show a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the γ-gas law equation of state for γ=2 and the general initial density in $H^5$. I derive a mixed space-time interpolation inequality which play a vital role in the energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$, which is different from the known results
Date: 2017-10-31
Speaker: Zeng Huihui 曾惠慧 [Tsinghua University]
Time: Tue 8:20-9:50
Place: Conf. room 3, Floor 2, Jin Chun Yuan West Bldg.
Title: On the Free Surface Motion of Highly Subsonic Heat-conducting Inviscid Flows
Description:
I will present a recent result joint with Tao Luo on the free surface problem of a highly subsonic heat-conducting inviscid flow. Adopting a geometric approach developed by Christodoulou and Lindblad in the study of the free surface problem of incompressible inviscid flows, we give the a priori estimates of Sobolev norms in 2-D and 3-D under the Taylor sign condition by identifying a suitable higher order energy functional. The estimates for some geometric quantities such as the second fundamental form and the injectivity radius of the normal exponential map of the free surface are also given. I will discuss the issues of the strong coupling of large variation of temperature, heat-conduction, compressibility of fluids and the evolution of free surface, loss of symmetries of equations, and loss of derivatives in closing the argument which is a key feature compared with Christodoulou and Lindblad's work.
Date: 2017-10-17
Speaker: Mats Ehrnstorm
Time: Tue 8:20-9:50
Place: Conf. room 3, Floor 2, Jin Chun Yuan West Bldg.
Title: Solitary and periodic waves of the Whitham equation
Description:
We will present the proof of Onsager's conjecture by Philip Isett, building on the work of Buckmaster, Daneri, De Lellis, Szekelyhidi, himself and others.
We study the nonlocal and nonlinear Whitham equation, which describes surface waves in shallow water. Due to its very weak dispersive properties, given by the exact dispersion relation for water waves, this equation displays interesting mathematical features and challenges, and may arguably be defended as one of the simplest models inherently capturing the nonlocal behaviour of the full water-wave problem.
In the first lecture, we give an overview of the equation and some of the results on it; in particular on solitary and large-amplitude periodic waves.
In the second and third lectures we go into depth of the steady theory and present the techniques used to construct the different travelling waves, as well as in determining their properties. These two aspects are in fact intertwined, as it turns out there exists a wave of greatest height, singular at it its crest. This is in parallel to the water-wave problem, where the steady periodic waves end with a singular sharp-peaked highest wave.
The methods covered in these lectures include constrained minimisation, concentration-compactness and bifurcation theory. Regularity theory in Zygmund ('Besov-Hölder’) spaces and basic harmonic analysis will also play a role in some of the proofs.
Date: 2017-09-19
Speaker: Luo Tianwen 罗天文 (Tsinghua University)
Time: Tue 8:20-9:50
Place: Conf. room 3, Floor 2, Jin Chun Yuan West B., THU
Title: Onsager Conjecture I
Abstract:
We will present the proof of Onsager's conjecture by Philip Isett, building on the work of Buckmaster, Daneri, De Lellis, Szekelyhidi, himself and others.
The talk consists of three parts.
Part I: General Introduction and Outline of Strategy.
Part II:Convex Integration using Beltrami flow as building blocks, leading to weak solution in C^{1/5-} class.