Geometric analysis in the development of shocks in compressible fluids
-
Demetrios Christodoulou
Abstract
We give an outline of the content of our recent monograph “The Shock Development Problem” which analyzes the development of shocks in fluids. The framework is that of Euler’s equations of the mechanics of compressible fluids. We first set up the mathematical problem as an initial-boundary value problem for a nonlinear hyperbolic system of partial differential equations with a free boundary and singular initial conditions. Then we describe the main mathematical methods which we introduce to solve the problem. Some of these methods are geometric and have to do with a geometric structure defined on spacetime by the fluid and its interaction with the background spacetime structure. Also, with the nature of the free boundary and the jump conditions there which include a non-linear jump condition. The central method however is analytic and arises from the need to handle singular integrals appearing in the energy identities.
Abstract
We give an outline of the content of our recent monograph “The Shock Development Problem” which analyzes the development of shocks in fluids. The framework is that of Euler’s equations of the mechanics of compressible fluids. We first set up the mathematical problem as an initial-boundary value problem for a nonlinear hyperbolic system of partial differential equations with a free boundary and singular initial conditions. Then we describe the main mathematical methods which we introduce to solve the problem. Some of these methods are geometric and have to do with a geometric structure defined on spacetime by the fluid and its interaction with the background spacetime structure. Also, with the nature of the free boundary and the jump conditions there which include a non-linear jump condition. The central method however is analytic and arises from the need to handle singular integrals appearing in the energy identities.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents XI
- Geometric analysis in the development of shocks in compressible fluids 1
- Ancient solutions to geometric flows 29
- Boundary value problems, medical imaging and the asymptotics of Riemann’s zeta function 51
- A short glimpse of the giant footprint of Fourier analysis and recent multilinear advances 79
- Fractional calculus and numerical methods for fractional PDEs 91
- Reinforcement learning: a comparison of UCB versus alternative adaptive policies 127
- Mathematics of computational modelling: some challenges of computing nonlinear phenomena 139
- Sharp estimates for dyadic-type maximal operators and stability 167
- Data structures for robust multifrequency imaging 181
- Theta and eta polynomials in geometry, Lie theory, and combinatorics 231
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents XI
- Geometric analysis in the development of shocks in compressible fluids 1
- Ancient solutions to geometric flows 29
- Boundary value problems, medical imaging and the asymptotics of Riemann’s zeta function 51
- A short glimpse of the giant footprint of Fourier analysis and recent multilinear advances 79
- Fractional calculus and numerical methods for fractional PDEs 91
- Reinforcement learning: a comparison of UCB versus alternative adaptive policies 127
- Mathematics of computational modelling: some challenges of computing nonlinear phenomena 139
- Sharp estimates for dyadic-type maximal operators and stability 167
- Data structures for robust multifrequency imaging 181
- Theta and eta polynomials in geometry, Lie theory, and combinatorics 231
