Shanghai Jiao Tong University

We consider a one-dimensional compressible Navier-Stokes model for reacting gas mixtures with the same γ-law in dynamic combustion. The unknowns of the PDE system consist of the inverse density, velocity, temperature, and mass fraction of the reactant (Z). First, we show that the graph of Z cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any time, based on a Bernis-type inequality by M. Winkler (2012) and the recent works by T. Cieślak et al (2023). In addition, we establish the global well-posedness theory of small BV weak solutions for initial data that are small perturbations around the constant equilibrium state (1, 0, 1, 0) in the L¹(R)∩BV(R)-norm, via an analysis of the Green's function of the linearised system. The large-time behaviour of the global BV weak solutions is also characterised. This is motivated by and extends the recent global well-posedness theory for BV weak solutions to the one-dimensional isentropic Navier-Stokes and Navier-Stokes-Fourier systems developed by T. Liu and S.-H. Yu (2022).

*Joint with Prof. Haitao Wang and Miss Jianing Yang (SJTU)

Quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators.  It is important to  to explore whether other problems in scientific computing, such as ODEs, PDEs, and  linear algebra that arise in both classical and quantum systems which are not unitary evolution,  can be handled by quantum computers.  

We will present a systematic way to develop quantum simulation algorithms for general differential equations. Our basic framework is dimension lifting, that transfers non-autonomous ODEs/PDEs systems to autonomous ones, nonlinear PDEs to linear ones, and linear ones to Schrodinger type PDEs—coined “Schrodingerization”—with uniform evolutions. Our formulation allows both qubit and qumode (continuous-variable) formulations, and their hybridizations, and provides the foundation for analog quantum computing which are easier to realize in the near term. We will also discuss  dimension lifting techniques for quantum simulation of stochastic DEs and PDEs with fractional derivatives. 

This talk is about a classical problem in differential geometry and global analysis: the isometric immersions of Riemannian manifolds into Euclidean spaces. We focus on the PDE approach to isometric immersions, i.e., the analysis of Gauss--Codazzi--Ricci equations, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges.

In this short course, we survey the celebrated weak and strong compactness theorems proved by Karen Uhlenbeck in 1982. These results are fundamental to the gauge theory and have found numerous applications to geometry, topology, and theoretical physics. The proof is based on the ingenious idea of putting connections into ``Uhlenbeck--Coulomb gauge'', which enables the use of standard elliptic and/or nonlinear PDE techniques, as well as involved local-to-global patching arguments. We aim at giving detailed explanation of the proof, and we shall also discuss the relation between Uhlenbeck's compactness and the classical geometric problem of isometric immersions of submanifolds into Euclidean spaces.

The development of high frequency and algorithmic trading allowed to considerably reduce the bid ask spread by increasing liquidity in limit order books. Beyond the problem of optimal placement of market and limit orders, the possibility to cancel orders for free leaves room for price manipulations, one of such being spoofing. Detecting spoofing from a regulatory viewpoint is challenging due to the sheer amount of orders and difficulty to discriminate between legitimate and manipulative flows of orders. However, it is empirical evidence that volume imbalance reflecting offer and demand on both sides of the limit order book has an impact on subsequent price movements. Spoofers use this effect to artificially modify the imbalance by posting limit orders and then execute market orders at subsequent better prices while canceling at a high speed their previous limit orders. In this work we set up a model to determine where a spoofer would place its limit orders to maximize its gains as a function of the imbalance impact on the price movement. We study the solution of this non local optimization problem as a function of the imbalance. With this at hand, we calibrate on real data from TMX the imbalance impact (as a function of its depth) on the resulting price movement. Based on this calibration and theoretical results, we then provide some methods and numerical results as how to detect in real time some eventual spoofing behavior based on Wasserstein distances. Joint work with Tao Xuan (SJTU), Ling Lan (SJTU) and Andrew Day (Western University)
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In this talk, we will introduce how to apply Green's function method to get  pointwise estimates for solutions of the Cauchy problem of nonlinear evolution equations with dissipative  structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and exhibit the generalized Huygen's principle. Then, for other nonlinear dissipative evolution equations, we will  introduce some recent results and give brief explanations. Our approach is based on the detailed analysis of the Green's function of the linearized system and micro-local analysis, such as frequency decomposition and so on.