# Schedule for: 23w5066 - Partial Differential Equations in Fluid Dynamics

Beginning on Sunday, August 6 and ending Friday August 11, 2023

All times in Hangzhou, China time, CST (UTC+8).

Sunday, August 6 | |
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14:00 - 17:30 | Check-in begins at 14:00 on Sunday and is open 24 hours (Front desk - Yuxianghu Hotel（御湘湖酒店前台)) |

18:00 - 20:30 |
Dinner ↓ A buffet dinner is served daily between 6:00pm and 8:30pm in the Xianghu Lake National Tourist Resort. (Dining Hall - Yuxianghu Hotel（御湘湖酒店餐厅)) |

Monday, August 7 | |
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07:00 - 09:00 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort (Dining Hall - Yuxianghu Hotel（御湘湖酒店餐厅)) |

09:30 - 09:45 |
Introduction and Welcome by IASM Staff ↓ A brief introduction to IASM with important logistical information, technology instruction, and opportunity for participants to ask questions. (Lecture Hall - Academic island（定山院士岛报告厅)) |

09:45 - 10:45 |
Gui-Qiang G. Chen: Cavitation and concentration in the solutions of the sompressible Euler equations and related nonlinear PDEs in fluid dynamics ↓ In this talk, we will discuss the intrinsic phenomena of cavitation/decavitation and concentration/deconcentration in the entropy solutions of the compressible Euler equations, the compressible Euler-Poisson equations, and related nonlinear PDEs, which are fundamental to the analysis of entropy solutions for nonlinear PDEs. We will start to discuss the formation process of cavitation and concentration in the entropy solutions of the isentropic Euler equations with respect to the initial data and the vanishing pressure limit. Then we will analyze a longstanding fundamental problem in fluid dynamics: Does the concentration occur generically so that the density develops into a Dirac measure at the origin in spherically symmetric entropy solutions of the multi-dimensional compressible Euler equations and related nonlinear PDEs? We will report our recent results and approaches developed for solving this problem for the Euler equations, the Euler-Poisson equations, and related nonlinear PDEs, and discuss its close connections with entropy methods and the theory of divergence-measure fields. Further related topics, perspectives, and open problems will also be addressed. (Lecture Hall - Academic island（定山院士岛报告厅)) |

10:45 - 11:00 | Coffee Break (Lecture Hall - Academic island（定山院士岛报告厅)) |

11:00 - 11:30 |
Yoshiyuki Kagei: Eckhaus instability of the compressible Taylor vortices ↓ This talk is concerned with the bifurcation and stability of
the compressible Taylor vortex. Consider the compressible Navier-Stokes
equations in a domain between two concentric infinite cylinders. If the
outer cylinder is at rest and the inner one rotates with sufficiently
small angular velocity, a laminar flow, called the Couette flow, is
stable. When the angular velocity of the inner cylinder increases,
beyond a certain value of the angular velocity, the Couette flow becomes
unstable and a vortex pattern, called the Taylor vortex, bifurcates and
is observed stably. This phenomenon is mathematically formulated as a
bifurcation and stability problem. In this talk, the compressible Taylor
vortices are shown to bifurcate near the criticality for the
incompressible problem when the Mach number is sufficiently small. The
localized stability of the compressible Taylor vortices is considered
under axisymmetric perturbations and it is shown that the Eckhaus
instability of compressible Taylor vortices occurs as in the case of the
incompressible ones. (Lecture Hall - Academic island（定山院士岛报告厅)) |

11:30 - 12:00 |
Yue-Jun Peng: Quasi-neutral limit in Euler-Poisson systems ↓ Euler-Poisson systems are widely used in the mathematical modeling of plasmas in which the Debye-length is a small parameter. The quasi-neutral limit leads to the quasi-neutrality of the plasma, which is realized as the parameter tends to zero. In this talk, I will present mathematical studies and results of the quasi-neutral limit in Euler-Poisson systems in stationary or non-stationary case. This concerns nonlinear partial differential equations of hyperbolic or elliptic type. Various mathematical tools are needed to justify the limit in mathematical frameworks. (Lecture Hall - Academic island（定山院士岛报告厅)) |

12:00 - 13:30 |
Lunch ↓ Lunch is served daily between 12:00am and 1:30pm in the Xianghu Lake National Tourist Resort (Dining Hall - Academic island（定山院士岛餐厅)) |

13:45 - 14:45 |
Eduard Feireisl: Glimm's method, convex Integration, and density of wild data for the Euler system of gas dynamics ↓ We adapt Glimm’s approximation method to the framework of convex
integration to show density of wild data for the (complete) Euler system of gas dynamics.
The desired infinite family of entropy admissible solutions emanating from the same initial
data is obtained via convex integration of suitable Riemann problems pasted with local smooth
solutions. In addition, the wild data belong to BV class. (Lecture Hall - Academic island（定山院士岛报告厅)) |

14:45 - 15:00 | Coffee Break(soft drink only) (Lecture Hall - Academic island（定山院士岛报告厅)) |

15:00 - 15:30 |
Kenneth Karlsen: Stochastic velocity averaging and a singular limit problem ↓ We address a singular limit problem involving stochastic conservation laws characterized by discontinuous flux, accompanied by vanishing diffusion and dynamic capillarity terms. To establish convergence, our investigation employs kinetic formulations, H-measures, and novel velocity averaging techniques for stochastic transport equations. Furthermore, we utilize almost-sure representations of random variables within carefully chosen quasi-Polish spaces. This talk is based on joint work M. Kunzinger and D. Mitrovic. (Zoom (Online)) |

15:30 - 16:00 |
Tao Luo: On the free boundary problems of 3-D compressible Euler equations coupled with a nonlinear Poisson equation ↓ For the problem of the non-isentropic compressible Euler Equations coupled with a nonlinear Poisson equation in three spatial dimensions with a general free boundary not restricting to a graph, we identify suitable stability conditions on the electric potential and the pressure under which we obtain a priori estimates on the Sobolev norms of the fluid and electric variables and bounds for geometric quantities of the free surface. In the isentropic case, the stability condition reduces to a single one. We also give the a priori estimates without losing derivatives for the free boundary problem of full compressible Euler equations with general variable entropy. This is a joint work with Konstantina Trivisa and Huihui Zeng. (Lecture Hall - Academic island（定山院士岛报告厅)) |

16:00 - 16:15 | Coffee Break (Lecture Hall - Academic island（定山院士岛报告厅)) |

16:15 - 16:45 |
Weizhu Bao: Modeling, analysis and simulation for degenerate dipolar quantum gas ↓ In this talk, I will present our recent work on mathematical
models, asymptotic analysis and numerical simulation for degenerate
dipolar quantum gas. As preparatory steps, I begin with the
three-dimensional Gross-Pitaevskii equation with a long-range
dipolar interaction potential which is used to model the degenerate
dipolar quantum gas and reformulate it as a Gross-Pitaevskii-Poisson
type system by decoupling the two-body dipolar interaction potential
which is highly singular into short-range (or local) and long-range
interactions (or repulsive and attractive interactions). Based on
this new mathematical formulation, we prove rigorously existence
and uniqueness as well as nonexistence of the ground states, and
discuss the existence of global weak solution and finite time blowup
of the dynamics in different parameter regimes of dipolar quantum gas.
In addition, a backward Euler sine pseudospectral method is presented
for computing the ground states and a time-splitting sine pseudospectral
method is proposed for computing the dynamics of dipolar BECs. Due to the
adoption of new mathematical formulation, our new numerical methods
avoid evaluating integrals with high singularity and thus they are
more efficient and accurate than those numerical methods currently used
in the literatures for solving the problem. In addition, new mathematical
formulations in two-dimensions and one dimension for dipolar quantum gas
are obtained when the external trapping potential is highly confined in
one or two directions. Numerical results are presented to confirm
our analytical results and demonstrate the efficiency and accuracy
of our numerical methods. Some interesting physical phenomena are
discussed too. (Zoom (Online)) |

16:45 - 17:15 |
Deng Zhang: Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: above the Lions exponent ↓ We are concerned with the 3D hyperdissipative Navier-Stokes equations, where the viscosity exponents can be larger than the Lions exponent. We will show that, even in this high dissipative regime, the uniqueness of weak solutions would fail in the supercritical mixed Lebesgue spaces. The non-uniqueness is sharp at the endpoints of the Ladyzenskaja-Prodi-Serrin criterion, and the constructed solutions admit the spatial regularity outside a fractal set of singular times with zero Hausdorff $H^{\eta}$ measure, where $\eta$ can be any given small positive constant. This talk is based on joint works with Yachun Li, Peng Qu and Zirong Zeng. (Lecture Hall - Academic island（定山院士岛报告厅)) |

18:00 - 20:30 | Dinner (Dining Hall - Yuxianghu Hotel（御湘湖酒店餐厅)) |

Tuesday, August 8 | |
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07:00 - 09:00 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort (Dining Hall - Yuxianghu Hotel（御湘湖酒店餐厅)) |

09:30 - 10:00 |
Alexis Vasseur: Inviscid limit from Navier-stokes to small BV solutions of Euler ↓ We show in this talk that small BV solutions to the isentropic Euler equations can be obtained as inviscid limits from the compressible barotropic Navier-Stokes equations.
This is a joint work with Geng Chen and Moon-Jin Kang. (Zoom (Online)) |

10:00 - 10:30 |
Dehua Wang: Euler equations and transonic flows ↓ In this talk, we will consider the Euler equations of gas
dynamics and applications in transonic flows. First the basic theory
of Euler equations will be reviewed. Then we will present the results
on the transonic flows past obstacles, transonic flows in the fluid
dynamic formulation of isometric embeddings, and the transonic flows
in nozzles. We will discuss global solutions and stability obtained
through various techniques and approaches. (Zoom (Online)) |

10:30 - 11:00 | Coffee Break (Lecture Hall - Academic island（定山院士岛报告厅)) |

11:00 - 11:30 |
Feimin Huang: Continued gravitational collapse for gaseous star and pressureless Euler-Poisson system ↓ The gravitational collapse of an isolated self-gravitating gaseous star for $\gamma-$law pressure is important. In this talk, I will introduce recent progress on the continued gravitational collapse in finite time for Euler-Poisson equation with self-gravitating gaseous star. (Lecture Hall - Academic island（定山院士岛报告厅)) |

11:30 - 12:00 |
Yao Yao: Small scale formations in fluid equations with gravity ↓ In this talk, we discuss some PDEs that describe fluid motion under the influence of gravity, including the incompressible porous media equation and incompressible Boussinesq equation in two dimensions. Using an interplay between various monotone and conserved quantities, we construct rigorous examples of small scale formations as time goes to infinity. These growth results work for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth. (Based on joint works with Alexander Kiselev and Jaemin Park). (Lecture Hall - Academic island（定山院士岛报告厅)) |

12:00 - 13:30 | Lunch (Dining Hall - Academic island（定山院士岛餐厅)) |

13:45 - 14:15 |
Tong Yang: Discussion on the Prandtl and Prandtl type operators ↓ In this talk, I will present some understanding on the classical Prandtl operator and some Prandtl type operators derived from fluid models in different settings. It aims to investigate the intrinsic mechanisms in the operators that lead to well-posedness theories in different function spaces. (Zoom (Online)) |

14:15 - 14:45 |
Shinya Nishibata: Stability of spherically symmetric stationary solutions for the compressible Navier-Stokes equation ↓ In the present talk, we discuss an asymptotic behavior
of a spherically symmetric solution on the exterior domain
of an unit ball for the compressible Navier-Stokes equation,
describing a motion of viscous barotropic gas.
Especially we study outflow problem, that is, the fluid blows out
a through boundary. Precisely we obtain
the property of the stationary solution and its convergence rate
as the spatial variable tends to infinity. Then we show the time global
existence of the solution and it converges to the stationary solution
as time tends to infinity. In this result, we do not assume the smallness of the initial condition if it belongs to the suitable Sobolev space. (Lecture Hall - Academic island（定山院士岛报告厅)) |

14:45 - 15:00 | Coffee Break(soft drink only) (Lecture Hall - Academic island（定山院士岛报告厅)) |

15:00 - 16:00 |
Denis Serre: Further a priori estimates in gas dynamics through Compensated Integrability ↓ In recent times, we elaborated a tool, called Compensated Integrability, which provides Strichartz-like inequalities for some systems of conservation laws. We were able to derive space-time estimates for Gas dynamics, in terms of internal variables like pressure and density. We shall describe new developments, in particular a multilinear version of C.I., and derive additional estimates. These involve either the velocity field, or singular integrals. (Zoom (Online)) |

16:00 - 16:30 | Coffee Break (Lecture Hall - Academic island（定山院士岛报告厅)) |

16:30 - 17:30 | Panel Discussion (Lecture Hall - Academic island（定山院士岛报告厅)) |

18:30 - 20:30 |
Outdoor dinner at Xianghu Lake ↓ Outdoor banquet (Academic island（定山院士岛)) |

Wednesday, August 9 | |
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07:00 - 09:00 | Breakfast (Dining Hall - Yuxianghu Hotel（御湘湖酒店餐厅)) |

09:30 - 10:00 |
Alexey Cheskidov: Turbulent solutions of fluid equations ↓ In the past couple of decades, mathematical fluid dynamics has been highlighted by numerous constructions of solutions to fluid equations that exhibit pathological or wild behavior. These include the loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness results in supercritical spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically, but had been difficult to produce analytically. In this talk I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations. (Lecture Hall - Academic island（定山院士岛报告厅)) |

10:00 - 10:30 |
Huijiang Zhao: Hilbert expansion for some nonrelativistic kinetic equations ↓ This talk is concerned with the hydrodynamic limits of some nonrelativistic kinetic equations, such as the Landau equation, the Vlasov-Maxwell-Landau system, and the Vlasov-Maxwell- Boltzmann system, in the whole space. Our main purpose is two-fold: the first one is to give a rigorous derivation of the compressible Euler equations from these kinetic equations via the Hilbert expansion; while the second one is to prove, still in the setting of Hilbert expansion, that the unique classical solutions of the Vlasov-Maxwell-Landau system and the Vlasov-Maxwell- Boltzmann system converge, which is shown to be globally in time, to the resulting global smooth solution of the Euler-Maxwell system, as the Knudsen number goes to zero. The main ingredient of our analysis is to derive some novel interplay energy estimates on the solutions of these kinetic equations which are small perturbations of both a local Maxwellian and a global Maxwellian, respectively. (Zoom (Online)) |

10:30 - 11:00 | Coffee Break (Lecture Hall - Academic island（定山院士岛报告厅)) |

11:00 - 12:00 |
Susan Friedlander: In Search of Euler equilibria via the MR equations ↓ The subject of “ geometric “ fluid dynamics flourished following the
seminal work of V.I. Arnold in the 1960s. A famous paper was
published in 1970 by David Ebin and Jerrold Marsden who used the
manifold structure of certain groups of diffeomorphisms to obtain sharp
existence and uniqueness results for the classical equations of fluid dynamics.
Of particular importance are the fixed points of the underlying dynamical system
and the “accessibility” of these Euler equilibria. In 1985 Keith Moffatt introduced
a mechanism for reaching these equilibria not through the Euler vortex dynamics
itself but via a topology preserving diffusion process called “Magnetic Relaxation”.
In this talk we will discuss some recent results for Moffatt’s MR equations which
are mathematically challenging not only because they are active vector equations
but also because they have a cubic nonlinearity. This is joint work with Rajendra Beckie, Adam Larios and Vlad Vicol. (Zoom (Online)) |

12:00 - 13:30 | Lunch (Dining Hall - Academic island（定山院士岛餐厅)) |

13:45 - 14:15 |
Seung Yeal Ha: Emergent dynamics of infinitely many Kuramoto oscillators ↓ In this talk, we propose an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row (or columm)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender's index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc. This is a joint work with Euntaek Lee (SNU) and Woojoo Shim (Kyungpook National University). (Zoom (Online)) |

14:15 - 14:45 |
Mimi Dai: Developments in fluid equations since Leray’s time ↓ Derived two hundred years ago, the Navier-Stokes equation (NSE) governs the motion of fluids. In 1930s, Leray established the theory of weak solutions for the NSE and raised questions, some of which still remain open and center around the well-posedness problem. In the talk, we will review some progresses in the effort to understand these classical questions. The emphasis will be on some recent results, sparked by empirical laws in physics (such as Kolmogorov’s phenomenological theory of turbulence) and techniques from other fields in mathematics (for instance the convex integration scheme). We will also discuss some ongoing interests in various problems and new perspectives opened up by these techniques. (Lecture Hall - Academic island（定山院士岛报告厅)) |

14:45 - 15:15 | Coffee Break (Academic island（定山院士岛)) |

15:15 - 15:45 |
Renjun Duan: Compressible Euler-Maxwell limit for global smooth solutions to the Vlasov-Maxwell-Boltzmann system ↓ Two fundamental models in plasma physics are given by the Vlasov-Maxwell-Boltzmann system and the compressible Euler-Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions at the kinetic and fluid levels, respectively. It has remained a long-standing open problem to rigorously justify the hydrodynamic limit from the former to the latter as the Knudsen number $\varepsilon$ tends to zero. In this paper we give an affirmative answer to the problem for smooth solutions to both systems near constant equilibrium in the whole space in case when only the dynamics of electrons is taken into account. The explicit rate of convergence in $\varepsilon$ over an almost global time interval is also obtained for well-prepared data. For the proof, one of main difficulties occurs to the cubic growth of large velocities due to the action of the classical transport operator on local Maxwellians and we develop new $\varepsilon$-dependent energy estimates basing on the macro-micro decomposition to characterize the asymptotic limit in the compressible setting. Joint with Dongcheng Yang and Hongjun Yu. (Lecture Hall - Academic island（定山院士岛报告厅)) |

15:45 - 16:15 |
Quoc-Hung Nguyen: Well-posedness theory for nonlinear evolution equations in fluids and geometry ↓ In this talk, I will introduce a new method to study critical well-posedness for local and nonlocal nonlinear evolution equations in fluids and geometry. (Zoom (Online)) |

16:15 - 16:30 | Coffee Break (Lecture Hall - Academic island（定山院士岛报告厅)) |

16:30 - 17:00 |
Zhifei Zhang: Linear inviscid damping and enhanced dissipation for shear flows ↓ The inviscid damping and enhanced dissipation play a crucial role in the hydrodynamic stability. Both stabilizing effects are due to the mixing mechanism induced by shear flows. In this talk, I will introduce some approaches to establish the inviscid damping and enhanced dissipation for the linearized 2-D Navier-Stokes system around shear flows in a finite channel. (Lecture Hall - Academic island（定山院士岛报告厅)) |

17:00 - 17:30 |
Yong Lu: Global solutions to isentropic compressible Navier-Stokes equations in 3D thin domains ↓ In this paper, we study the global well-posedness of isentropic compressible Navier-Stokes equations in three dimensional periodic thin domains of type $\mathbb{T}\times(\delta \mathbb{T})^2$, where $0 < \delta < 1$ is a small parameter. We apply Littlewood-Paley decomposition theory to the periodic thin domain $\mathbb{T}\times(\delta \mathbb{T})^2$ and show some Bernstein-type inequalities with specific dependence on parameter δ. This allows us to establish various embedding inequalities in Besov spaces in $\mathbb{T}\times(\delta \mathbb{T})^2$ as well as the interpolation inequalities of Gagliardo- Nirenberg type. Together with D. Hoff’s idea, we prove that the compressible Navier-Stokes equations in $\mathbb{T}\times(\delta \mathbb{T})^2$ admits a unique global regular solution when the thickness $\delta$ of the domain is sufficiently small, even if the initial data $(\rho_{0,\delta}, u_{0,\delta})$ is large in the sense that
$||(\nabla^2 \rho_{0,\delta}, \nabla^2 u_{0,\delta})||_{L^2(\mathbb{T}\times(\delta \mathbb{T})^2)} \sim\delta^{-\kappa}$
with $\kappa \in (0,1/3)$. (Lecture Hall - Academic island（定山院士岛报告厅)) |

17:30 - 17:45 | Group Photo at lakeside (Academic island（定山院士岛)) |

18:00 - 20:30 | Dinner (Dining Hall - Yuxianghu Hotel（御湘湖酒店餐厅)) |

Thursday, August 10 | |
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07:00 - 09:00 | Breakfast (Dining Hall - Yuxianghu Hotel（御湘湖酒店餐厅)) |

09:30 - 10:00 |
Stefano Bianchini: On spiral strategies for blocking fire ↓ The fire blocking problem can be stated as follows: the fire is
propagating in all directions with speed $1$ starting from a nonempty
open set $\Omega_0 \subset \mathbb{R}^2$, and a barrier $\Gamma(t)$ is
constructed with speed $\sigma > 0$, i.e. $L(\Gamma(t)) \leq \sigma t$.
The question is if there a strategy to build a barrier $\Gamma(t)$ which
encloses the fire in finite time. It is known that for $\sigma \leq 1$ this is impossible, while for
$\sigma > 2$ there is an admissible strategy. An open conjecture is that
for $\sigma \leq 2$ the fire cannot be blocked. A somewhat simplified conjecture is that if the blocking barrier is
spiral-like, then one can block the fire only if the speed of building
the barrier is $\sigma > 2.61...$. The positive part of this conjecture
is known. In this talk I will present how we partially prove the negative part, i.e. if
$\sigma \leq 2$ then every admissible spiral is exponentially
diverging, and in particular the fire cannot be blocked. (Zoom (Online)) |

10:00 - 10:30 |
Ya-Guang Wang: On the controllability of the incompressible MHD systems ↓ In this talk, we shall introduce our recent study on the controllability of the initial boundary value problem for the incompressible magnetohydrodynamic systems. For the two-dimensional ideal incompressible MHD system, we obtained the global exact controllability by using the return method, and for the two- and three-dimensional viscous MHD systems with coupled Navier slip boundary condition, we deduced the global approximate controllability. This is a joint work with Manuel Rissel. (Lecture Hall - Academic island（定山院士岛报告厅)) |

10:30 - 10:45 | Coffee Break (Lecture Hall - Academic island（定山院士岛报告厅)) |

10:45 - 11:15 |
Hermano Frid: On short wave-long wave interactions in the relativistic context: application to the relativistic Euler equations ↓ On short wave-long wave interactions in the relativistic context: application to the relativistic Euler equations
\\Abstract: This talk is about recent works on short wave-long wave interactions in the relativistic context. In particular, we introduce a model of relativistic short wave-long wave interaction where the short waves are described by the massless 1 + 3-dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the 1 + 3- dimensional relativistic Euler equations. The interaction coupling terms are modeled by a potential proportional to the relativistic specific volume in the Dirac equation and an external force proportional to the square modulus of the Dirac wave function in the relativistic Euler equation. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this paper is a clear formulation of the relativistic Lagrangian transformation. This is done by means of the introduction of natural auxiliary dependent variables, render- ing the discussion totally similar to the non-relativistic case. As far as the authors know the definition of the Lagrangian transformation given in this paper is new. Finally, we establish the short-time existence and uniqueness of a smooth solution of the Cauchy problem for the regularized model. This follows through the symmetrization of the relativistic Euler equation introduced by Makino and Ukai (1995) and requires a slight extension of a well known theorem of T. Kato (1975) on quasi-linear symmetric hyperbolic systems. This is a joint work with JOÃO PAULO DIAS. (Zoom (Online)) |

11:15 - 11:45 |
Myoungjean Bae: The steady Euler-Poisson system and accelerating flows with $C^1-$ transonic transitions ↓ In this talk, I present the recent result[2] on the study of accelerating flows with $C^1-$ transonic transitions given as a classical solution to the steady Euler-Poisson system. The key is to establish the well-posedness of a boundary value problem for a linear second order system that consists of an elliptic-hyperbolic mixed type equation with a degeneracy occurring on an interface of codimension 1, and an elliptic equation weakly coupled together. Most importantly, the solutions constructed in this work are classical solution to Euler-Poisson system, thus their sonic interfaces are not weak discontinuities in the sense that all the flow variables are $C^1$ across the interfaces. This shows a sharp contrast with a sonic arc appearing in self-similar flows governed by the unsteady Euler system (see [1]). This feature combined with the result from [1] indicates that there are at least two types of sonic interfaces: a weak discontinuity type [1], and a regular type[2].
This talk is based on a collaboration with B. Duan(Jilin Univ.) and C. Xie (Shanghai Jiao Tong Univ.). (Zoom (Online)) |

12:00 - 13:30 | Lunch (Dining Hall - Academic island（定山院士岛餐厅)) |

13:30 - 20:30 | Free afternoon （IASM will offer a city tour including dinner） (Academic island（定山院士岛)) |

Friday, August 11 | |
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07:00 - 09:00 | Breakfast (Dining Hall - Yuxianghu Hotel（御湘湖酒店餐厅)) |

09:30 - 10:30 |
Constantine Dafermos: Hyperbolic Systems of Balance Laws with Stiff Source ↓ The method of redistribution of damping will be employed for solv- ing the Cauchy problem, in the BV setting, for hyperbolic systems of balance laws with partially dissipative source that becomes stiff as the relaxation time shrinks to zero. One may then pass to the zero relaxation limit. (Zoom (Online)) |

10:30 - 10:45 | Coffee Break (Lecture Hall - Academic island（定山院士岛报告厅)) |

10:45 - 11:15 |
Xinwei Yu: On the very weak solutions of the 3D MHD equations ↓ On the very weak solutions of the 3D MHD equations
\\Abstract: For a typical PDE in fluid mechanics there are many
different ways to define weak solutions, from the "very weak
solutions" that only makes the equation meaningful in the sense of
distributions, to solutions with more regularity such as the so-called
Leray-Hopf solutions. In recent years a new methodology has been
developed for the 3D Navier-Stokes equations connecting the very weak
solutions to the Leray-Hopf solutions. In this talk we present similar
results for the 3D Magneto-hydrodynamical equations through
application of this methodology. This is joint work with Mr. Mark
Pineau. (Zoom (Online)) |

11:15 - 11:45 |
Shaohua Chen: Self-similar Blow-up Solutions of the Nonlinear Schrodinger Equation with Moving Mesh methods ↓ This paper deals with the initial-value problem for the radially symmetric nonlinear Schr\"{o}dinger equation with cubic nonlinearity (NLS) in $d = 2$ and $3$ space dimensions. A very simple, robust and efficient moving mesh method is proposed for numerically solving the radially symmetric nonlinear Schr\"{o}dinger equation. The first numerical simulation is to reproduce the stable self-similar blowup solution for $d=3$. The computed data is used to compare it with the exact blowup solution. The two solutions are overlapped when the amplitude of the solution is less than $100,000$. Next, a typical initial function is used to simulate the blowup solution for $d=3$ and compared it with the corresponding exact blowup solution. The graphs of the two solutions are almost overlapped when the amplitude of the solution reaches $10^60$ and the adjacent mesh points near $0$ are as small as $10^{−61}$. The two solution curves are smooth and showing slow oscillations both in $r$ and $t$ directions. (Lecture Hall - Academic island（定山院士岛报告厅)) |

11:45 - 13:30 | Lunch (Dining Hall - Academic island（定山院士岛餐厅)) |