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

A buffet dinner is served daily between 6:00pm and 8:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A brief introduction to IASM with important logistical information, technology instruction, and opportunity for participants to ask questions.

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.

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.

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.

Lunch is served daily between 12:00am and 1:30pm in the Xianghu Lake National Tourist Resort

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.

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.

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.

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.

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.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

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.

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.

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.

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).

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.

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.

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.

Outdoor banquet

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.

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.

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.

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).

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.

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.

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.

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.

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 $\kappa \in (0,1/3)$.

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.

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.

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.

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.).

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.

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.

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.