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As studied in ​many examples, higher order correction added to the Newtonian potential often provides more realistic and accurate quasi-homogeneous models in astrophysics. Important examples include the Schwarzschild and the Manev potentials. The quasi-homogeneous N-body problem aims to study the interaction between N point particles under a prescribed potential. The classical (Newtonian) Hill’s lunar problem aims to improve the solution accuracy of the lunar motion obtained by solving the two-body (Earth-Moon) system. Hill's lunar equation under the Newtonian or homogeneous potentials has been derived from the Hamiltonian of the three-body problem in a uniform rotating coordinate system with angular speed $\omega$, by using symplectic scaling and heuristic arguments on various physical quantities. In this talk, we first introduce a new variational method characterizing relative equilibria with minimal energy. This enables us to classify the dynamic in terms of global existence and singularity for all possible ranges of the parameters. Then we derive Hill’s lunar problem with quasi-homogenous potential, and finally, we implement the same ideas to demonstrate the existence of ``black hole effect" for a certain range of the parameters: below and at some energy threshold, invariant sets (in the phase space) with non-zero Lebesgue measure that either contain global solutions or solutions with singularity are constructed.

The primitive equation is an important model for large scale fluid model including oceans and atmosphere. While solutions to the viscous model enjoy global regularity, inviscid solutions may develop singularities in finite time. In this talk, I will review the methods to show blowup, and share more recent progress on the qualitative properties of the singularity formation. Most notably, I will provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations. This is a joint work with C. Collot and Q. Lin.

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Abstract: Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). While it is well-known that the PEs with full viscosity are globally well-posed in Sobolev spaces, the study of well-posedness of the inviscid PEs is more challenging. In this talk, I will first review some results on the deterministic inviscid PEs. Then I will discuss the differences and challenges in the stochastic case and show the existence of local martingale solutions and pathwise uniqueness. Finally I will discuss some interesting open problems. This is a joint work with Ruimeng Hu.

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Abstract: We describe several aspects of an analytic/geometric framework for the three-dimensional Navier-Stokes regularity problem, which is directly inspired by the morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of three-dimensional turbulence. Among these, we present our proof that the scaling gap in the 3D Navier-Stokes regularity problem can be reduced by an algebraic factor within an appropriate functional setting incorporating the intermittency of the spatial regions of high vorticity.

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Abstract: For the 3D incompressible Euler equation, a distinguished solution is the flow leading to rigid rotation about an axis u(x,y,z)=(-y,x,0). We show that this solution is stable under small axisymmetric perturbations, and that the perturbations exist globally and scatter linearly back to the equilibrium through a dispersive effect induced by the linearized Euler-Coriolis equations. This is joint work with Y. Guo and K. Widmayer.

Abstract: The 2d incompressible Euler equation is globally well posed for smooth initial data. However the long term dynamics of general solutions is difficult to understand due to the lack of global relaxation mechanisms. Numerical simulations and physical experiments show that vortices (steady solutions with radial vorticity functions) play an important role in the global dynamics, through a process called vortex symmetrization of small perturbations. In this talk, I will discuss some recent progress on this problem, including a full nonlinear symmetrization result near a special point vortex and precise linearized symmetrization result near general vortices. Difficulties of full nonlinear vortex symmetrization around general vortices will also be discussed.​

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Abstract: We consider the classical problem of water waves on the surface of an ideal fluid in 2D and in particular, we study the growth of perturbations to nearly limiting Stokes waves at infinite depth. The stability of Stokes waves is determined by the linearization of the equations of motion of fluid around a Stokes wave.

The eigenvalue problem is solved numerically via shift and invert method.

We identify each positive eigenvalue with an unstable eigenmode in the equations of motion, and find that real positive eigenvalues of the linearized problem converge to a selfsimilar curve as a function of steepness. The power law is suggested for unstable eigenmodes in the immediate vicinity of the limiting Stokes wave. We will discuss the topic of Stokes waves in the finite depth fluid.

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Abstract: Using the idea of ground states in nonlinear dispersive equations (e.g. Klein-Gordon and Schr\"odinger equations), we can characterize the solutions of the N-body problem with strong force under some energy constraints.In this talk, I will explore this method to a restricted 3-body problem (Hill’s lunar problem), and talk about the dynamics of the solutions below, at, and (slightly) above the ground state energy threshold. This is a joint work with Slim Ibrahim.​

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Abstract: We use a generic two-stage population model to derive an adaptive dynamical system for the evolution of reproduction age, and study how this evolution is driven by the harvest of adults. We consider the tradeoffs between maturation rate and fecundity, juvenile mortality, and adult mortality. We analyze the benefit and cost of faster maturation under each tradeoff that drives the evolution. We found that harvesting adults affects the evolution of maturation by affecting the benefit. For the tradeoff between maturation and juvenile mortality, harvesting adults does not affect the benefit, and thus does not affect optimal maturation strategy. For the other two tradeoffs, harvesting adults affects the benefit through the equilibrium adult/juvenile ratio, which is determined by the density dependence of juveniles. Harvesting adults causes a slower maturation only if it significantly reduces this ratio, which can only happen with very strong adult protection to juveniles. Otherwise, harvesting adults always causes a faster maturation.