Nonlinear asymptotic stability of inhomogeneous steady solutions to boundary problems of Vlasov-Poisson equation

We consider an ensemble of mass collisionless particles, which interact mutually either by an attraction of Newton's law of gravitation or by an electrostatic repulsion of Coulomb's law, under a background downward gravity in a horizontally-periodic 3D half-space, whose inflow distribution at the boundary is prescribed. We investigate a nonlinear asymptotic stability of its generic steady states in the dynamical kinetic PDE theory of the Vlasov-Poisson equations. We construct Lipschitz continuous space-inhomogeneous steady states and establish exponentially fast asymptotic stability of these steady states with respect to small perturbation in a weighted Sobolev topology. In this proof, we crucially use the Lipschitz continuity in the velocity of the steady states. Moreover, we establish well-posedness and regularity estimates for both steady and dynamic problems.Comment: reorganize the main theorems and simplify the proofs, 38 page

The viscous surface-internal wave problem: global well-posedness and decay

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh-Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.Comment: 70 pages; v2: typos and minor errors correcte

Three essays in macroeconomics

This thesis consists of three independent papers on macroeconomics. Chapter one provides the introduction of the papers. \\ Chapter two investigates the response of firm-level markup to aggregate shocks and the role of intangible capital in markup determination. A markup is a key object in understanding a firm’s pricing behavior. Using a panel version of local projection, I document noble evidence that firm-level markup is countercyclical to aggregate productivity and monetary policy shock. To explain the empirical evidence, I combine Hopenhayn (1992) firm dynamics model with habit accumulation at a good level (Ravn et al., 2006). After calibrating the model to US data, I find that the model can quantitatively match the empirical evidence. Furthermore, the model endogenously matches the age-dependent growth rate and the exit rate, which the profession had difficulty with. \\ Chapter three asks, “how a firm responds to tax shock in the short run?”. Differently from the existing literature, I exploit narratively identified shock and study the firm-level response over the business cycle using local projection instrumental variable approach. I find that intangible and tangible investment and labor use increase, while leverage goes down. Firm revenue productivity and markup increase, but firm churn is stable in the short run. Extrapolating the estimates, I project that the effect of the 2017 tax cut is significant. \\ Chapter four studies the effect of forward guidance under an incomplete market. In standard New Keynesian models, there is a peculiar property called “forward guidance puzzle”: if a central bank promises to cut its policy rate from a farther future, the effect of promise strengthens. There exists debate that the introduction of an incomplete market can solve the puzzle, or it additionally requires procyclical income risk. Building on Ravn and Sterk (2020), my paper analytically proves that income risk cyclicality matters

Boltzmann Equation with a Large Potential in a Periodic Box

The stability of the Maxwellian of the Boltzmann equation with a large amplitude external potential $\Phi$ has been an important open problem. In this paper, we resolve this problem with a large $C3-$potential in a periodic box $\mathbb{T}^d$, $d \geq 3$. We use [1] in $L^p-L^{\infty}$ framework to establish the well-posedness and the $L^{\infty}-$stability of the Maxwellian $\mu_E(x,v)=\exp\{-\frac{|v|^2}{2}-\Phi(x)\}$