158 research outputs found
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 has been an important open problem. In this
paper, we resolve this problem with a large potential in a periodic box
, . We use [1] in framework to
establish the well-posedness and the stability of the Maxwellian
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