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Abstract We study the Kardar–Parisi–Zhang (KPZ) equation on the half-line x ⩾ 0 with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter u of the dynamics, and the drift − v of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time t , it is natural to rescale parameters as ( u , v ) = t −1/3 ( a , b ) to study the critical region. In the special case a + b = 0, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase ( a < 0) but not in the unbound phase. For instance, starting from the flat or droplet initial condition, the height field near the boundary converges to the stationary process with a > 0 and b = 0, which is not Brownian. For a + b ⩾ 0, we determine exactly the large time distribution F a , b stat of the height function h (0, t ). As an application, we obtain the exact covariance of the height field in a half-line at two times 1 ≪ t 1 ≪ t 2 starting from stationary initial condition, as well as estimates, when starting from droplet initial condition, in the limit t 1 / t 2 → 1.
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This content will become publicly available on June 1, 2026
On the propulsion of a rigid body in a viscous liquid under the action of a time-periodic force
Arigid body B moves in an otherwise quiescent viscous liquid filling the whole space outside B, under the action of a time-periodic force f of period T applied to a given point of B and of fixed direction. We assume that the average of f over an interval of length T does not vanish, and that the amplitude, δ,off is sufficiently small. Our goal is to investigate when B executes a nonzero net motion; that is, B is able to cover any prescribed distance in a finite time. We show that, at the order δ, this happens if and only if f and B satisfy a certain condition. We also show that this is always the case if B is prevented from spinning. Finally, we provide explicit examples where the condition above is satisfied or not. All our analysis is performed in a general class of weak solutions to the coupled system body-liquid problem.
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- Award ID(s):
- 2307811
- PAR ID:
- 10586668
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Research in the Mathematical Sciences
- Volume:
- 12
- Issue:
- 2
- ISSN:
- 2522-0144
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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