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Abstract Taking place naturally in a gas subject to a given wall temperature distribution, the “ghost effect” exhibits a rare kinetic effect beyond the prediction of classical fluid theory and Fourier law in such a classical problem in physics. As the Knudsen number goes to zero, the finite variation of temperature in the bulk is determined by an infinitesimal, ghost‐like velocity field, created by a givenfinitevariation of the tangential wall temperature as predicted by Maxwell's slip boundary condition. Mathematically, such a finite variation leads to the presence of a severe singularity and a Knudsen layer approximation in the fundamental energy estimate. Neither difficulty is within the reach of any existing PDE theory on the steady Boltzmann equation in a general 3D bounded domain. Consequently, in spite of the discovery of such a ghost effect from temperature variation in as early as 1960s, its mathematical validity has been a challenging and intriguing open question, causing confusion and suspicion. We settle this open question in affirmative if the temperature variation is small but finite, by developing a new framework with four major innovations as follows: (1) a key ‐Hodge decomposition and its corresponding local ‐conservation law eliminate the severe bulk singularity, leading to a reduced energy estimate; (2) a surprising gain in via momentum conservation and a dual Stokes solution; (3) the ‐conservation, energy conservation, and a coupled dual Stokes–Poisson solution reduces to an boundary singularity; (4) a crucial construction of ‐cutoff boundary layer eliminates such boundary singularity via new Hardy's and BV estimates. 

In this paper we study the dynamics of an incompressible viscous fluid evolving in an open-top container in two dimensions. The fluid mechanics are dictated by the Navier–Stokes equations. The upper boundary of the fluid is free and evolves within the container. The fluid is acted upon by a uniform gravitational field, and capillary forces are accounted for along the free boundary. The triple-phase interfaces where the fluid, air above the vessel, and solid vessel wall come in contact are called contact points, and the angles formed at the contact point are called contact angles. The model that we consider integrates boundary conditions that allow for full motion of the contact points and angles. Equilibrium configurations consist of quiescent fluid within a domain whose upper boundary is given as the graph of a function minimizing a gravity-capillary energy functional, subject to a fixed mass constraint. The equilibrium contact angles can take on any values between 0 and\pidepending on the choice of capillary parameters. The main thrust of the paper is the development of a scheme of a priori estimates that show that solutions emanating from data sufficiently close to the equilibrium exist globally in time and decay to equilibrium at an exponential rate. 

Abstract Let the viscosity for the 2D steady Navier‐Stokes equations in the region and with no slip boundary conditions at . For , we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in ε are achieved through a fixed‐point scheme:for solving the Navier‐Stokes equations, where are the tangential and normal velocities at , DNS stands for of the vorticity equation for the normal velocityv, and the compatibility ODE for at . 

Abstract In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of selfsimilar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying non-autonomous dynamical system to which the problem reduces after a selfsimilar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the so-called Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in double-null gauge to flatten the selfsimilar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity.