For a fractal Schrodinger operator with a continuous potential and a coupling parameter, we obtain an analog of a semi-classical asymptotic formula for the number of bound states as the parameter tends to infinity. We also study Bohr's formula for Schrodinger operators on blowups of self-similar sets. For a Schrodinger operator defined by a fractal measure and a locally bounded potential that tends to infinity, we derive an analog of Bohr's formula under various assumptions. We demonstrate how these results can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, a family of convolutions of Cantor-type measures, and a family of measures that we call essentially of finite type.

In this paper, we study the inviscid limit of the free surface incompressible Navier-Stokes equations with or without surface tension. By delicate estimates, we prove the weak boundary layer of the velocity of the free surface Navier-Stokes equations and the existence of strong or weak vorticity layer for different conditions. When the limit of the difference between the initial Navier-Stokes vorticity and the initial Euler vorticity is nonzero, or the tangential projection on the free surface of the Euler strain tensor multiplying by normal vector is nonzero, there exists a strong vorticity layer. Otherwise, the vorticity layer is weak. We estimate convergence rates of tangential derivatives and the first order standard normal derivative in energy norms, we show that not only tangential derivatives and standard normal derivative have different convergence rates, but also their convergence rates are different for different Euler boundary data. Moreover, we determine regularity structure of the free surface Navier-Stokes solutions with or without surface tension, surface tension changes regularity structure of the solutions.

This paper proves the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant γ lies in the stability range (4/3, 2). It is shown that for small perturbations of a LaneEmden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is C 1/2 -H¨older continuous across the vacuum boundary provided that γ lies in (4/3, 2). The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane-Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary. In particular, it is shown that every spherical surface moving with the fluid converges to the sphere enclosing the same mass inside the domain of the Lane-Emden solution with a uniform convergence rate and the large time asymptotic states for the vacuum free boundary problem (1.1.2) are determined by the initial mass distribution and the total mass. To overcome the difficulty caused by the degeneracy and singular behavior near the vacuum free boundary and coordinates singularity at the symmetry center, the main ingredients of the analysis consist of combinations of some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and space-time weighted energy estimates. The constructions of these weighted nonlinear functionals and space-time weights depend crucially on the structures of the Lane-Emden solution, the balance of pressure and gravitation, and the dissipation. Finally, the uniform boundedness of the acceleration of the vacuum boundary is also proved.

This paper is concerned with the classical motion of a string in global AdS3. The initially static string stretches between two antipodal points on the boundary circle. Both endpoints are perturbed which creates cusps at a steady rate. The cusps propagate towards the interior where they collide. The behavior of the string depends on the strength of forcing. Three qualitatively different phases can be distinguished: transparent, gray, and black. The transparent region is analogous to a standing wave. In the black phase, there is a horizon on the worldsheet and cusps never reach the other endpoint. The string keeps folding and its length grows linearly over time. In the gray phase, the string still grows linearly. However, cusps do cross to the other side. The transparent and gray regions are separated by a transition point where a logarithmic accumulation of cusps is numerically observed. A simple model reproduces the qualitative behavior of the string in the three phases.

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