We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in$H$^{1}close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].

Based on the fully compressible Navier-Stokes equations, the linear stability of thermal convection in rapidly rotating spherical shells of various radius ratios eta is studied for a wide range of Taylor number Ta, Prandtl number Pr and the number of density scale height N-rho. Besides the classical inertial mode and columnar mode, which are widely studied by the Boussinesq approximation and anelastic approximation, the quasi-geostrophic compressible mode is also identified in a wide range of N-rho and Pr for all eta considered, and this mode mainly occurs in the convection with relatively small Pr and large N.. The instability processes are classified into five categories. In general, for the specified wavenumber m, the parameter space (Pr, N-rho) of the fifth category, in which the base state loses stability via the quasi-geostrophic compressible mode and remains unstable, shrinks as eta increases. The asymptotic scaling behaviours of the critical Rayleigh numbers Ra-c and corresponding wavenumbers m(c) to Ta are found at different eta for the same instability mode. As eta increases, the flow stability is strengthened. Furthermore, the linearized perturbation equations and Reynolds-Orr equation are employed to quantitatively analyse the mechanical mechanisms and flow instability mechanisms of different modes. In the quasi-geostrophic compressible mode, the time-derivative term of disturbance density in the continuity equation and the diffusion term of disturbance temperature in the energy equation are found to be critical, while in the columnar and inertial modes, they can generally be ignored. Because the time-derivative term of the disturbance density in the continuity equation cannot be ignored, the anelastic approximation fails to capture the instability mode in the small-Pr and large-N-rho system, where convection onset is dominated by the quasi-geostrophic compressible mode. However, all the modes are primarily governed by the balance between the Coriolis force and the pressure gradient, based on the momentum equation. Physically, the most important difference between the quasi-geostrophic compressible mode and the columnar mode is the role played by the disturbance pressure. The disturbance pressure performs negative work for the former mode, which appears to stabilize the flow, while it destabilizes the flow for the latter mode. As eta increases, in the former mode the relative work performed by the disturbance pressure increases and in the latter mode decreases.

The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if ||z|−1|≥τ for arbitrarily small τ>0, the circular law is valid around z up to scale $N^{-1/2+ \e}$ for any $\e > 0$ under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

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