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We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set$K$defined on the boundary of the unit ball.

We prove the existence of reaction-diffusion traveling fronts in mean zero space-time periodic shear flows for nonnegative reactions including the classical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPP nonlinearity, the minimal front speed is characterized by a variational principle involving the principal eigenvalue of a space-time periodic parabolic operator. Analysis of the variational principle shows that adding a mean-zero space time periodic shear flow to an existing mean zero space-periodic shear flow leads to speed enhancement. Computation of KPP minimal speeds is performed based on the variational principle and a spectrally accurate discretization of the principal eigenvalue problem. It shows that the enhancement is monotone decreasing in temporal shear frequency, and that the total enhancement from pure reaction-diffusion obeys quadratic and linear laws at small and large shear amplitudes.

In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of $z$-weak solution is established for any such initial datum that itself and its vertical derivative belong to $L^2$. This not only extends the results in \cite{Cao5} from the spatially periodic case to general cylindrical domains but also weakens the regularity assumptions on the initial data which are required to be $H^2$ there.

Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation limit which can be solved numerically much more efficiently. For the Jin–Xin relaxation system in such a two-scale setting, we establish its wellposedness and singular limit as the (smaller) relaxation time goes to zero. The limit is a multiscale coupling problem which couples the original Jin–Xin system on the domain when the relaxation time is O(1) with its relaxation limit in the other domain through interface conditions which can be derived by matched interface layer analysis.As a result, we also establish the well-posedness and regularity (such as boundedness in sup norm with bounded total variation and L 1-contraction) of the coupling problem, thus providing a rigorous mathematical foundation, in the general nonlinear setting, to the multiscale domain decomposition method for this two-scale problem originally proposed in Jin et al. in Math. Comp. 82, 749–779, 2013.