Global H1 solvability of the 3D logarithmic Schrödinger equation

In this paper we investigate the existence of a unique global mild solution in $H^1\left({\mathbb{R}}^3\right)$ of the initial-boundary value problem associated with the logarithmic Schrödinger equation $i{?}_t\psi =?D\Delta \psi +\sigma \log \left(\right|\psi {|}^2)\psi$, with $D>0$ and $\sigma \in \mathbb{R}?\left\{0\right\}$.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

For the vorticity–velocity–pressure form of the Navier–Stokes equations on a bounded plane domain with corners

We show existence and regularity for the boundary value problems of the Navier–Stokes equations with non-standard BCs on a bounded plane domain with non-convex corners. We assign the vorticity value $\omega ={\omega }_0$ and the velocity normal component $u?n=u_0?n$ over the non-convex corner, the dynamic pressure value $p+{\left|u\right|}^2/2=p_0$ over inflow and outflow boundaries, and so on. We construct the corner singularity functions for the Stokes operator with zero vorticity and velocity normal component BCs, subtract its leading singularity from the solution by defining the coefficient of the singularity and show increased regularity for the remainder. The solution is determined by the smoother part and the coefficients of the singularities. It is seen from the singularity that the dynamic pressure has a transition layer that changes the sign (at $\theta =\pi /2$ in the domain). The obtained results can be applied to general polygonal domains and the cavity flows.