Related papers: The Case Against Smooth Null Infinity II: A Logari…

We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^3$ for a one-parameter family of explicit scale-invariant axi-symmetric initial data, which is smooth away from the origin and invariant under the…

We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce…

We construct solutions with prescribed radiation fields for wave equations with polynomially decaying sources close to the lightcone. In this setting, which is motivated by semi-linear wave equations satisfying the weak null condition,…

For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem.…

An exact solution of the Einstein field equations is found under the assumption of spherically symmetry and the existence of one-parameter group of homothetic motions. This solution has a singularity at $r = 0$, and has non-vanishing…

Application of the "hairy ball theorem" to the analysis of the surface instabilities inherent for liquid/vapor interfaces is reported. When a continuous tangential velocity field exists on the surface of the liquid sample which is…

We consider spherically symmetric space-times in GR under the unconventional assumptions that the spherical radius $r$ is either a constant or has a null gradient in the $(t,x)$ subspace orthogonal to the symmetry spheres (i.e., $(\partial…

The asymptotic charges of spin-1 and spin-2 fields are studied near spatial infinity. We evaluate the charges at the critical sets where spatial infinity meets null infinity with the aim of finding the relation between the charges at future…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\Delta_g^{k/2}…

The linear instability study of the viscous Rayleigh-Taylor model in the neighborhood of a laminar smooth increasing density profile $\rho_0(x_3)$ amounts to the study of the following ordinary differential equation of order 4:…

We study the scalar, conformally invariant wave equation on a four-dimensional Minkowski background in spherical symmetry, using a fully pseudospectral numerical scheme. Thereby, our main interest is in a suitable treatment of spatial…

Let $(\mathcal{M}^{3+1},g)$ be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion $\mathscr{E}$ and no future event horizon $\mathcal{H}^{+}$. On such spacetimes, Friedman provided a heuristic argument…

Let $M^\circ$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^\circ$, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on…

Given sets $\Phi_1=\{\phi_{11},...,\phi_{1u(1)}\}, ...,\Phi_{z}=\{\phi_{z1},...,\phi_{zu(z)}\}$ of boolean formulas, a formula $\omega$ follows from the conjunction $\bigwedge\Phi_i= \bigwedge \phi_{ij}$ iff $\neg \omega\wedge…

We consider the nonlinear problem \[(P) \;\; I u=f(x,u) \text{ in $\Omega$,} \;\; u=0 \text{ on $\mathbb{R}^{N}\setminus\Omega$ }\] in an open bounded set $\Omega\subset\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may be…

The rigidity of the Riemannian positive mass theorem for asymptotically hyperbolic manifolds states that the total mass of such a manifold is zero if and only if the manifold is isometric to the hyperbolic space. This leads to study the…

For manifolds with a distinguished asymptotically flat end, we prove a density theorem which produces harmonic asymptotics on the distinguished end, while allowing for points of incompleteness (or negative scalar curvature) away from this…

For the physical vacuum free boundary problem with the sound speed being $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the…

We study the asymptotic behavior, as $\lambda\rightarrow 0^+$, of the state-constraint Hamilton--Jacobi equation $\phi(\lambda) u_\lambda(x) + H(x,Du_\lambda(x)) = 0$ in $(1+r(\lambda))\Omega$ and the corresponding additive eigenvalues, or…

This paper is devoted to the geometric analysis of the incompressible averaged Euler equations on compact Riemannian manifolds with boundary. The equation also coincides with the model for a second-grade non-Newtonian fluid. We study the…