Coarse Graining Nonisothermal Microswimmer Suspensions

We investigate coarse-grained models of suspended self-thermophoretic microswimmers. Upon heating, the Janus spheres, with hemispheres made of different materials, induce a heterogeneous local solvent temperature that causes the self-phoretic particle propulsion. Starting from molecular dynamics simulations that schematically resolve the molecular composition of the solvent and the microswimmer, we verify the coarse-grained description of the fluid in terms of a local molecular temperature field, and its role for the particle’s thermophoretic self-propulsion and hot Brownian motion. The latter is governed by effective nonequilibrium temperatures, which are measured from simulations by confining the particle position and orientation. They are theoretically shown to remain relevant for any further spatial coarse-graining towards a hydrodynamic description of the entire suspension as a homogeneous complex fluid.

Local regularity for concave homogeneous complex degenerate elliptic equations dominating the Monge–Ampère equation

In this paper, we establish a local regularity result for $$W^{2,p}_{{\mathrm {loc}}}$$ W loc 2 , p solutions to complex degenerate nonlinear elliptic equations $$F(D^2_{\mathbb {C}}u)=f$$ F ( D C 2 u ) = f when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called k-Monge–Ampère equation.

Effective and efficient approaches for calculating seismic ray velocity and attenuation in viscoelastic anisotropic media

In viscoelastic anisotropic media, the elastic moduli, slowness vector, phase, and ray velocity are all complex-valued quantities in the frequency domain. Solving the complex eikonal equation becomes computationally complex and time-consuming. We have developed two approximate methods to effectively calculate the ray velocity vector, attenuation, and quality factor in viscoelastic transversely isotropic media with a vertical symmetry axis (VTI) and in orthorhombic (ORT) anisotropy. The first method is based on the perturbation theory (PER) under the assumption of a homogeneous complex ray vector, which is obtained by applying the elastic background and viscoelastic perturbations to the real and imaginary components of the modulus tensor, respectively. The perturbations of the slowness vectors of the three wave modes (qP, qSV, and qSH) are determined through the vanishing Hamiltonian function. The second method is derived by applying a real slowness direction (RSD) to the inhomogeneous complex slowness vector and then approximately calculating the complex ray velocity vector with the condition of the homogeneous complex vector. The numerical results verify that the two approaches can produce accurate ray velocity vector, attenuation, and quality factors of the qP-wave in viscoelastic VTI and ORT media. The RSD method can yield high accuracies of ray velocity for the qSV- and qSH-wave in viscoelastic VTI models even at triplication of the qSV wavefronts, as well as qS1 and qS2 in a weak ORT medium ([Formula: see text] > 20), except for near the cusp of the qS1 wavefronts (errors approximately 6%) where the PER has more than 10% error.

Pluricomplex Green's functions and Fano manifolds

We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau. Comment: EpiGA Volume 3 (2019), Article Nr. 9

The Cauchy problem for the homogeneous Monge–Ampère equation, III. Lifespan

We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge–Ampère equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the

Quaternionic-like manifolds and homogeneous twistor spaces

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the ‘quaternionic-like manifolds’. These contain, as particular subclasses, the CR quaternionic and the ρ -quaternionic manifolds. Moreover, the notion of ‘heaven space’ finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a ρ -quaternionic manifold. We, also, give a natural construction of homogeneous complex manifolds endowed with embedded spheres, thus, emphasizing the abundance of the quaternionic-like manifolds.