Analytical smoothing effect of solution for the boussinesq equations

In this paper, we study the analytical smoothing effect of Cauchy problem for the incompressible Boussinesq equations. Precisely, we use the Fourier method to prove that the Sobolev H 1-solution to the incompressible Boussinesq equations in periodic domain is analytic for any positive time. So the incompressible Boussinesq equation admet exactly same smoothing effect properties of incompressible Navier-Stokes equations

Probing the properties of the pulsar wind via studying the dispersive effects in the pulses from the pulsar companion in a double neutron-star binary system

The velocity and density distribution of $e^\pm$ in the pulsar wind are crucial distinction among magnetosphere models, and contains key parameters determining the high energy emission of pulsar binaries. In this work, a direct method is proposed, which might probe the properties of the wind from one pulsar in a double-pulsar binary. When the radio signals from the first-formed pulsar travel through the relativistic $e^\pm$ flow in the pulsar wind from the younger companion, the components of different radio frequencies will be dispersed. It will introduce an additional frequency-dependent time-of-arrival delay of pulses, which is function of the orbital phase. In this paper, we formulate the above-mentioned dispersive delay with the properties of the pulsar wind. As examples, we apply the formula to the double pulsar system PSR J0737-3039A/B and the pulsar-neutron star binary PSR B1913+16. For PSR J0737-3039A/B, the time delay in 300\,MHz is $\lesssim10\mu$s near the superior-conjunction, under the optimal pulsar wind parameters, which is $\sim$ half of the current timing accuracy. For PSR B1913+16, with the assumption that the neutron star companion has a typical spin down luminosity of $10^{33}$\,ergs/s, the time delay is as large as $10\sim20\mu$s in 300\,MHz. The best timing precision of this pulsar is $\sim5\mu$s in 1400\,MHz. Therefore, it is possible that we can find this signal in archival data. Otherwise, we can set an upper-limit on the spin down luminosity. Similar analysis can be apply to other eleven known pulsar-neutron star binariesComment: 6 pages, 6 figures, accepted for publication in MNRA