The nonlinear heat equation involving highly singular initial values and new blowup and life span results

In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = \Delta u +a |u|^\alpha u, \; t\in(0,T),\; x=(x_1,\,\cdots,\, x_N)\in {\mathbb R}^N,\; a = \pm 1,\; \alpha>0;$ with initial value $u(0)\in L^1_{\rm{loc}}\left({\mathbb R}^N\setminus\{0\}\right)$, anti-symmetric with respect to $x_1,\; x_2,\; \cdots,\; x_m$ and $|u(0)|\leq C(-1)^m\partial_{1}\partial_{2}\cdot \cdot \cdot \partial_{m}(|x|^{-\gamma})$ for $x_1>0,\; \cdots,\; x_m>0,$ where $C>0$ is a constant, $m\in \{1,\; 2,\; \cdots,\; N\},$ $0<\gamma<N$ and $0<\alpha<2/(\gamma+m).$ This gives a local existence result with highly singular initial values. As an application, for $a=1,$ we establish new blowup criteria for $0<\alpha\leq 2/(\gamma+m)$, including the case $m=0.$ Moreover, if $(N-4)\alpha<2,$ we prove the existence of initial values $u_0 = \lambda f,$ for which the resulting solution blows up in finite time $T_{\max}(\lambda f),$ if $\lambda>0$ is sufficiently small. We also construct blowing up solutions with initial data $\lambda_n f$ such that $\lambda_n^{[({1\over \alpha}-{\gamma+m\over 2})^{-1}]}T_{\max}(\lambda_n f)$ has different finite limits along different sequences $\lambda_n\to 0$. Our result extends the known "small lambda" blow up results for new values of $\alpha$ and a new class of initial data.Comment: Submitte

Large time behavior of solutions to the nonlinear heat equation with absorption with highly singular antisymmetric initial values

In this paper we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption, $u_t - \Delta u + |u|^\alpha u =0$, where $u=u(t,x)\in {\mathbb R},$ $(t,x)\in (0,\infty)\times{\mathbb R}^N$ and $\alpha>0$. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables $x_1,\; x_2,\; \cdots,\; x_m$ for some $m\in \{1,2, \cdots, N\}$, such as $u_0 = (-1)^m\partial_1\partial_2 \cdots \partial_m|\cdot|^{-\gamma} \in {{\mathcal S'}({\mathbb R}^N)}$, $0 < \gamma < N$. In fact, we show global well-posedness for initial data bounded in an appropriate sense by $u_0$, for any $\alpha>0$. Our approach is to study well-posedness and large time behavior on sectorial domains of the form $\Omega_m = \{x \in {{\mathbb R}^N} : x_1, \cdots, x_m > 0\}$, and then to extend the results by reflection to solutions on ${{\mathbb R}^N}$ which are antisymmetric. We show that the large time behavior depends on the relationship between $\alpha$ and $2/(\gamma+m)$, and we consider all three cases, $\alpha$ equal to, greater than, and less than $2/(\gamma+m)$. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions

Electromagnetic Simulation and Design of a Novel Waveguide RF Wien Filter for Electric Dipole Moment Measurements of Protons and Deuterons

The conventional Wien filter is a device with orthogonal static magnetic and electric fields, often used for velocity separation of charged particles. Here we describe the electromagnetic design calculations for a novel waveguide RF Wien filter that will be employed to solely manipulate the spins of protons or deuterons at frequencies of about 0.1 to 2 MHz at the COoler SYnchrotron COSY at J\"ulich. The device will be used in a future experiment that aims at measuring the proton and deuteron electric dipole moments, which are expected to be very small. Their determination, however, would have a huge impact on our understanding of the universe.Comment: 10 pages, 10 figures, 4 table

Diversité des méthodes utilisées par les laboratoires français pour la surveillance des infections à cytomégalovirus humain

Monitoring cytomegalovirus circulating viral load is an important parameter of the follow-up in immunocompromised patients. It can be measured either by DNAemia or by pp65 antigenemia. The French national reference center for cytomegaloviruses organized an investigation of practice in 37 teacher hospital virology laboratories to assess the situation in France in 2010. Methods A questionnaire was sent to collect following information: method used in routine for monitoring of circulating viral load of CMV, assay used, sample matrix and extraction method. Results Thirty-six over thirty-seven laboratories filled the questionnaire. Among these, 67% used the quantitative PCR in routine, 11% antigenemia and 22% antigenemia or quantitative PCR; 87% of the laboratories use whole blood for quantitative PCR, whereas 10% and 3% use plasma and leukocytes respectively. Among the laboratories using DNAemia, 100% used real-time PCR assays, 91% use an automated extraction and 9% a manual extraction. Conclusion Thus in France, measurement of DNAemia by real-time PCR is a tool, which gradually replaces the antigenemia for the monitoring of cytomegalovirus infection among immunocompromised patients. The very great diversity of the methods used justifies the installation of a national quality control on total blood, matrix used by 87% of the laboratories

Spin tune mapping as a novel tool to probe the spin dynamics in storage rings

Precision experiments, such as the search for electric dipole moments of charged particles using storage rings, demand for an understanding of the spin dynamics with unprecedented accuracy. The ultimate aim is to measure the electric dipole moments with a sensitivity up to 15 orders in magnitude better than the magnetic dipole moment of the stored particles. This formidable task requires an understanding of the background to the signal of the electric dipole from rotations of the spins in the spurious magnetic fields of a storage ring. One of the observables, especially sensitive to the imperfection magnetic fields in the ring is the angular orientation of stable spin axis. Up to now, the stable spin axis has never been determined experimentally, and in addition, the JEDI collaboration for the first time succeeded to quantify the background signals that stem from false rotations of the magnetic dipole moments in the horizontal and longitudinal imperfection magnetic fields of the storage ring. To this end, we developed a new method based on the spin tune response of a machine to artificially applied longitudinal magnetic fields. This novel technique, called \textit{spin tune mapping}, emerges as a very powerful tool to probe the spin dynamics in storage rings. The technique was experimentally tested in 2014 at the cooler synchrotron COSY, and for the first time, the angular orientation of the stable spin axis at two different locations in the ring has been determined to an unprecedented accuracy of better than $2.8\mu$rad.Comment: 32 pages, 15 figures, 7 table

Phase Measurement for Driven Spin Oscillations in a Storage Ring

This paper reports the first simultaneous measurement of the horizontal and vertical components of the polarization vector in a storage ring under the influence of a radio frequency (rf) solenoid. The experiments were performed at the Cooler Synchrotron COSY in J\"ulich using a vector polarized, bunched $0.97\,\textrm{GeV/c}$ deuteron beam. Using the new spin feedback system, we set the initial phase difference between the solenoid field and the precession of the polarization vector to a predefined value. The feedback system was then switched off, allowing the phase difference to change over time, and the solenoid was switched on to rotate the polarization vector. We observed an oscillation of the vertical polarization component and the phase difference. The oscillations can be described using an analytical model. The results of this experiment also apply to other rf devices with horizontal magnetic fields, such as Wien filters. The precise manipulation of particle spins in storage rings is a prerequisite for measuring the electric dipole moment (EDM) of charged particles