Narrow Results

The boson star filled with two interacting scalar fields is investigated. The scalar fields can be considered as a gauge condensate formed by SU(3) gauge field quantized in a nonperturbative manner. The corresponding solution is regular everywhere, has a finite energy and can be considered as a quantum SU(3) version of the Bartnik–McKinnon particle-like solution.

This paper deals with a parabolic–elliptic chemotaxis-consumption system with tensor-valued sensitivity $S(x,n,c)$ under no-flux boundary conditions for $n$ and Robin-type boundary conditions for $c$. The global existence of bounded classical solutions is established in dimension two under general assumptions on tensor-valued sensitivity $S$. One of the main steps is to show that $\nabla c(\cdot ,t)$ becomes tiny in $L^2(B_r(x)\cap \mathrm{\Omega })$ for every $x\in \overline{\mathrm{\Omega }}$ and $t$ when $r$ is sufficiently small, which seems to be of independent interest. On the other hand, in the case of scalar-valued sensitivity $S=\chi (x,n,c)𝕀$, there exists a bounded classical solution globally in time for two and higher dimensions provided the domain is a ball with radius $R$ and all given data are radial. The result of the radial case covers scalar-valued sensitivity $\chi$ that can be singular at $c=0$.

It is shown that formally regular solutions in 5D Kaluza–Klein gravity have singularities. This phenomenon is connected with the existence of a minimal length in nature. The calculation of the derivative of the G₅₅ metric component leads to the appearance of the Dirac δ-function. In this case, the Ricci scalar becomes singular since there is a square of this derivative.

Elastic nucleon–nucleus scattering in all partial waves is studied using Deng–Fan model of nuclear interaction. Analytical expressions for regular solution, binding energy and normalization constant are expressed in their maximal reduced form. Exploiting the asymptotic behavior of the solution the scattering phase shifts are computed for the neutron–Carbon system and are found to be in order with previous calculations.

We develop linear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (LSSs) that unravel quantum master equations in Lindblad form into quantum trajectories. More precisely, this paper establishes the existence and uniqueness of the smooth strong solution X_t to a LSS with regular initial condition. Moreover, we obtain that the mean value of the square norm of X_t is constant. We also treat the approximation of LSSs by ordinary stochastic differential equations. We apply our results to:

(i) models of quantum measurements of position and momentum; and

(ii) a system formed by fermions.

The regular solutions of the isentropic Euler equations with degenerate linear damping for a perfect gas are studied in this paper. And a critical degenerate linear damping coefficient is found, such that if the degenerate linear damping coefficient is larger than it and the gas lies in a compact domain initially, then the regular solution will blow up in finite time; if the degenerate linear damping coefficient is less than it, then under some hypotheses on the initial data, the regular solution exists globally.

We study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schrödinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation.