765 research outputs found
New type of solutions for the critical Lane-Emden system
In this paper, we consider the critical Lane-Emden system \begin{align*}
\begin{cases} -\Delta u=K_1(y)v^p,\quad y\in \mathbb{R}^N,&\\ -\Delta
v=K_2(y)u^q,\quad y\in \mathbb{R}^N,&\\ u,v>0, \end{cases} \end{align*} where
, with
, and are positive
radial potentials. Under suitable conditions on and , we
construct a new family of solutions to this system, which are centred at points
lying on the top and the bottom circles of a cylinder
Partial regularity of solutions to the 3D chemotaxis-Navier-Stokes equations at the first blow-up time
In this note, we investigate partial regularity of weak solutions of the
three dimensional chemotaxis-Navier-Stokes equations, and obtain the
-dimensional Hausdorff measure of the possible singular set is
vanishing at the first blow-up time. The new ingredients are to establish
certain type of local energy inequality and deal with the non-scaling invariant
quantity of , which seems to be the first description for the singular
set of weak solutions of the chemotaxis-fluid model, which is motivated by
Caffarelli-Kohn-Nirenberg's partial regularity theory \cite{CKN}
Capacity sharing, product differentiation and welfare
This article constructs a duopoly market with product differentiation
and analyses profits, consumer surplus and social welfare
under three conditions: (a) two enterprises have sufficient capacity;
(b) one enterprise has insufficient capacity, and another
enterprise has excess capacity that is not shared; and (c) one
enterprise has insufficient capacity, and another enterprise has
excess capacity and engages in capacity sharing. Through comparison,
the implementation conditions for and effects of capacity
sharing and the role of product differentiation are revealed. The
results show that capacity sharing helps increase producer surplus
and social welfare. Capacity constraints reduce social welfare but
can be solved by capacity sharing. Capacity sharing can only be
realised when both enterprises are profitable, and the charge for
capacity sharing should not be too high or too low. Product differentiation
has impacts on output, profit, consumer surplus and
social welfare, and these impacts are restricted by the existence
of capacity constraints and capacity sharing
Global existence of suitable weak solutions to the 3D chemotaxis-Navier-Stokes equations
In 2004, Dombrowski et al. showed that suspensions of aerobic bacteria often
develop flows from the interplay of chemotaxis and buoyancy, which is described
as the chemotaxis-Navier-Stokes model, and they observed self-concentration
occurs as a turbulence by exhibiting transient, reconstituting, high-speed
jets, which entrains nearby fluid to produce paired, oppositely signed
vortices. In order to investigate the properties of these vortices (singular
points), one approach is to follow the partial regularity theory of
Caffarelli-Kohn-Nirenberg to study the singularity properties of suitable weak
solutions. In this paper, we established the existence of suitable weak
solution for the three dimensional chemotaxis-Navier-Stokes equations, where
the main difficulty is to establish appropriate local energy inequalities
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