1,130 research outputs found
On some properties of weak solutions to elliptic equations with divergence-free drifts
We discuss the local properties of weak solutions to the equation . The corresponding theory is well-known in the case , where is the dimension of the space. Our main interest is focused on
the case . In this case the structure assumption turns out to be crucial
Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations
In this paper we consider a -dimensional () parabolic-elliptic
Keller-Segel equation with a logistic forcing and a fractional diffusion of
order . We prove uniform in time boundedness of its solution
in the supercritical range , where is an explicit
constant depending on parameters of our problem. Furthermore, we establish
sufficient conditions for , where
is the only nontrivial homogeneous solution. Finally, we
provide a uniqueness result
Holder continuity for a drift-diffusion equation with pressure
We address the persistence of H\"older continuity for weak solutions of the
linear drift-diffusion equation with nonlocal pressure u_t + b \cdot \grad u
- \lap u = \grad p,\qquad \grad\cdot u =0 on ,
with . The drift velocity is assumed to be at the critical
regularity level, with respect to the natural scaling of the equations. The
proof draws on Campanato's characterization of H\"older spaces, and uses a
maximum-principle-type argument by which we control the growth in time of
certain local averages of . We provide an estimate that does not depend on
any local smallness condition on the vector field , but only on scale
invariant quantities
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