Hölder Continuous Solutions of Active Scalar Equations
We consider active scalar equations ∂ t θ + ∇ · ( u θ ) = 0 , where u = T [ θ ] is a divergence-free velocity field, and T is a Fourier multiplier operator with symbol m . We prove that when m is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, wi...
Saved in:
Published in | Annals of PDE Vol. 1; no. 1; p. 2 |
---|---|
Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Cham
Springer International Publishing
01.12.2015
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | We consider active scalar equations
∂
t
θ
+
∇
·
(
u
θ
)
=
0
, where
u
=
T
[
θ
]
is a divergence-free velocity field, and
T
is a Fourier multiplier operator with symbol
m
. We prove that when
m
is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with Hölder regularity
C
t
,
x
1
/
9
-
. In fact, every integral conserving scalar field can be approximated in
D
′
by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when the multiplier
m
is odd, weak limits of solutions are solutions, so that the
h
-principle for odd active scalars may not be expected. |
---|---|
ISSN: | 2199-2576 2524-5317 2199-2576 |
DOI: | 10.1007/s40818-015-0002-0 |