Abstract
AbstractWe apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension $$d\ge 3$$
d
≥
3
. For the power index q below the compactness threshold, i.e. $$q \in (1, \frac{2d}{d+2})$$
q
∈
(
1
,
2
d
d
+
2
)
, we show ill-posedness of Leray–Hopf solutions. For a wider class of indices $$q \in (1, \frac{3d+2}{d+2})$$
q
∈
(
1
,
3
d
+
2
d
+
2
)
we show ill-posedness of distributional (non-Leray–Hopf) solutions, extending the seminal paper of Buckmaster & Vicol [10]. In this wider class we also construct non-unique solutions for every datum in $$L^2$$
L
2
.
Funder
European Research Council
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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