We study a doubly parabolic Keller-Segel system in one spatial dimension, with diffusions given by fractional laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we need certain smallness assumptions). We also study dynamical properties of the system with added logistic term. Then, this model exhibits a spatio-temporal chaotic behavior, where a number of peaks emerge. In particular, we prove the existence of an attractor and provide an upper bound on the number of peaks that the solution may develop. Finally, we perform a numerical analysis suggesting that there is a finite time blow up if the diffusion is weak enough, even in presence of a damping logistic term. Our results generalize on one hand the results for local diffusions, on the other the results for the parabolic-elliptic fractional case.
We 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
.
We consider a two dimensional parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order α. We obtain existence of global in time regular solution for arbitrary initial data with no size restrictions and c < α ≤ 2, where c ∈ (0, 2) depends on the equation's parameters. For an even wider range of α ′ s, we prove existence of global in time weak solution for general initial data.
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