˖ڍመڙጲ
http://www.paper.edu.cn
0.3 Main results
Now we give our main result of propagation of Gevrey regularity in spatially homogeneous
case as follows:
Theorem 1. Let ν > 1(which is independent of s) and assume that 0 < s ≤ 1/2, −5/2 < γ < 0.
Let f(t, v) be a smooth Maxwellian decay solution to the Cauchy problem (0.2). If there exist
ρ
0
, δ
0
such that
sup
α
ρ
0|α|
ke
δ
0
hvi
2
∂
α
v
f(0)k
L
2
{α!}
ν
< +∞, (0.5)
then there exist ρ > 0 and δ, κ > 0 with δ > κT such that
sup
t∈(0,T ]
sup
α
ρ
|α|
ke
(δ−κt)hvi
2
∂
α
v
f(t)k
L
2
{α!}
ν
< +∞. (0.6)
Remark 1. It should be noted that the above theorem is similar as Theorem 1.2 in [17] and
Theorem 1.3 in [18], but we consider here Φ = |v|
γ
and γ ∈ (−5/2, 0).
Arguing as in Section 4 of [17], and thanks to the interpolation inequality with weight
(Lemma 3.7 of [21]), for γ ∈ [−2, 0),
kW
l
F
(α,β)
k
2
H
s/2
. kW
l+γ/2
F
(α,β)
k
H
s
kW
l−γ/2
F
(α,β)
k
L
2
. kW
l+γ/2
F
(α,β)
k
H
s
kW
l+1
F
(α,β)
k
L
2
,
we would obtain the Gevrey smoothing effect of order 1/s as follows:
Theorem 2. Assume that 0 < s ≤ 1/2, γ ∈ [−2, 0). Let ν = 1/s and let f(t, v) be a smooth
Maxwellian decay solution to the Cauchy problem (0.2), then for any t
0
∈ (0, T ), there exist
ρ > 0 and δ, κ > 0 with δ > κT such that
sup
t∈[t
0
,T ]
sup
α
ρ
|α|
ke
(δ−κt)hvi
2
∂
α
v
f(t)k
L
2
{α!}
ν
< +∞. (0.7)
Remark 2. From the above interpolation inequality, we know that the additional requirement
γ ≥ −2 is natural, which is different from the assumption γ ∈ (−5/2, 0) in Theorem 1.
Remark 3. We believe that these results are also valid for spatially inhomogeneous case, as
we did in [19].
0.4 The structure of the paper
The remainder of the paper proceeds as follows. In the next section we give some prelimi-
naries and a main lemma, by using which we can complete the proof of Theorem 1 immediately.
In Section 3 we prove a coercivity estimate for collision operator which is different from what
we used before. The proof of the main lemma will be given in Section 4.
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