Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 261, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
WELL-POSEDNESS AND BLOWUP CRITERION OF
GENERALIZED POROUS MEDIUM EQUATION IN
BESOV SPACES
XUHUAN ZHOU, WEILIANG XIAO, TAOTAO ZHENG
Abstract. We study the generalized porous medium equation of the form
ut + νΛβ u = ∇ · (u∇P u) where P is an abstract operator. We obtain the local
well-posedness in Besov spaces for large initial data, and show the solution
becomes global if the initial data is small. Also, we prove a blowup criterion
for the solution.
1. Introduction
In this article, we study the equation in Rn of the form
ut + νΛβ u = ∇ · (u∇P u);
u(0, x) = u0 .
(1.1)
Here u = u(t, x) ia a real-valued function, represents a density or concentration.
The dissipative coefficient ν > 0 corresponds to the viscous case, while ν = 0
corresponds to the inviscid case. The fractional operator Λβ is defined by Fourier
transform as (Λβ u)∧ = |ξ|β û, and P is an abstract operator.
The general form of equation (1.1) has good suitability in many cases. The
simplest case ν = 0, P u = u comes from a model in groundwater infiltration,
that is, ut = ∆u2 (see [2, 33]). We call (1.1) generalized porous medium equation
(GPME) inspired by Caffarelli and Vázquez [12], in which they introduced the
fractional porous medium flow (FPME) when ν = 0 and P u = Λ−2s u, 0 < s < 1.
When P u = Λ−2 u, it is the mean field equation, which is first studied by Lin and
Zhang [28]. Some results on the well-posedness and regularity on those equations
can be seen [7, 8, 13, 14, 30, 31, 34, 39] and the references therein.
Another similar model occurs in the aggregation equation, which is an important
equation arising in physics, biology, chemistry, population dynamics, etc.([15, 10,
21, 32]). In this model, the operator P is a convolution operator with kernel K;
that is, P u = K ∗ u. The typical kernels are |x|γ , see [9, 22, 27], and −e−|x| , see
[3, 6, 25, 26]. For more results on this equation, we refer to [4, 5, 11, 23, 24] and
2010 Mathematics Subject Classification. 35K15, 35K55, 35Q35, 76S05.
Key words and phrases. Generalized porous medium equation; well-posedness;
blowup criterion; Besov spaces.
c
2015
Texas State University.
Submitted June 25, 2015. Published October 7, 2015.
1
2
X. ZHOU, W. XIAO, T. ZHENG
EJDE-2015/261
the references therein. Besides, if we rewrite the equation (1.1) with same initial
data as
ut + νΛβ u + v · ∇u = −u(∇ · v);
(1.2)
v = −∇P u,
then it is a kind of special transport type equation. Furthermore, if we assume that
v is divergence-free vector function (∇ · v = 0), the form (1.2) can contain the 2 − D
quasi-geostrophic (Q-G) equation [17, 18, 19, 35].
In this article we study the well-posedness of equation (1.1) in homogenerous
Besov spaces under a general condition
s+σ .
k∇P ukḂ s ≤ CkukḂp,q
(1.3)
p,q
It is widespread adopted in the case of FPME, Q-G equation, or aggregation equation with its usual kernel |x|γ and it plays somewhat key role in the well-posedness
and regularity of those equations in Besov spaces or Sobolev spaces. Based on the
ideas used in [29, 36, 37, 38], we prove the following theorem.
Theorem 1.1. Assume P satisfies (1.3), β ∈ (0, 2], p ≥ 1 and σ + 1 < β <
n
+σ−β
p
σ +n min(2/p, 1). Then for any initial data u0 ∈ Ḃp,1
problem (1.1) admits a unique solution
n
+σ−β
p
u ∈ C([0, T ); Ḃp,1
n
+σ−β+1
p
∩ Ḃp,1
n
+σ−β+1
p
∩ Ḃp,1
n
+σ
p
) ∩ L1 ([0, T ); Ḃp,1
n
, the Cauchy
+σ+1
p
∩ Ḃp,1
).
Moreover, if T ∗ denotes the maximal time of existence of u,
(i) there exists a constant C0 > 0 such that if ku0 k
∞;
R T∗
(ii) if T ∗ < ∞, then 0 ku(t)kḂ n/p+σ+1 dt = ∞.
n +σ−β+1
p
Ḃp,1
< C0 , then T ∗ =
p,1
Remark 1.2. In the case of aggregation equation, Wu and Zhang [36] proved a
similar result under the condition ∇K ∈ W 1,1 , β ∈ (0, 1). Corresponding to their
case we prove same result for σ = 0, that is ∇K ∈ L1 , β ∈ (1, 2), and also a similar
result for σ = −1; that is, ∇K ∈ Ẇ 1,1 , β ∈ (0, 1).
Throughout this article, C denotes a positive constant that may differ line by
line, the notation A . B means A ≤ CB, and A ∼ B denotes A . B and B . A.
2. Preliminaries
Let us recall some basic knowledge on Littlewood-Paley theory and Besov spaces.
Let S (Rn ) be the Schwartz class and S 0 be its dual space. Given f ∈ S (Rn ), we
use its Fourier transform Ff = fˆ as
Z
−n/2
ˆ
f = (2π)
e−ix·ξ f (x)dx.
Rn
Let ϕ ∈
and
Cc∞ (Rn )
be a radial real-valued smooth function such that 0 ≤ ϕ(ξ) ≤ 1
supp ϕ ⊂ {ξ ∈ Rn :
3
8
≤ |ξ| ≤ },
4
3
X
j∈Z
ϕ(2−j ξ) = 1 for any ξ 6= 0.
EJDE-2015/261
POROUS MEDIUM EQUATION IN BESOV SPACES
3
We denote ϕj (ξ) = ϕ(2−j ξ) and P the set of all polynomials. Setting h = F −1 ϕ,
we define the frequency localization operators as follows:
Z
X
∆j u = ϕ(2−j D)u = 2jn
h(2j y)u(x − y)dy, Sj f =
∆k u.
Rn
k≤j−1
Definition 2.1. For s ∈ R, p, q ∈ [1, ∞], we define the homogeneous Besov space
s
Ḃp,q
as
X
1/q
s
Ḃp,q
= {f ∈ S 0 /P : kf kḂ s =
2jsq k∆j f kqLp
< ∞}.
p,q
j∈Z
Here the norm changes normally when p = ∞ or q = ∞.
Definition 2.2. In this paper, we need two kinds of mixed time-space norm defined
as follows: For s ∈ R, 1 ≤ p, q ≤ ∞, I = [0, T ), T ∈ (0, ∞], and X a Banach space
with norm k · kX where
1/r
Z
kf (τ, ·)krX dτ
,
kf (t, x)kLr (I;X) :=
I
X
1/q
.
2jsq k∆j f kqLr (I;Lp )
kf (t, x)kLr (I;Ḃ s ) :=
p,q
j∈Z
By Minkowski’ inequality, there holds
s
s
Lr (I; Ḃp,q
) ,→ Lr (I; Ḃp,q
),
if r ≤ q,
s
s
Lr (I; Ḃp,q
) ,→ Lr (I; Ḃp,q
),
if r ≥ q.
(2.1)
Now we state some basic properties about the homogeneous Besov spaces.
Proposition 2.3 ([1]). For s ∈ R, 1 ≤ p, q ≤ ∞, the following hold:
(i) Let β ∈ R, we have the equivalence of norms: kΛβ f kḂ s ∼ kf kḂp,q
s+β .
p,q
(ii)
(iii)
s−n(1/p −1/p2 )
.
If p1 ≤ p2 , q1 ≤ q2 , then Ḃps1 ,q1 ,→ Ḃp2 ,q2 1
Let 1 ≤ p, q ≤ ∞, s1 , s2 < np when q > 1 (or s1 , s2
≤
n
p
when q = 1), and
s1 + s2 > 0, there holds
kuvk
s1 +s2 − n
p
Ḃp,q
s1 kvk s2 ,
≤ CkukḂp,q
Ḃp,q
where C > 0 be a constant depending on s1 , s2 , p, q, n.
Lemma 2.4 (Bernstein’s inequalities [17]). Set B to be a ball and C to be an
annulus, and let 1 ≤ p ≤ q ≤ ∞, α ∈ ({0} ∪ N)n , then the following estimates hold:
(i) If supp fb ∈ 2j B, β + |α| ≥ 0, then
kΛβ Dα f kLq ≤ C2j(β+|α|+n(1/p−1/q)) kf kLp .
(ii) If supp fb ∈ 2j C, then
C2j(β+|α|) kf kLp ≤ kΛβ Dα f kLp ≤ C 0 2j(β+|α|) kf kLp ,
where C ≤ C 0 are positive constants independent of j.
Lemma 2.5 ([1]). Let C ba an annulus. If supp fb ⊂ 2j C, then positive constant
c > 0 exists such that for any t > 0, there holds
β
jβ
ke−tΛ f kLp ≤ Ce−ct2 kf kLp .
4
X. ZHOU, W. XIAO, T. ZHENG
EJDE-2015/261
Lemma 2.6 ([29]). Let s ∈ R and 1 ≤ p ≤ p1 ≤ ∞. Set Rj := (Sj−1 v − v) ·
∇∆j u − [∆j , v · ∇]u. There exists a constant C = C(n, s) such that
X
0
2js kRj kLp ≤ C
kSj 0 −1 ∇vkL∞ 2j s k∆j 0 ukLp
|j−j 0 |≤4
0
X
+
2j−j k∆j 0 ∇vkL∞ 2js k∆j ukLp
j 0 ≥j−3
X
+
2(j−j
00
)(s−1− pn ) j 0 pn
2
1
1
00
k∆j 0 ∇vkLp1 2j s k∆j 00 ukLp
|j 0 −j|≤4
j 00 ≤j 0 −2
X
+
(j−j 0 )(s+n min( p1 , p10 )) j 0 pn
2
2
1
1
0
(2j−j k∆j 0 ∇vkLp1
j 0 ≥j−3
|j 00 −j 0 |≤1
00
+ k∆j 0 ∇ · vkLp1 )2j s k∆j 00 ukLp .
Now we recall a priori estimates needed in our proof. Consider the transportdiffusion equation
∂t u + v · ∇u + νΛβ u = f,
u(0, x) = u0 (x).
(2.2)
Lemma 2.7 ([29]). Let 1 ≤ r1 ≤ r ≤ ∞, 1 ≤ p ≤ p1 ≤ ∞ and 1 ≤ q ≤ ∞. Assume
s ∈ R satisfies the following conditions:
n
n
(or s ≤ , if q = 1),
s<1+
p1
p1
n n
n n
s > − min ( , 0 ) (or s > −1 − min ( , 0 ), if div u = 0).
p1 p
p1 p
There exists a positive constant C = C(n, β, s, p, p1 , q) such that for any smooth
solution u of (2.2) with ν ≥ 0, the following a priori estimate holds:
ν 1/r kukLr Ḃ s+β/r ≤ CeCZ(T ) ku0 kḂ s + ν 1/r1 −1 kf kLr1 Ḃ s−β+β/r1 ,
T
where Z(T ) =
RT
0
p,q
p,q
T
p,q
k∇v(t)kḂ n/p1 ∩L∞ dt.
p1 ,∞
3. Local and global well-posedness
In this section we prove our main theorem. We first rewrite (1.1) as
ut + Λβ u + v · ∇u = u(∆P u);
v = −∇P u;
u(0, x) = u0 .
Step 1: Approximate solutions. In this step we construct approximate equaβ
tions, and prove the boundedness of the approximate solutions. Set u0 = e−tΛ u0 (x)
m+1
and let u
be the solution of the linear equation
um+1
+ Λβ um+1 + v m · ∇um+1 = um (∆P um );
t
v m = −∇P um ;
u
m+1
(0, x) = u0 .
(3.1)
EJDE-2015/261
n
POROUS MEDIUM EQUATION IN BESOV SPACES
+σ−β
p
Set X = Ḃp,1
n
+σ−β+1
p
∩ Ḃp,1
n
+σ
p
and Y = Ḃp,1
n
+σ+1
p
∩ Ḃp,1
5
. It is easy to show
u0 ∈ L∞ (R+ ; X) ∩ L1 (R+ ; Y ).
Now by induction, we deduce um belongs to the same spaces. In fact, by Lemma
2.7
kum+1 kL∞ Ḃ n/p+σ−β+1 + kum+1 kL1 Ḃ n/p+σ+1
p,1
T
.e
c
RT
0
p,1
T
k∇v m (t)k
m
dt
n/p
Ḃp,∞ ∩L∞
m
ku0 kḂ n/p+σ−β+1
p,1
+ ku (∆P u )kL1 Ḃ n/p+σ−β+1
p,1
T
c
.e
RT
0
k∇v m (t)k
n/p dt
Ḃp,1
(3.2)
ku0 kḂ n/p+σ−β+1
p,1
+ kum kL∞ Ḃ n/p+σ−β+1 k∆P um kL1 Ḃ n/p
p,1
T
ckum k
.e
T
n/p+σ+1
L1 Ḃp,1
T
p,1
ku0 kḂ n/p+σ−β+1
p,1
+ kum kL∞ Ḃ n/p+σ−β+1 kum kL1 Ḃ n/p+σ+1 .
p,1
T
T
p,1
Similarly, we conclude that
kum+1 kL∞ Ḃ n/p+σ−β + kum+1 kL1 Ḃ n/p+σ
T
ckum k
.e
p,1
T
n/p+σ+1
L1 Ḃp,1
T
p,1
ku0 kḂ n/p+σ−β + kum kL∞ Ḃ n/p+σ−β kum kL1 Ḃ n/p+σ+1 .
p,1
m
Thus for all m ∈ N , we have u
T
∞
+
p,1
1
(3.3)
p,1
T
+
∈ L (R ; X) ∩ L (R ; Y ).
Step 2: Uniform estimates. We prove the key uniform estimates during this
step. Set uj := ∆j u, fj := ∆j (um ∆P um ). Then we can obtain
Claim 3.1. There exists T1 ≤ T , such that for all t ∈ [0, T1 ]
X
jβ 1/r
kum+1 kLr Ḃ s+β/r .
1 − e−ctr2
2js ku0,j kLp
t
p,1
j∈Z
+
XZ
j∈Z
t
2js kfj kLp + kRj kLp dτ,
0
where Rj := (Sj−1 v m − v m ) · ∇um+1
− [∆j , v m · ∇]um+1 .
j
We postpone the proof of this claim to the appendix. Taking s = np + σ − β + 1
and ρ large enough such that σ + 1 + β/ρ ≤ β. Then by Proposition 2.3 with
s1 = n/p + σ + 1 − β + β/r, s2 = n/p − β/r, there holds
XZ t
2j(n/p+σ+1−β) kfj kLp dτ . kum kLρ Ḃ n/p+σ+1−β+β/ρ kum kLρ0 Ḃ n/p+σ+1−β/ρ .
j∈Z
t
0
p,1
t
p,1
Taking ρ large enough and using Lemma 2.6 with s = n/p + 1 + σ − β, p1 = p,
XZ t
2j(n/p+σ+1−β) kRj kLp dτ
j∈Z
0
. kum+1 kLρ Ḃ n/p+σ+1−β+β/ρ kum kLρ0 Ḃ n/p+σ+1−β/ρ .
t
p,1
t
p,1
6
X. ZHOU, W. XIAO, T. ZHENG
EJDE-2015/261
Hence for r ≥ 1 and ρ large enough, we have
kum+1 kLr Ḃ n/p+σ+1−β+β/r
t p,1
X
jβ 1/r
.
1 − e−ctr2
2j(n/p+σ+1−β) ku0,j kLp
(3.4)
j∈Z
+ kum kLρ Ḃ n/p+σ+1−β+β/ρ kum kLρ0 Ḃ n/p+σ+1−β/ρ
t
p,1
t
p,1
+ kum+1 kLρ Ḃ n/p+σ+1−β+β/ρ kum kLρ0 Ḃ n/p+σ+1−β/ρ .
t
p,1
t
p,1
0 jβ
Now by (3.4) with r = ρ and the fact that 1 − e−ctρ 2
for ρ large,
1/ρ0
jβ
≤ 1 − e−ctρ2
1/ρ
kum+1 kLρ Ḃ n/p+σ+1−β+β/ρ + kum+1 kLρ0 Ḃ n/p+σ+1−β/ρ
t p,1
t
p,1
X
−ctρ2jβ 1/ρ j(n/p+σ+1−β)
.
1−e
2
ku0,j kLp
j∈Z
+ kum kLρ Ḃ n/p+σ+1−β+β/ρ kum kLρ0 Ḃ n/p+σ+1−β/ρ
t
m+1
+ ku
p,1
t
p,1
m
kLρ Ḃ n/p+σ+1−β+β/ρ ku kLρ0 Ḃ n/p+σ+1−β/ρ .
t
p,1
t
p,1
By Lebesgue dominated convergence theorem, for ρ < ∞, we have
X
jβ 1/ρ
1 − e−ctρ2
2j(n/p+σ+1−β) ku0,j kLp = 0.
lim+
t7→0
j∈Z
Now we set
X
jβ 1/ρ
T = sup t > 0 : c
1 − e−ctρ2
2j(n/p+σ+1−β) ku0,j kLp ≤ η ,
j∈Z
for some η > 0 sufficiently small. By definition of u0 , ∀t ≤ T , we have
ku0 kLρ Ḃ n/p+σ+1−β+β/ρ ≤ η,
t
p,1
ku0 kLρ0 Ḃ n/p+σ+1−β/ρ ≤ η.
t
p,1
Choosing η small enough such that cη ≤ 1/2,
ku1 kLρ Ḃ n/p+σ+1−β+β/ρ + ku1 kLρ0 Ḃ n/p+σ+1−β/ρ ≤ 3η.
t
p,1
t
(3.5)
p,1
If we assume that (3.5) holds for um and further take η small enough such that
3cη ≤ 1/3, we obtain
kum+1 kLρ Ḃ n/p+σ+1−β+β/ρ + kum+1 kLρ0 Ḃ n/p+σ+1−β/ρ ≤ 3η.
t
p,1
t
(3.6)
p,1
Thus by induction, we prove the uniform boundedness (3.6) for some suitable η and
jβ jβ 1/ρ
, we have
∀t ≤ T . Let r = 1 in (3.4), since 1 − e−ct2
≤ 1 − e−ctρ2
X
jβ 1/ρ
kum+1 kL1 Ḃ n/p+σ+1 .
1 − e−ctρ2
2j(n/p+σ+1−β) ku0,j kLp
t
p,1
j∈Z
+ kum kLρ Ḃ n/p+σ+1−β+β/ρ kum kLρ0 Ḃ n/p+σ+1−β/ρ
t
m+1
+ ku
p,1
t
p,1
m
kLρ Ḃ n/p+σ+1−β+β/ρ ku kLρ0 Ḃ n/p+σ+1−β/ρ .
t
p,1
t
p,1
This and (3.6) imply
kum+1 kL1 Ḃ n/p+σ+1 ≤ η + 18cη 2 ≤ 3η.
t
p,1
(3.7)
EJDE-2015/261
POROUS MEDIUM EQUATION IN BESOV SPACES
n/p+σ+1−β
Next we prove that um are uniformly bounded in L∞
t Ḃp,1
7
. In fact, by (3.2)
1
kum+1 kL∞ Ḃ n/p+σ+1−β ≤ ce1/3 ku0 kḂ n/p+σ+1−β + e1/3 kum kL∞ Ḃ n/p+σ+1−β .
t
t
p,1
p,1
p,1
3
By induction, we conclude
kum kL∞ Ḃ n/p+σ+1−β
t
p,1
1 − ( 13 e1/3 )m 1/3
ce ku0 kḂ n/p+σ+1−β
p,1
1 − 13 e1/3
1/3
(3ce + 1)ku0 kḂ n/p+σ+1−β .
p,1
≤
≤
+
em/3 0
ku kL∞ Ḃ n/p+σ+1−β
t
p,1
3m
(3.8)
n/p+σ+1−β
Thus approximate solutions are uniformly bounded in the space L∞
T Ḃp,1
n/p+σ+1
L1T Ḃp,1
∩
. Now we return to (3.3) and by the uniform estimate (3.7)
kum+1 kL∞ Ḃ n/p+σ−β + kum+1 kL1 Ḃ n/p+σ
T
ckum k
.e
p,1
T
n/p+σ+1
L1 Ḃp,1
T
p,1
ku0 kḂ n/p+σ−β + kum kL∞ Ḃ n/p+σ−β kum kL1 Ḃ n/p+σ+1
p,1
T
p,1
T
p,1
(3.9)
1
≤ ce1/3 ku0 kḂ n/p+σ−β + e1/3 kum kL∞ Ḃ n/p+σ−β .
t
p,1
p,1
3
Hence, similarly to (3.8), by induction again,
kum kL∞ Ḃ n/p+σ−β ≤ (3ce1/3 + 1)ku0 kḂ n/p+σ−β .
t
p,1
p,1
Substituting this into (3.9) we conclude
kum kL1 Ḃ n/p+σ ≤ (4ce1/3 + 1)ku0 kḂ n/p+σ−β .
t
p,1
p,1
1
Thus we conclude that (u )m∈N is uniformly bounded in L∞
T X ∩ LT Y .
m
Step 3: Strong convergence. Let (m, k) ∈ N2 , m > k and um,k = um −
uk , v m,k = v m − v k . Then um,k satisfies the equation
um+1,k+1
+ Λβ um+1,k+1 + v k · ∇um+1,k+1
t
= um,k (∆P um ) + uk (∆P um,k ) − v m,k · ∇um+1 ;
v m,k = −∇P um,k ;
um+1,k+1 (0, x) = 0.
Set U m+1,k+1 (T ) = kum+1,k+1 kLρ Ḃ n/p+σ−β/ρ0 +kum+1,k+1 kLρ0 Ḃ n/p+σ−β/ρ . Applying
T
T
p,1
p,1
Lemma 2.7 with s = n/p + σ − β, there holds
U m+1,k+1 (T )
ck∇v k k
.e
n/p
L1 Ḃp,1
T
kum,k (∆P um )kL1 Ḃ n/p+σ−β
T
(3.10)
p,1
+ kuk (∆P um,k )kL1 Ḃ n/p+σ−β + kv m,k · ∇um+1 kL1 Ḃ n/p+σ−β .
T
p,1
T
p,1
Now applying Proposition 2.3 with s1 = n/p − β/ρ, s2 = n/p + σ − β/ρ0 ,
kv m,k · ∇um+1 kL1 Ḃ n/p+σ−β . kum,k kLρ0 Ḃ n/p+σ−β/ρ kum+1 kLρ Ḃ n/p+σ+1−β/ρ0 .
T
p,1
T
p,1
T
p,1
8
X. ZHOU, W. XIAO, T. ZHENG
EJDE-2015/261
Similarly, for ρ large enough such that σ + 1 + β/ρ ≤ β, we conclude
U m+1,k+1 (T ) . e
ck∇v k k
n/p
L1 Ḃp,1
T
kum,k kLρ0 Ḃ n/p+σ−β/ρ kum+1 kLρ Ḃ n/p+σ+1−β/ρ0
T
p,1
T
p,1
+ kuk kLρ Ḃ n/p+σ+1−β/ρ0 kum,k kLρ0 Ḃ n/p+σ−β/ρ
+ ku
k
p,1
T
p,1
T
m,k
0 n/p+σ−β/ρ
Lρ
T Ḃp,1
m
ku kLρ Ḃ n/p+σ+1−β/ρ0
T
p,1
≤ cηU m,k (T ).
Choosing η small enough such that cη < 1 and by induction, we conclude that {um }
0
n/p+σ−β/ρ0
n/p+σ−β/ρ
is a Cauchy sequence in LρT Ḃp,1
∩ LρT Ḃp,1
. So um hence converges
strongly to some u in it. Now by taking r = 1 and r = ∞ in (3.4), respectively,
and by passing to the limit into the approximation equation, we can get a solution
1
to in L∞
T X ∩ LT Y .
1
Step 4: Uniqueness. Let u1 , u2 ∈ L∞
T X ∩ LT Y be two solutions of (1.1) with
the same initial data. Let u1,2 = u1 − u2 , then
∂t u1,2 + Λβ u1,2 + v2 · ∇u1,2 = u1,2 (∆P u1 ) + u2 (∆P u1,2 ) − v1,2 · ∇u1 ;
v1,2 = −∇P u1,2 ;
u1,2 (0, x) = 0.
According to Lemma 2.7,
ku1,2 kL∞ Ḃ n/p+σ−β
p,ρ
t
.e
cku2 k
n/p+σ+1
L1
t Ḃp,1
ku1,2 (∆P u1 )kLρ Ḃ n/p+σ−2β+β/ρ
p,ρ
t
+ ku2 (∆P u1,2 )kLρ Ḃ n/p+σ−2β+β/ρ + kv1,2 · ∇u1 kLρ Ḃ n/p+σ−2β+β/ρ .
t
p,ρ
t
p,ρ
By a similar argument as in Step 3, we have
ku1,2 kρ ∞
n/p+σ−β
Lt Ḃp,ρ
cρku2 k
n/p+σ+1
L1 Ḃp,1
T
Z
t
ku1,2 kρ ∞ n/p+σ−β ku1 (τ )kρ n/p+σ+1−β+β/ρ
Lτ Ḃp,ρ
Ḃp,ρ
0
ρ
+ ku2 (τ )k n/p+σ+1−β+β/ρ dτ.
.e
Ḃp,ρ
s
s
Since the inclusion Ḃp,1
⊂ Ḃp,ρ
holds for any ρ ∈ [1, ∞]. Thus the Minkowski’s
inequality and Gronwall’s inequality give that u1 = u2 , ∀t ∈ [0, T ].
Step 5: Continuity in time. For all t, t0 ∈ [0, T ), we have
ku(t) − u(t0 )kḂ n/p+σ−β
p,1
X
X
j(n/p+σ−β)
p.
≤
2
kuj (t) − uj (t0 )kLp + 2
2j(n/p+σ−β) kuj kL∞
T L
j>N
j≤N
Since u ∈
n/p+σ−β
L∞
,
T Ḃp,1
for any > 0, we can choose N large enough such that
X
p ≤
.
2j(n/p+σ−β) kuj kL∞
T L
4
j>N
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POROUS MEDIUM EQUATION IN BESOV SPACES
9
On the other hand,
X
X
p
2j(n/p+σ−β) kuj (t) − uj (t0 )kLp ≤ |t − t0 |
2j(n/p+σ−β) k∂t uj kL∞
T L
j≤N
j≤N
0
≤ |t − t |2N β k∂t ukL∞ Ḃ n/p+σ−2β .
p,1
T
Now write
ut = −Λβ u − v · ∇u + u(∆P u)
and v = −∇P u.
Obviously, we have
kΛβ ukL∞ Ḃ n/p+σ−2β ≤ kukL∞ Ḃ n/p+σ−β .
p,1
T
T
p,1
Applying Proposition 2.3 with s1 = n/p − β, s2 = n/p + σ − β,
kv · ∇ukL∞ Ḃ n/p+σ−2β ≤ kukL∞ Ḃ n/p+σ−β kukL∞ Ḃ n/p+σ+1−β .
p,1
T
p,1
T
p,1
T
Similarly, we have
ku(∆P u)kL∞ Ḃ n/p+σ−2β ≤ kukL∞ Ḃ n/p+σ−β kukL∞ Ḃ n/p+σ+1−β .
T
0
Nβ
Thus for |t − t | ≤ (2
p,1
p,1
T
k∂t ukL∞ Ḃ n/p+σ−2β )−1 2 ,
p,1
T
T
p,1
we conclude
ku(t) − u(t0 )kḂ n/p+σ−β ≤ .
p,1
Hence u ∈
n/p+σ−β
C([0, T ); Ḃp,1
).
n/p+σ+1−β
Similarly we obtain u ∈ C([0, T ); Ḃp,1
).
Step 6: Blowup criterion. We give a blowup criterion as follows:
Proposition 3.2. Let T ∗ denote the maximal time of existence of a solution u in
C([0, T ∗ ); X) ∩ L1 ([0, T ∗ ); Y ). If T ∗ < ∞, then
Z T∗
ku(t)kḂ n/p+σ+1 dt = ∞.
p,1
0
R T∗
Proof. Supposing T ∗ < ∞ and
0
ku(t)kḂ n/p+σ+1 dt < ∞, and using Lemma 2.7
p,1
with ρ = ∞, we have
kukL∞ Ḃ n/p+σ+1−β . e
T∗
c
R T∗
0
ku(t)k
n/p+σ+1 dt
Ḃp,1
ku0 kḂ n/p+σ+1−β
p,1
p,1
Z
+
0
T
∗
ku(t)kḂ n/p+σ+1−β ku(t)kḂ n/p+σ+1 dt .
p,1
p,1
Hence by Gronwall’s inequality we have
kukL∞ Ḃ n/p+σ+1−β . e
T∗
c
R T∗
0
ku(t)k
n/p+σ+1 dt
Ḃp,1
p,1
ku0 kḂ n/p+σ+1−β < ∞.
(3.11)
ku0 kḂ n/p+σ−β < ∞.
(3.12)
p,1
By a similar argument there also holds
kukL∞ Ḃ n/p+σ−β . e
T∗
c
R T∗
0
ku(t)k
n/p+σ+1 dt
Ḃp,1
p,1
p,1
From Step 5, for all t, t0 ∈ [0, T ∗ ), we have
ku(t) − u(t0 )kX → 0
as t → t0 .
This means that u(t) satisfies the Cauchy criterion at T ∗ . So there exists an element
u∗ in X such that u(t) → u∗ in X as t → T ∗ . Now set u(T ∗ ) = u∗ and consider the
10
X. ZHOU, W. XIAO, T. ZHENG
EJDE-2015/261
equation with initial data u∗ . By the well-posedness we obtain a solution existing
on a larger time interval than [0, T ∗ ), which is a contradiction.
Step 7: Global solution. To obtain global well-posedness for small initial data,
it is sufficient to bound
Z t
ku(τ )kḂ n/p+σ+1 dτ.
F (t) :=
p,1
0
Lemma 2.7 gives
kukL1 Ḃ n/p+σ+1 . e
t
cF (t)
Z
ku0 kḂ n/p+σ+1−β +
p,1
p,1
0
t
ku(τ )kḂ n/p+σ+1−β ku(τ )kḂ n/p+σ+1 dτ .
p,1
p,1
A similar argument with (3.11) gives
kukL∞ Ḃ n/p+σ+1−β . ecF (t) ku0 kḂ n/p+σ+1−β .
t
p,1
p,1
Hence we conclude
F (t) ≤ CeCF (t) (1 + F (t))ku0 kḂ n/p+σ+1−β .
p,1
Since F (t) is continuous and F (0) = 0, we obtain that: if the initial data satisfies
e−C
ku0 kḂ n/p+σ+1−β < 1+C
, then
p,1
2
F (t) ≤ CeC (1 + C)ku0 kḂ n/p+σ+1−β .
p,1
By the blow-up criterion, the solution is global.
4. Appendix
We now give the proof of Claim 3.1. We list some lemmas which will be used in
our proof.
Lemma 4.1 ([20]). Let v be a smooth vector field, and ψt be the solution to
Z t
ψt (x) = x +
v(τ, ψτ (x))dτ.
0
Then for all t ∈ R+ , the flow ψt is a C 1 diffeomorphism over Rn and there holds
k∇ψt±1 kL∞ ≤ eV (t) ,
k∇ψt±1 − IdkL∞ ≤ eV (t) − 1,
Z t
k∇2 ψt±1 kL∞ ≤ eV (t)
k∇2 v(τ )kL∞ eV (τ ) dτ,
0
where V (t) =
Rt
0
k∇v(τ )kL∞ dτ .
Lemma 4.2 ([20]). Let χ ∈ S (Rn ). There exists a constant C = C(χ, n) such
that for all C 2 diffeomorphism ψ over Rn with inverse φ, and for all u ∈ S 0 , p ∈
[1, +∞], (j, j 0 ) ∈ Z2 ,
0
kχ(2−j D)(∆j u ◦ ψ)kLp
1/p
≤ CkJφ kL∞ k∆j ukLp (2−j kDJφ kL∞ kJψ kL∞ + 2j
0
−j
kDφkL∞ ).
EJDE-2015/261
POROUS MEDIUM EQUATION IN BESOV SPACES
11
Lemma 4.3 ([16]). Let v ∈ L1loc (R+ ; Lip) be a fixed vector field. For j ∈ Z, set
β
uj = ∆j u, ψj be the flow of the regularized vector field Sj−1 v. Then for u ∈ Ḃp,∞
with β ∈ [0, 2), p ∈ [1, ∞], there holds
β
kΛβ (uj ◦ ψj ) − (Λβ uj ) ◦ ψj kLp ≤ CeCV (t) V 1− 2 (t)2jβ kuj kLp ,
and when β = 2,
kΛ2 (uj ◦ ψj ) − (Λ2 uj ) ◦ ψj kLp ≤ CeCV (t) V (t)22j kuj kLp ,
Rt
where V (t) = 0 k∇v(τ )kL∞ dτ and C = C(β, p) > 0.
Proof of Claim 3.1. Applying ∆j to (3.1) we have
∂t um+1
+ Sj−1 v m · ∇um+1
+ Λβ um+1
= fj + Rj ,
j
j
j
(4.1)
where Rj := (Sj−1 v m − v m ) · ∇um+1
− [∆j , v m · ∇]um+1 . Let ψj be the flow of the
j
m
regularized vector field Sj−1 v . Denote ūj := uj ◦ ψj , then (4.1) becomes
∂t ūm+1
+ Λβ ūm+1
= f¯j + R̄j + Gj ,
j
j
(4.2)
where Gj := Λβ (um+1
◦ ψj ) − (Λβ um+1
) ◦ ψj .
j
j
Applying ∆k on the equivalent integral equation of (4.2), we have
k∆k ūm+1
(t)kLp
j
kβ
. e−ct2 k∆k u0,j kLp
Z t
kβ
+
e−c(t−τ )2 k∆k f¯j kLp + k∆k R̄j kLp + k∆k Gj kLp dτ.
(4.3)
0
Lemma 4.3 implies
k∆k Gj (t)kLp . ecV (t) V 1−β/2 (t)2jβ kum+1
kLp ,
j
with V (t) =
Rt
0
k∇v m (τ )kL∞ dτ . From Bernstein inequality and Lemma 4.1
k∆k f¯j kLp . 2−k k∇∆k f¯j kLp . 2−k k(∇fj ) ◦ ψj kLp k∇ψj kL∞ . 2j−k ecV (t) kfj kLp .
A similarly argument implies
k∆k R̄j (t)kLp . 2j−k ecV (t) kRj kLp .
Taking the Lr norm over [0, t] on (4.3) and plugging the above estimates give
kβ
2j(s+β/r) k∆k ūm+1
kLrt Lp . 2(j−k)β/r (1 − e−ctr2 )1/r 2js k∆k u0,j kLp
j
+ 2j(s+β/r) 2(j−k)β ecV (t) V 1−β/2 (t)kum+1
kLrt Lp
j
(4.4)
Z t
+ 2(j−k)(1+β/r) ecV (t)
2js kfj kLp + kRj kLp dτ.
0
Let M0 ∈ Z to be fixed later. Decomposing
um+1
= Sj−M0 ūm+1
◦ ψj−1 +
j
j
X
∆k ūm+1
◦ ψj−1 .
j
k≥j−M0
For all t ∈ [0, T ], there holds
kum+1
kLrt Lp . ecV (t) kSj−M0 ūm+1
kLrt Lp +
j
j
X
k≥j−M0
k∆k ūm+1
kLrt Lp .
j
(4.5)
12
X. ZHOU, W. XIAO, T. ZHENG
EJDE-2015/261
By Lemma 4.1 and Lemma 4.2,
kSj−M0 ūm+1
kLrt Lp . ecV (t) ecV (t) − 1 + 2−M0 kum+1
kLrt Lp .
j
j
(4.6)
0
Since ∆k u0,j = 0 for |k − j| ≥ 2 and ecV (t) − 1 + 2−M0 . e−c V (t) 2−M0 , multiplying
(4.5) by 2j(s+β/r) and using (4.4) and (4.6), we obtain
jβ 1/r
2j(s+β/r) kum+1
kLrt Lp . ecV (t) 2M0 β/r 1 − e−ctr2
2js ku0,j kLp
j
+ ecV (t) 2j(s+β/r) 2−M0 + 2M0 β V 1−β/2 (t) kum+1
kLrt Lp
j
Z t
+ ecV (t) 2M0 (1+β/r)
2js kfj kLp + kRj kLp dτ.
0
Now we choose M0 to be the unique integer such that 2c2−M0 ∈ (1/8, 1/4] and
T1 ≤ T be the largest real number such that
2−M0 β 2 cV (T1 ) ≤ min ln 2, ( β/2 ) 2−β .
8c
Thus for t ∈ [0, T1 ],
Z t
jβ 1/r
m+1
j(s+β/r)
−ctr2
js
2
kuj kLrt Lp . 1 − e
2 ku0,j kLp +
2js kfj kLp + kRj kLp dτ.
0
1
Taking the l -norm we conclude that
kum+1 kLr Ḃ s+β/r
t
.
X
j∈Z
p,1
jβ
1 − e−ctr2
1/r
2js ku0,j kLp +
XZ
j∈Z
t
2js kfj kLp + kRj kLp dτ.
0
4.1. Acknowledgments. This research was supported by the NNSF of China under grant 11271330.
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14
X. ZHOU, W. XIAO, T. ZHENG
EJDE-2015/261
Xuhuan Zhou
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
E-mail address: zhouxuhuan@163.com
Weiliang Xiao
chool of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing
210023, China
E-mail address: xwltc123@163.com
Taotao Zheng
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou
310023, China
E-mail address: taotzheng@126.com