d+ 1,d≥2, the equation has a unique local in time solution for any initial data inBα1

Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 199, pp. 1–14.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

FRACTIONAL POROUS MEDIUM AND MEAN FIELD EQUATIONS IN BESOV SPACES

XUHUAN ZHOU, WEILIANG XIAO, JIECHENG CHEN

Abstract. In this article, we consider the evolution model

∂tu− ∇ ·(u∇P u) = 0, P u= (−∆)^−su, 0< s≤1, x∈R^d, t >0.

We show that whens∈[1/2,1),α > d+ 1,d≥2, the equation has a unique local in time solution for any initial data inB^α_1,∞. Moreover, in the critical cases= 1, the solution exists inB_p,∞^α , 2≤p≤ ∞,α > d/p,d≥3.

1. Introduction Let 0< s≤1, we consider the evolution equation

∂tu− ∇ ·(u∇P u) = 0, P u= (−∆)^−su, u(x,0) =u0(x). (1.1) wherex∈R^d,t >0. u=u(x, t) is a real-valued function, representing the density or concentration. P represents the pressure.

When 0< s <1, we refer the equation as a fractional porous medium equation.

It was first introduced by Caffarelli and V´azquez [2]. They proved the existence of a weak solution when u₀ is a bounded function and has exponential decay at infinity. Caffarelli, Soria, and V´azquez [1] studied its regularity theory of the weak solution withu0∈L¹∩L^∞and the continuity of bounded solutions.

Whens= 1, the equation leads to a mean field equation

ut=∇ ·(u∇P u), P u= (−∆)⁻¹u, u(x,0) =u0(x). (1.2) Equation that was first studied by Lin and Zhang [7]. They proved the existence and uniqueness of positiveL^∞ solution in two dimensions. When d≥3, V´azquez and Serfaty [11] studied the existence and uniqueness of the weak solution inL^∞ spaces for (1.2) by taking the limit s → 1 in the weak solutions of (1.1). Since papers in literature only addressed the weak solution of (1.2), in this article we show that the strong solution exists inB^α_p,∞,α >0.

In this article, we are interested in finding the strong solutions of (1.1) in the Besov spaces B_p,∞^α . We will show that when d ≥ 2, s ∈ [1/2,1), α > d+ 1, equation (1.1) has a unique local in time solution for any initial data in B^α_1,∞. Whens= 1, the solution extends toB_p,∞^α ,α > d/p, 2≤p≤ ∞,d≥3. The idea

2000Mathematics Subject Classification. 35K55, 35K65, 76S05.

Key words and phrases. Fractional porous medium equation; mean field equation;

local solution; Besov space.

c

2014 Texas State University - San Marcos.

Submitted April 10, 2014. Published September 23, 2014.

1

of our proof is inspired by the methods used in [3, 12], where the authors studied the quasi-geostrophic equation. In our proof, we construct two commutators, give three estimates of them, and construct a function sequence fitting our equation.

The rest of this article is divided in four parts. Section 2 recalls the definition and some properties of the Besov spaces, the Bernstein inequality involving both integer and fractional derivatives, as well as some properties of the fractional Laplacian.

In Section 3 we prove three estimates about the constructed commutators and a priori estimate of the solution. Section 4 proves the existence and uniqueness of the fractional porous medium equation. Section 5 is devoted to the mean field equation.

The main results are the following two theorems.

Theorem 1.1. Let d≥2, s∈[1/2,1], α > d+ 1. Assume that the initial data u0∈ B_1,∞^α . Then we can findT =T(ku0kB_1,∞^α ), such that a unique solutionuto(1.1)on [0, T]×R^dexists. And the solution belongs toC¹([0, T];B_1,∞^α+2s−2)∩L^∞([0, T];B_1,∞^β ), andβ∈[α+ 2s−2, α].

Theorem 1.2. Let d≥3, α > d/p when2≤p≤ ∞. Assume that the initial data u0∈B_p,∞^α . Then we can find T =T(ku0kB^α_p,∞), such that a unique solution uto the given mean field equation (1.2) on[0, T]×R^d exists. And the solution belongs toC¹([0, T];B_p,∞^β )∩L^∞([0, T];B_p,∞^α ),β ∈(d/p, α),2≤p≤ ∞.

2. Preliminaries

In this section, we recall the definition of the Besov space. We start with a dyadic decomposition ofR^d.

Supposeχ∈C₀^∞(R^d), φ∈C₀^∞(R^d\ {0}) satisfying suppχ⊂ {ξ∈R^d :|ξ| ≤ 4

3}, suppφ⊂ {ξ∈R^d: 3

4 <|ξ|<8 3}, χ(ξ) +X

j≥0

φ(2^−jξ) = 1, ξ∈R^d, X

j∈Z

φ(2^−jξ) = 1, ξ∈R^d\{0}.

Define the operators

∆ju=φ(2^−jD)u= 2^jd Z

h(2^jy)u(x−y)dy, Sjf = X

k≤j−1

∆kf =χ(2^−jD)u= 2^jd Z

g(2^jy)u(x−y)dy,

(2.1)

whereg=χ^∨andh=φ^∨are the inverse Fourier transform ofχandφ, respectively.

It can be easily verified that with our choice ofφ,

∆j∆kf ≡0, if |j−k| ≥2, ∆j(S_k−1f∆kf)≡0, if|j−k| ≥5. (2.2) Definition 2.1. For anyα∈R, andp, q∈[1,∞], the homogeneous Besov spaces B˙_p,q^r are defined as

B˙^α_p,q={f ∈ Z⁰(R^d) :kfkB˙^α_p,q<∞}.

Here,

kfkB˙^α_p,q =h X

j∈Z

2^jαqk∆_jfk^q_Lp

i1/q

, whereq <∞.

Whenq=∞,

kfkB˙^α_p,∞ = sup

j∈Z

2^jαk∆jfkL^p.

Z⁰(R^d) denotes the dual space ofZ(R^d) ={f ∈ S(R^d) :∂^γfˆ(0) = 0,∀γ∈N^d} and can be identified by the quotient space ofS⁰/P with the polynomials spaceP.

Definition 2.2. For anyα∈R, andp, q∈[1,∞], the inhomogeneous Besov space B_p,q^r is defined as

B^α_p,q={f ∈ S⁰(R^d) :kfkB^α_p,q<∞}.

Here,

kfkB^α_p,q=X^∞

j≥0

2^jαqk∆jfk^q_Lp

^1/q

+kS0(f)kL^p, whenq <∞.

Whenq=∞,

kfkB_p,∞^α = sup

j≥0

2^jαk∆jfkL^p+kS0(f)kL^p. Let us state some basic properties of the Besov spaces.

Proposition 2.3. Let s∈R,1≤p≤ ∞,1≤q≤ ∞.

(i) If α >0, thenB_p,q^α = ˙B^α_p,q∩L^p, andkfkB^α_p,q=kfkB˙_p,q^α +kfkL^p;

(ii) If α1 ≤α2, thenB^α_p,q² ⊂B_p,q^α1. If 1≤q1≤q2 ≤ ∞, thenB˙_p,q^α ₁ ⊂B˙_p,q^α ₂ and B_p,q^α ₁⊂B_p,q^α ₂;

(iii) If α > ^d_p, thenB_p,q^α ,→L^∞. If p₁≤p₂, α₁−_p^d

1 > α₂−_p^d

2, thenB_p^α1

1,q₁ ,→ B_p^α2

2,q₂,B_p,min(p,2)^α ,→H_p^α,→B^α_p,max(p,2); (iv) If α >0,p≥1, thenkuvk_Bα

p,∞ ≤Ckuk_L^∞kvk_Bα

p,∞+Ckuk_Bα

p,∞kvk_L^∞. We now turn to Bernstein’s inequalities. When the Fourier transform of a func- tion is supported on a ball or an annulus, the L^p-norms of the derivatives of the function can be bounded in terms of theL^p-norms of the function itself. And it also exists when one replaces the derivatives by the fractional derivatives (see [6, 14]).

Proposition 2.4. Let 1≤p≤q ≤ ∞, γ ∈N^d. (1) If α≥0 andsupp ˆf ⊂ {ξ∈ R^d:|ξ| ≤K2^j} for someK >0 and integerj, then

k(−∆)^αD^γfkL^q ≤C2j(2α+|γ|)+jd(¹_p−¹_q)

kfkL^p.

(2) If α∈Rand supp ˆf ⊂ {ξ∈R^d :K12^j ≤ |ξ| ≤K22^j} for some K1, K2>0 and integerj, then

C2j(2α+|γ|)+jd(_p¹−¹_q)

kfkL^p≤ k(−∆)^αD^γfkL^q ≤C2e j(2α+|γ|)+jd(¹_p−_q¹)

kfkL^p, whereC andCe are positive constants independent of j.

Next we state two pointwise inequalities which were proved in [5, 13].

Proposition 2.5. Let 0 ≤α≤ 1, f ∈C²(R^d) decay sufficiently fast at infinity.

Then for anyx∈R^d,

2f(x)(−∆)^αf(x)≥(−∆)^αf²(x).

Proposition 2.6. Let 0 ≤α ≤ 1, p1 = ^k_l¹

1 ≥0, p2 = ^k_l²

2 ≥ 1 be rational num- bers with l₁, l₂ odd, and k₁l₁+k₂l₂ even. Then for any f ∈ C²(R^d) that decays sufficiently fast at infinity, and for anyx∈R^d,

(p1+p2)f^p1(x)(−∆)^αf^p2(x)≥p2(−∆)^αf^p1+p2(x).

3. A priori estimate

Proposition 3.1. Let α > 0, s ∈ (0,1), p∈ [1,∞] be given. Assume r > d/p.

Then there exists some constantC such that

2^jαk[∆j, ∂_i(−∆)^−su]∂_ivkL^p ≤Ckvk_B^r+1_p,∞kuk_Bp,∞^α+1−2s+Ckuk_Bp,∞^r+2−2skvkB_p,∞^α , (3.1) where the brackets[,]represents the commutator, namely

[∆j, ∂i(−∆)^−su]∂iv= ∆j(∂i(−∆)^−su)∂iv)−∂i(−∆)^−su∆j(∂iv).

Proof. Using Bony’s para-product decomposition, we have [∆_j, ∂_i(−∆)^−su]∂_iv=L₁+L₂+L₃+L₄+L₅, where

L1= X

|k−j|≤4

∆j[S_k−1(∂i(−∆)^−su)∆k(∂iv)]−S_k−1(∂i(−∆)^−su)∆k(∆j(∂iv)), L₂= X

|k−j|≤4

∆_j[S_k−1(∂_iv)∆_k(∂_i(−∆)^−su)], L3= X

k≥j−2

∆j(∆k(∂i(−∆)^−su)∆ek(∂iv)), L4=X

k

Sk−1(∆j(∂iv))∆k(∂i(−∆)^−su), L5= X

|j⁰−j⁰⁰|≤1

∆j⁰(∆j(∂iv))∆j⁰⁰(∂i(−∆)^−su).

We shall estimate the above terms separately. First observe L1= X

|k−j|≤4

2^jd Z

h(2^j(x−y))

S_k−1(∂i(−∆)^−su)(y)

−S_k−1(∂_i(−∆)^−su)(x)

∆_k(∆_j(∂_iv))(y)dy.

By Young’s inequality and Bernstein’s inequality, kL1kL^p≤C X

|k−j|≤4

2^−jk∇∂i(−∆)^−sukL^∞k∆j(∂iv)kL^p

Z

|y||h(y)|dy

≤C2^−j2^jk(−∆)^1−sukL^∞k∆jvkL^p

≤Ck∆jvkL^pkuk_B^r+2−2s

p,∞ .

Similarly,

kL2kL^p≤C2^j(1−2s)k∇vkL^∞k∆jukL^p≤C2^−jαkvk_B^r+1

p,∞kuk_B^α+1−2s

p,∞ .

We can also estimate, kL3kL^p ≤C X

k≥j−2

k∆k(∂i(−∆)^−su)kL^pk∇vkL^∞

≤C2^−jα X

k≥j−2

2^(j−k)α2^k(α+1−2s)k∆kukL^pkvk_B^r+1_p,∞

≤C2^−jαkuk_Bα+1−2s

p,∞ kvk_Br+1 p,∞.

To estimateL₄, by the definition ofS_jand ∆_j, we can observe that onlyksatisfying k≥j survive. Thus

kL₄k_Lp≤X

k≥j

Ck∇vk_L^∞2^k(1−2s)k∆_kuk_Lp

≤C2^−jαX

k≥j

2^(j−k)αkvk_B^r+1

p,∞kuk_B^α+1−2s

p,∞

≤C2^−jαkvk_Br+1

p,∞kuk_Bα+1−2s

p,∞ .

Since ∆_k∆_j = 0, for|j−k| ≥2, we have

kL5kL^p≤Ck∇vkL^∞2^j(1−2s)k∆jukL^p≤C2^−jαkvk_Br+1

p,∞kuk_Bα+1−2s

p,∞ .

Collecting the estimates above, we obtain 2^jαk[∆j, ∂i(−∆)^−su]∂ivkL^p

≤C2^jαk∆_jvk_Lpkuk_Br+2−2s

p,∞ + 2^j(α+1−2s)kvk_Br+1

p,∞k∆_juk_Lp+kuk_Bα+1−2s

p,∞ kvk_Br+1 p,∞

≤ kvk_Bα

p,∞kuk_Br+2−2s

p,∞ +kuk_Bα+1−2s

p,∞ kvk_Br+1 p,∞.

This completes the proof.

Proposition 3.2. Let α, s, p, r be as in Proposition 3.1. Then there exists a con- stantC such that

2^jαk[∆j, ∂_i(−∆)^−su]∂_ivkL^p ≤CkvkB^r_p,∞kuk_Bp,∞^α+2−2s+Ckuk_Bp,∞^r+2−2skvkB_p,∞^α . (3.2) Whenp=∞, this inequality becomes: for anyr >0,

2^jαk[∆j, ∂i(−∆)^−su]∂ivkL^∞≤CkvkB^r_∞,∞kuk_B^α+2−2s

∞,∞ +Ckuk_B^r+2−2s

∞,∞ kvkB^α_∞,∞. (3.3) Proof. We want to give a new estimate of the commutator in Proposition 3.1.

Following the above proof, the estimate ofL1unchanged, we give different bounds forL2, L3, L4, L5. First,

L₂= X

|k−j|≤4

2^jd Z

h(2^j(x−y))(S_k−1∂_iv)(y)∆_k(∂_i(−∆)^−su)(y)dy

= X

|k−j|≤4

2^jd Z

∂_ih(2^j(x−y))2^j(S_k−1v)(y)∆_k(∂_i(−∆)^−su)(y)dy

− X

|k−j|≤4

2^jd Z

h(2^j(x−y))(S_k−1v)(y)∆_k(∂_ii(−∆)^−su)(y)dy.

So we obtain

kL2kL^p ≤C2^2−2s(k∂ihkL¹+khkL¹)kvkL^∞k∆jukL^p

≤C2^−jαkuk_B^α+2−2s

p,∞ kvkB^r_p,∞.

Similarly,

kL3kL^p≤C X

k≥j−2

2^jd Z

∂ih(2^j(x−y))2^j∆k(∂i(−∆)^−su)(y)∆kv(y)dy

−2^jd Z

h(2^j(x−y))∆k(∂ii(−∆)^−su)(y)∆kv(y)dy.

Hence we obtain, kL3kL^p ≤C X

k≥j−2

(2^{(j−k)(α+1)}+ 2^(j−k)α)2^k(α+2−2s)kvkL^∞k∆kukL^p

≤C2^−jαkuk_B^α+2−2s

p,∞ kvkB_p,∞^r . Also,

kL4kL^p≤X

k≥j

C2^jk∆jvkL^∞2^k(1−2s)k∆kukL^p

≤C2^−jαX

k≥j

2^{(j−k)(α+1)}2^k(α+2−2s)kvkB_p,∞^r k∆kukL^p

≤C2^−jαkvkB_p,∞^r kuk_B^α+2−2s

p,∞ .

Finally,

kL₅k_Lp≤Ck∆_jvk_L^∞2^j(2−2s)k∆_juk_Lp≤C2^−jαkvk_Br

p,∞kuk_Bα+2−2s

p,∞ .

Collecting the estimates above, we can obtain

2^jαk[∆j, ∂i(−∆)^−su]∂ivkL^p≤C2^jαk∆jvkL^pkuk_B^r+2−2s

p,∞ +kvkB_p,∞^r kuk_B^α+2−2s

p,∞

≤ kvkB_p,∞^α kuk_B^r+2−2s

p,∞ +kuk_B^α+2−2s

p,∞ kvkB_p,∞^r .

This completes the proof.

Proposition 3.3. Let α >0,s ∈(0,1),p∈ [1,∞]. Assume r > ^d_p. Then there exists a constantC such that

2^jαk[∆j, v](−∆)^1−sukL^p ≤Ckvk_Br+1

p,∞kuk_Bα+1−2s

p,∞ +Ckuk_Br+2−2s

p,∞ kvkB_p,∞^α , (3.4) where the brackets[,]represents the commutator,

[∆_j, v](−∆)^1−su= ∆_j(v(−∆)^1−su)−v∆_j((−∆)^1−su). (3.5) Proof. This proposition is proved similarly to Proposition 3.1. We start the proof by writing

[∆j, v](−∆)^1−su=I1+I2+I3+I4+I5, where

I1= X

|k−j|≤4

∆j(S_k−1v∆k(−∆)^1−su)−S_k−1v∆k(∆j(−∆)^1−su), I₂= X

|k−j|≤4

∆_j(S_k−1(−∆)^1−su)∆_kv), I3= X

k≥j−2

∆j(∆kv)e∆k(−∆)^1−su), I4=X

k

S_k−1(∆j(−∆)^1−su)∆kv,

I₅= X

|j⁰−j⁰⁰|≤1

∆_j⁰v∆_j⁰⁰(∆_k(−∆)^1−su).

Similar to the proof for Proposition 3.1, first we observe that kI₁k_Lp =k X

|k−j|≤4

2^jd Z

h(2^j(x−y))(S_k−1v(y)−S_k−1v(x))∆_k(−∆)^1−su)(y)dyk_Lp

≤C X

|k−j|≤4

2^−jk∇vk_L^∞k∆_j(−∆)^1−suk_Lp

Z

|y||h(y)|dy

≤C2^−j2^2j(1−s)k∇vkL^∞k∆jukL^p

≤C2^j(1−2s)k∆jukL^pkvk_Br+1 p,∞. Also we obtain

kI2kL^p≤Ck(−∆)^1−sukL^∞k∆jvkL^p≤Ckuk_Br+2−2s

p,∞ k∆jvkL^p, kI3kL^p≤C X

k≥j−2

k∆kvkL^p2^k(2−2s)k∆kukL^∞

≤C2^−jα X

k≥j−2

2^(j−k)α2^kαk∆kvkL^p2^k(2−2s)k∆kukL^∞

≤C2^−jαkvk_Bα

p,∞kuk_Br+2−2s

p,∞ .

Similarly, we estimate kI4kL^p≤CX

k≥j

k(−∆)^1−sukL^∞k∆kvkL^p

≤C2^−jαX

k≥j

2^(j−k)αk(−∆)^1−sukL^∞2^kαk∆jvkL^p

≤C2^−jαkuk_B^r+2−2s_p,∞ kvkB^α_p,∞,

kI5kL^p≤Ck(−∆)^1−sukL^∞k∆jvkL^p≤Ckuk_B^r+2−2s

p,∞ k∆jvkL^p. Collecting the estimates above, we obtain

2^jαk[∆_j, v](−∆)^1−suk_Lp≤C2^j(α+1−2s)k∆_juk_Lpkvk_Br+1 p,∞

+C2^jαkuk_Br+2−2s

p,∞ k∆_jvk_Lp+Ckvk_Bα

p,∞kuk_Br+2−2s p,∞

≤Ckvk_Br+1

p,∞kuk_Bα+1−2s

p,∞ +Ckvk_Bα

p,∞kuk_Br+2−2s

p,∞ .

This completes the proof.

Proposition 3.4. Lets∈[1/2,1],p=k/lbe a rational number withkeven,lodd, andα > ^d_p + 1. Assume that u(x, t)∈B^α_p,∞ is a solution of (1.1)with u0 ∈B^α_p,∞

fort∈[0, T]. Then, when u(x, t)≥0, we can find someC=C(p, α), that for any t≤T,

kukB^α_p,∞ ≤Cku0kB_p,∞^α exp{C Z t

0

kukB_p,∞^α dτ}. (3.6) Proof. Applying ∆j on (1.1), we obtain

∂_t∆_ju=X

[∆_j, ∂_i(−∆)^−su]∂_iu+∇(−∆)^−su∆_j(∇u)

−[∆_j, u](−∆)^1−su−u∆_j((−∆)^1−su.

Multiplying both sides by p∆ju|∆ju|^p−2 and integrating over R^d, the equation becomes

d

dtk∆juk^p_Lp

=XZ

p∆ju|∆ju|^p−2[∆j, ∂i(−∆)^−su]∂iu+ Z

p∆ju|∆ju|^p−2∇(−∆)^−su∆j(∇u)

− Z

p∆_ju|∆_ju|^p−2[∆_j, u](−∆)^1−su− Z

p∆_ju|∆_ju|^p−2u∆_j((−∆)^1−su

=J₁+J₂+J₃+J₄.

From Propositions 3.1 and 3.3, we obtain the estimates

J₁≤C2^−jαk∆juk^p−1_Lp kukB^α_p,∞kuk_B^α+1−2s_p,∞ ≤C2^−jαk∆juk^p−1_Lp kuk²_Bα p,∞, J₃≤C2^−jαk∆juk^p−1_Lp kukB^α_p,∞kuk_B^α+1−2s_p,∞ ≤C2^−jαk∆juk^p−1_Lp kuk²_Bα

p,∞. It is easy to see that

J2= Z

∇(−∆)^−su)∇(|∆j(u)|^p) = Z

(−∆)^1−su|∆ju|^pdx

≤Ck(−∆)^1−suk_B^α−1

p,∞k∆juk^p_Lp≤Ckuk_B^α+1−2s

p,∞ k∆juk^p−1_Lp k∆jukL^p

≤C2^−jαkuk_Bα+1−2s p,∞ kuk_Bα

p,∞k∆_juk^p−1_Lp .

Using the fact thatu≥0 and Propositions 2.5 and 2.6, we estimate J₄≤ −

Z

p|∆ju|^p−2u(−∆)^1−s|∆ju|²≤ − Z

u(−∆)^1−s|∆ju|^p

=− Z

(−∆)^1−su|∆ju|^p≤Ck(−∆)^1−suk_B^α−1

p,∞k∆juk^p_Lp

≤C2^−jαkuk_Bp,∞^α+1−2skukB_p,∞^α k∆juk^p−1_Lp .

Combining the above four estimates, we set _dt^dkukB˙_p,∞^α ≤Ckuk²_Bα

p,∞. Sinceu≥0, it follows

d

dtkuk_Lp= d dt

Z

u^pdx=p Z

u^p−1u_tdx

=p Z

u^p−1∇ ·(u∇(−∆)^−su) =−(p−1) Z

∇u^p·(∇(−∆)^−su)

=−(p−1) Z

u^p((−∆)^1−su)

≤Ckuk^p−1_Lp kukB_p,∞^α kuk_B^α+1−2s

p,∞ .

This implies

d

dtkukB_p,∞^α ≤CkukB_p,∞^α kukB_p,∞^α ,

which with the Gronwall’s inequality yield (3.6).

4. Proof of Theorem 1.1

From the definition of Riesz potentials (−∆)^−su=c(n, s)|x|^−d+2s∗u, 0<2s < d.

So whend≥2, ¹₂≤s <1, we definePu=c(n, s)(|x|^−d+2s∗σ)∗u= (−∆)^−s(σ∗ u) = (−∆)^−su, u =σ∗u. Here σ ∈ C_c^∞, is nonnegative, radially symmetric

and decreasing, R

σ = 1, σ =ε^−dσ(x/). Then we construct a sequence {u⁽ⁿ⁾}, defined recursively by solving the following equations

u⁽¹⁾=S₂(u₀)

∂_tu⁽ⁿ⁺¹⁾=∇ ·(u⁽ⁿ⁺¹⁾∇(Pu⁽ⁿ⁾)) u⁽ⁿ⁺¹⁾(x,0) =uⁿ⁺¹₀ =Sn+2u0.

(4.1)

Sinceu⁽ⁿ⁾ solves the linear system, we can always find the sequence. Similarly to the proof of Proposition 3.4, taking ∆j on (4.1), we obtain

∂t∆ju⁽ⁿ⁺¹⁾=X

[∆j, ∂i(−∆)^−su⁽ⁿ⁾ ]∂iu⁽ⁿ⁺¹⁾+X

∂i(−∆)^−su⁽ⁿ⁾ ∆j(∂iu⁽ⁿ⁺¹⁾)

−[∆j, u⁽ⁿ⁺¹⁾](−∆)^1−su⁽ⁿ⁾ −u⁽ⁿ⁺¹⁾∆j((−∆)^1−su⁽ⁿ⁾ ).

(4.2) Multiplying both sides by ^∆ju(n+1)

|∆ju⁽ⁿ⁺¹⁾|, integrating over R^d, we denote each corre- sponding part in the right side byJ₁⁰, J₂⁰, J₃⁰, J₄⁰. Now we obtain the estimates

J₁⁰ ≤C2^−jαku⁽ⁿ⁺¹⁾k_B^α

1,∞ku⁽ⁿ⁾k_Bα+1−2s 1,∞ , J₃⁰ ≤C2^−jαku⁽ⁿ⁺¹⁾kB^α_1,∞ku⁽ⁿ⁾k_B^α+1−2s

1,∞ ,

(4.3)

where we used the fact thatku⁽ⁿ⁾ k_B^α+1−2s

1,∞ ≤ ku⁽ⁿ⁾k_B^α+1−2s

1,∞ . Also, we have J₂⁰ =

Z

∇(−∆)^−su⁽ⁿ⁾ )∇(|∆j(u⁽ⁿ⁺¹⁾)|)

= Z

(−∆)^1−su⁽ⁿ⁾ |∆_ju⁽ⁿ⁺¹⁾|dx

≤C2^−jαku⁽ⁿ⁺¹⁾k_B1,∞^α ku⁽ⁿ⁾k_Bα+1−2s 1,∞ ,

(4.4)

and

J₄⁰ =−

Z ∆ju⁽ⁿ⁺¹⁾

|∆ju⁽ⁿ⁺¹⁾|u⁽ⁿ⁺¹⁾∆j((−∆)^1−su⁽ⁿ⁾ )

=

Z ∆ju⁽ⁿ⁺¹⁾

|∆ju⁽ⁿ⁺¹⁾|u⁽ⁿ⁺¹⁾∆_j(∆(−∆)^−su⁽ⁿ⁾ )

=−

Z ∆_ju⁽ⁿ⁺¹⁾

|∆ju⁽ⁿ⁺¹⁾|∇u⁽ⁿ⁺¹⁾∆_j(∇(−∆)^−su⁽ⁿ⁾ )

≤C2^−jαku⁽ⁿ⁾k_Bα+1−2s

1,∞ ku⁽ⁿ⁺¹⁾kB_1,∞^α .

(4.5)

On the other hand, d

dt Z

|u⁽ⁿ⁺¹⁾|dx=

Z u⁽ⁿ⁺¹⁾

|u⁽ⁿ⁺¹⁾|u⁽ⁿ⁺¹⁾_t dx= Z

∇ ·(|u⁽ⁿ⁺¹⁾|∇(−∆)^−su⁽ⁿ⁾ ) = 0.

Collecting the estimates above, whens≥1/2, we obtain d

dtku⁽ⁿ⁺¹⁾k_B1,∞^α ≤Cku⁽ⁿ⁺¹⁾k_B1,∞^α ku⁽ⁿ⁾k_Bα+1−2s 1,∞

≤Cku⁽ⁿ⁺¹⁾kB_1,∞^α ku⁽ⁿ⁾kB^α_1,∞.

(4.6)

Now from Gronwall’s inequality we obtain ku⁽ⁿ⁺¹⁾k_B^α

1,∞ ≤ ku⁽ⁿ⁺¹⁾₀ k_B^α

1,∞exp{C Z t

0

ku⁽ⁿ⁾k_B^α

1,∞)dτ}

≤Cku0kB_1,∞^α exp{C Z t

0

ku⁽ⁿ⁾kB_1,∞^α )dτ}, withC independent ofn. Defining X_T :=C([0, T];B_1,∞^α ), we have

ku⁽ⁿ⁺¹⁾kX_T ≤Cku0kB_1,∞^α exp(CTku⁽ⁿ⁾kX_T).

Thus, by the standard induction argument we find sup

0≤t≤T₀

ku⁽ⁿ⁾kB_1,∞^α ≤2Cku0kB^α_1,∞, (4.7) for all n ≥ 1, if exp(2CT₀ku₀k_B^α_1,∞) ≤ 2; namely, T₀ ≤ _2C ^{ln 2}

0(1+ku0k_Bα

1,∞). Using Proposition 3.2, 3.3, and (4.4), (4.5), we see

d

dtku⁽ⁿ⁺¹⁾k_Bα+2s−2

1,∞ ≤Cku⁽ⁿ⁺¹⁾k_Bα+2s−2

1,∞ ku⁽ⁿ⁾k_B^α

1,∞. (4.8)

Thus, from the uniform estimate (4.7), we have sup

0≤t≤T₀

k∂

∂tu⁽ⁿ⁾k_Bα+2s−2

1,∞ ≤Cku₀k²_Bα

1,∞. (4.9)

LetY_T :=C([0, T];B_1,∞^α+2s−2). We will prove that the sequence{u⁽ⁿ⁾} is Cauchy in YT₁ for some T1∈(0, T0). Considering the differenceu⁽ⁿ⁺¹⁾−u⁽ⁿ⁾,

∂_t(u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾)

=∇u⁽ⁿ⁺¹⁾· ∇(−∆)^−su⁽ⁿ⁾ −u⁽ⁿ⁺¹⁾(−∆)^1−su⁽ⁿ⁾ − ∇u⁽ⁿ⁾· ∇(−∆)^−su⁽ⁿ⁻¹⁾ +u⁽ⁿ⁾(−∆)^1−su⁽ⁿ⁻¹⁾

=∇ ·((u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾)∇(−∆)^−su⁽ⁿ⁾ ) +∇ ·(u⁽ⁿ⁾∇(−∆)^−s(u⁽ⁿ⁾ −u⁽ⁿ⁻¹⁾ )), with initial datum (u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾)(x,0) = ∆n+1u0. Proceeding as in the proof of (4.6), we obtain the estimate

k∇ ·(u⁽ⁿ⁾∇(−∆)^−s(u⁽ⁿ⁾ −u⁽ⁿ⁻¹⁾ ))k_Bα+2s−2

1,∞ ≤Cku⁽ⁿ⁾−u⁽ⁿ⁻¹⁾k_Bα−1

1,∞ku⁽ⁿ⁾kB_1,∞^α . Proceeding as in the proof of (4.8), we obtain the estimate

k∇ ·((u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾)∇(−∆)^−su⁽ⁿ⁾ )k_B^α+2s−2

1,∞ ≤Cku⁽ⁿ⁺¹⁾−u⁽ⁿ⁾k_B^α+2s−2

1,∞ ku⁽ⁿ⁾kB_1,∞^α . Then we obtain

d

dtku⁽ⁿ⁺¹⁾−u⁽ⁿ⁾k_B^α+2s−2

1,∞ ≤Cku⁽ⁿ⁺¹⁾−u⁽ⁿ⁾k_B^α+2s−2

1,∞ ku⁽ⁿ⁾kB^α_1,∞

+Cku⁽ⁿ⁾−u⁽ⁿ⁻¹⁾k_B^α+2s−2

1,∞ ku⁽ⁿ⁾kB^α_1,∞, and

k(u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾)(x,0)k_Bα+2s−2 1,∞

=k∆n+1u₀k_B^α+2s−2

1,∞ = sup

j

2^j(α+2s−2)k∆j∆_n+1u₀kL¹

≤C2^n(2s−2) sup

n≤j≤n+2

2^jαk∆_ju₀k_L1

≤C2^n(2s−2)ku0kB^α_1,∞.

The above inequalities and Gronwall’s inequality imply that ku⁽ⁿ⁺¹⁾(t)−u⁽ⁿ⁾(t)k_Bα+2s−2

1,∞

≤ k(u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾)(x,0)k_B^α+2s−2

1,∞ exp(Cku0kB^α_1,∞) +

Z t

0

exp(Cku0kB_1,∞^α (t−s))ku⁽ⁿ⁾−u⁽ⁿ⁻¹⁾k_Bα+2s−2 1,∞ (s)ds

≤C⁰2^n(2s−2)+C⁰ku⁽ⁿ⁾−u⁽ⁿ⁻¹⁾kYT1(expC⁰T1−1),

(4.10)

whereT1∈[0, T0], and the constantC⁰=C⁰(ku0kB^α_1,∞). Thus, ifC⁰(expT1−1)<

1

2, we can deduce that u⁽ⁿ⁾ converges to u ∈ L^∞([0, T1];B_1,∞^α+2s−2) in YT₁. By the well-known interpolation inequality in the Besov spaces we have uⁿ → u in L^∞([0, T₁];B_1,∞^β ) for allβ ∈[α+ 2s−2, α]. Moreover, the estimate (4.8) allows us to conclude thatu∈C¹([0, T1];B_1,∞^α+2s−2). Letting→0, n→ ∞,we find thatuis a solution to (1.1) inB^β_1,∞.

Next we prove that the solution is unique. Suppose thatu,vinL^∞([0, T1];B_1,∞^β ) are two solutions of (1.1) associated with the initial condition u0, v0. Thenu−v satisfies the equation

u−v=∇ ·[(u−v)∇(−∆)^−su] +∇ ·[v∇(−∆)^−s(u−v)], (u−v)(x,0) =u0−v0.

Working similarly with (4.10), we can obtain k(u−v)(t)k_Bβ

1,∞≤M2^j(β−α)k(u−v)(x,0)k_Bβ 1,∞

+Mku−vk_C([0,T

1];B^β_1,∞)(expC⁰⁰T1−1), forM =M(k(u0k_Bβ

1,∞,kv0k_Bβ

1,∞). IfM(expC⁰⁰T1−1)<1, then we get the unique- ness of solutions inC([0, T₁];B_1,∞^β ). The proof is complete.

5. Proof of Theorem 1.2

Whend≥3, we know (−∆)⁻¹u=c(n, s)|x|^−d+2∗u. LetPu=c(n, s)(|x|^−d+2∗ σ)∗u= (−∆)⁻¹(σ∗u) = (−∆)⁻¹u, whereσis defined in the beginning of Section 4. Consider the successive approximation sequence{u⁽ⁿ⁾}satisfying

u⁽¹⁾=S2(u0)

∂_tu⁽ⁿ⁺¹⁾=∇ ·(u⁽ⁿ⁺¹⁾∇(Pu⁽ⁿ⁾ )) =∇u⁽ⁿ⁺¹⁾· ∇(−∆)⁻¹u⁽ⁿ⁾ −u⁽ⁿ⁺¹⁾u⁽ⁿ⁾ u⁽ⁿ⁺¹⁾(x,0) =uⁿ⁺¹₀ =S_n+2u₀.

(5.1)

Appling ∆j on (5.1),

∂t∆ju⁽ⁿ⁺¹⁾=X

[∆j, ∂i(−∆)^−su⁽ⁿ⁾ ]∂iu⁽ⁿ⁺¹⁾

+∇(−∆)^−su⁽ⁿ⁾ ∇∆j(u⁽ⁿ⁺¹⁾)−∆j(u⁽ⁿ⁺¹⁾u⁽ⁿ⁾ ).

(5.2) Whenp≥2, multiplying both sides of (5.2) byp∆ju⁽ⁿ⁺¹⁾|∆ju⁽ⁿ⁺¹⁾|^p−2 and inte- grating overR^d, we obtain

d

dtk∆ju⁽ⁿ⁺¹⁾k^p_Lp =XZ

p∆ju⁽ⁿ⁺¹⁾|∆ju⁽ⁿ⁺¹⁾|^p−2[∆j, ∂i(−∆)⁻¹u⁽ⁿ⁾ ]∂iu⁽ⁿ⁺¹⁾

+ Z

p∆ju⁽ⁿ⁺¹⁾|∆ju⁽ⁿ⁺¹⁾|^p−2∇(−∆)⁻¹u⁽ⁿ⁾ ∇∆j(u⁽ⁿ⁺¹⁾)

− Z

p∆_ju⁽ⁿ⁺¹⁾|∆_ju⁽ⁿ⁺¹⁾|^p−2∆_j(u⁽ⁿ⁺¹⁾u⁽ⁿ⁾_ε ) =J₁⁰⁰+J₂⁰⁰+J₃⁰⁰. By H¨older’s inequality,

J₁⁰⁰≤Ck∆ju⁽ⁿ⁺¹⁾k^p−1_Lp k[∆j, ∂i(−∆)⁻¹u⁽ⁿ⁾ ]∂iu⁽ⁿ⁺¹⁾kL^p. From the proof of Proposition 2.3 (iv),

J₃⁰⁰≤ k∆ju⁽ⁿ⁺¹⁾k^p−1_Lp [ku⁽ⁿ⁾kL^∞k∆ju⁽ⁿ⁺¹⁾kL^p+ku⁽ⁿ⁺¹⁾kL^∞k∆ju⁽ⁿ⁾kL^p].

Also we obtain J₂⁰⁰=

Z

∇(−∆)⁻¹u⁽ⁿ⁾ ∇(|∆j(u⁽ⁿ⁺¹⁾)|^p)

= Z

u⁽ⁿ⁾ |∆ju⁽ⁿ⁺¹⁾|^pdx≤ ku⁽ⁿ⁾kL^∞k∆ju⁽ⁿ⁺¹⁾k^p_Lp. Now we obtain

d

dtk∆ju⁽ⁿ⁺¹⁾kL^p≤Ck[∆j, ∂i(−∆)⁻¹u⁽ⁿ⁾]∂iu⁽ⁿ⁺¹⁾kL^p

+Cku⁽ⁿ⁾k_L^∞k∆_ju⁽ⁿ⁺¹⁾k_Lp+Cku⁽ⁿ⁺¹⁾k_L^∞k∆_ju⁽ⁿ⁾k_Lp. When,α > d/p,

d

dtku⁽ⁿ⁺¹⁾kB˙^α_p,∞ ≤Cku⁽ⁿ⁺¹⁾k_Bα

p,∞ku⁽ⁿ⁾k_Bα p,∞. Lettingp→ ∞,

d

dtk∆ju⁽ⁿ⁺¹⁾kL^∞≤Ck[∆j, ∂i(−∆)⁻¹u⁽ⁿ⁾]∂iu⁽ⁿ⁺¹⁾kL^∞

+Cku⁽ⁿ⁾kL^∞k∆ju⁽ⁿ⁺¹⁾kL^∞+Ck∆j(u⁽ⁿ⁺¹⁾u⁽ⁿ⁾)kL^∞. So for anyα >0,

d

dtku⁽ⁿ⁺¹⁾kB˙^α_∞,∞ ≤Cku⁽ⁿ⁺¹⁾k_Bα

∞,∞ku⁽ⁿ⁾k_Bα

∞,∞. It is easy to prove that whenp= 1,

d dt

Z

|u⁽ⁿ⁺¹⁾|dx=

Z u⁽ⁿ⁺¹⁾

|u⁽ⁿ⁺¹⁾|u⁽ⁿ⁺¹⁾_t dx

= Z

∇ ·(|u⁽ⁿ⁺¹⁾|(|x|^−d+2∗σ)∗u⁽ⁿ⁾) = 0.

Notingu⁽ⁿ⁺¹⁾(x,0) =uⁿ⁺¹₀ =S_n+2u₀, we knowku⁽ⁿ⁺¹⁾kL¹≤C.

Whenp≥2, d dt

Z

|u⁽ⁿ⁺¹⁾|^pdx=p Z

|u⁽ⁿ⁺¹⁾|^p−2u⁽ⁿ⁺¹⁾u⁽ⁿ⁺¹⁾_t dx

=p Z

|u⁽ⁿ⁺¹⁾|^p−2u⁽ⁿ⁺¹⁾∇ ·(u⁽ⁿ⁺¹⁾∇(−∆)⁻¹u⁽ⁿ⁾ )

=−(p−1) Z

∇|u⁽ⁿ⁺¹⁾|^p·(∇(−∆)⁻¹u⁽ⁿ⁾ )

= (p−1) Z

|u⁽ⁿ⁺¹⁾|^pu⁽ⁿ⁾ ≤Cku⁽ⁿ⁺¹⁾k^p_Lpku⁽ⁿ⁾kL^∞.

It means d

dtku⁽ⁿ⁺¹⁾kL^p≤ ku⁽ⁿ⁺¹⁾kL^pku⁽ⁿ⁾kL^∞ ≤Cku⁽ⁿ⁺¹⁾kB^α_p,∞ku⁽ⁿ⁾kB_p,∞^α . Again lettingp→ ∞, we obtain

d

dtku⁽ⁿ⁺¹⁾k_L^∞ ≤Cku⁽ⁿ⁺¹⁾k_Bα

∞,∞ku⁽ⁿ⁾k_Bα

∞,∞. Collecting the estimates above, we now obtain

d

dtku⁽ⁿ⁺¹⁾kB^α_p,∞≤Cku⁽ⁿ⁺¹⁾kB_p,∞^α ku⁽ⁿ⁾kB^α_p,∞, p <∞, α > d p, d

dtku⁽ⁿ⁺¹⁾kB^α_∞,∞ ≤Cku⁽ⁿ⁺¹⁾kB^α_∞,∞ku⁽ⁿ⁾kB_∞,∞^α , p=∞, α >0.

For anyβ∈(d/p, α),p <∞, orβ ∈(0, α),p=∞we have d

dtku⁽ⁿ⁺¹⁾k_Bβ

p,∞ ≤ ku⁽ⁿ⁺¹⁾k_Bβ

p,∞ku⁽ⁿ⁾k_Bβ p,∞, and

d

dtku⁽ⁿ⁺¹⁾−u⁽ⁿ⁾k_Bβ

p,∞≤ ku⁽ⁿ⁺¹⁾−u⁽ⁿ⁾k_Bβ

p,∞ku⁽ⁿ⁾k_Bβ p,∞

+ku⁽ⁿ⁾−u⁽ⁿ⁻¹⁾k_Bβ

p,∞ku⁽ⁿ⁾k_Bβ p,∞. The rest of the proof is similar to the proof of Theorem 1.1, we omit it.

Acknowledgments. This research was supported by the National Natural Science Foundation of China (11201103).

References

[1] L. A. Caffarelli, F. Soria, J. L. V´azquez; Regularity of solutions of the fractional porous medium flow, arXiv preprint arXiv:1201.6048, 2012.

[2] L. A. Caffarelli, J. L. V´azquez;Nonlinear porous medium flow with fractional potential pres- sure, Arch.Rational Mech.Anal, 202(2011), 537-565.

[3] D. Chae; The quasi-geostrophic equation in the TriebelCLizorkin spaces, Nonlinearity, 16(2003), 479-495.

[4] Q. Chen, C. Miao, Z. Zhang; A new Bernstein’s inequality and the 2D dissipative quasi- geostrophic equation, Communications in mathematical physics, 271(2007), 821-838.

[5] A. C´ordoba, D. C´ordoba;A maximum principle applied to quasi-geostrophic equations, Com- munications in mathematical physics, 249(2004), 511-528.

[6] P. G. Lemar¨ı-Rieusset;Recent development in the navier-stokes problem, CRC Press, 2002.

[7] F. Lin, P. Zhang;On the hydrodynamic limit of Ginzburg-Landau wave vortices, Communi- cations on pure and applied mathematics, 55(2002), 831-856.

[8] H. Triebel;Theory of Function Spaces. Monograph in Mathematics, Vol.78, Basel: Birkhauser Verlag, 1983.

[9] J. L. V´azquez;The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, 2006.

[10] J. L. V´azquez; Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations. Springer Berlin Heidelberg, 2012: 271-298.

[11] J. L. V´azquez, S. Serfaty;A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calculus of Variations and Partial Differential Equations, 49 (2014), 1091-1120.

[12] J. Wu; Solutions of the 2D quasi-geostrophic equation in H¨older spaces, Nonlinear Anal,62(2005), 579-594.

[13] J. Wu; Lower bounds for an integral involving fractional Laplacians and the general- ized Navier-Stokes equations in Besov spaces, Communications in mathematical physics, 263(2006), 803-831.

[14] J. Wu;Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation, Nonlinear Anal,67(2007), 3013-3036.

Xuhuan Zhou

Department of Mathematics, Zhejiang University, 310027 Hangzhou, China E-mail address:[email protected]

Weiliang Xiao

Department of Mathematics, Zhejiang University, 310027 Hangzhou, China E-mail address:[email protected]

Jiecheng Chen

Department of Mathematics, Zhejiang Normal University, 321004 Jinhua, China E-mail address:[email protected]