Introduction We consider the porous medium type equation ∂tu=∇ ·(u∇p), p= (−∆)−su, 0&lt

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

WELL-POSEDNESS OF A POROUS MEDIUM FLOW WITH FRACTIONAL PRESSURE IN SOBOLEV SPACES

XUHUAN ZHOU, WEILIANG XIAO

Abstract. We prove the existence of a non-negative solution for a linear degenerate diffusion transport equation from which we derive the existence and uniqueness of the solution for the fractional porous medium equation in Sobolev spacesH^αwith nonnegative initial data,α >^d₂+ 1. We also correct a mistake in our previous paper [14].

1. Introduction We consider the porous medium type equation

∂_tu=∇ ·(u∇p), p= (−∆)^−su, 0< s <1, u(x,0)≥0, (1.1) where x ∈ Rⁿ, n ≥ 2, and t > 0 and the fractional Laplacian (−4)^s/2 := Λ^s is given by the psuedo differential operator with symbol|ξ|^s, that is:

(−4)^s/2f = Λ^sf =F⁻¹|ξ|^sFf.

Using the Riesz potential, one can also define this operator as (−4)^s/2f(x) = Λ^sf(x) =cn,s

Z

Rⁿ

f(x)−f(y)

|y|^n+s dy.

This model is based on Darcy’s law with pressure, p, is given by an inverse fractional Laplacian operator. It was first introduced by Caffarelli and V´azquez [4], in which they proved the existence of a weak solution whenu0is a bounded function with exponential decay at infinity. Forα= _n+2−2sⁿ , Caffarelli, Soria and V´azquez [3] proved that the bounded nonnegative solutions areC^α continuous in a strip of space-time fors6= 1/2. And same conclusion for the indexs= 1/2 was proved by Caffarelli and V´azquez in [5]. [7, 6, 13] give a detailed description of the large-time asymptotic behaviour of the solutions of (1.1). [2, 12] consider degenerate cases and show the existence and properties of self-similar solutions. Allen, Caffarelli and Vasseur [1] study the equation with fractional time derivative, and proved the H¨older continuity for its weak solutions.

In this paper, we study the existence and uniqueness of solutions of (1.1) in Sobolev spaces. Unlike considering the existence of weak solution in L^∞ or con- structing approximate solutions of linear transport systems, we solve equation (1.1) by constructing solutions to a linear degenerate diffusion transport systems. The

2010Mathematics Subject Classification. 35K55, 35K65, 76S05.

Key words and phrases. Fractional porous medium equation; Sobolev space;

degenerate diffusion transport equation.

c

2017 Texas State University.

Submitted December 22, 2016. Published October 3, 2017.

1

well-posedness and properties of the linear degenerate diffusion transport are inter- esting results by themselves and lead us to proving that for s∈[¹₂,1), α > ^d₂+ 1, u0∈H^α(Rⁿ) nonnegative, and some T0>0, the equation (1.1) inRⁿ×[0, T0] has a unique solutions. Besides, using the methods and results in this paper, we correct a mistake in our previous paper [14].

2. Preliminaries Defineρ∈C_c^∞(Rⁿ) by

ρ(x) =

(c0exp(−_1−|x|¹ 2),|x|<1, 0,|x| ≥1,

wherec₀is selected such thatR

ρ(x)dx= 1. LetJ be defined by Ju=ρ∗u=⁻ⁿρ(·

)∗u.

This operator satisfies the following properties.

Proposition 2.1. (1) Λ^sJu=JΛ^su,s∈R.

(2) For allu∈L^p(Rⁿ),v∈H^α(Rⁿ), with ¹_p+¹_q = 1,R

(Jf)g=R

f(Jg).

(3) For allu∈H^α(Rⁿ),

→0limkJu−ukH^α = 0, lim

→0kJu−ukH^α−1 ≤CkukH^α. (4) For allu∈H^α(Rⁿ),s∈R,k∈Z∪ {0}, then

kJuk_Hα+k≤ Cαk

^k kukH^α, kJD^kukL^∞ ≤ Ck

^n2+kkukH^α

The following propositions can be found in [8, 9].

Proposition 2.2. Suppose that s >0 and1 < p <∞. If f, g ∈ S, the Schwartz class, then we have

kΛ^s(f g)−fΛ^sgkL^p≤ck∇fkL^p1kgkH˙^s−1,p2+ckgkL^p4kfkH˙^s,p3, kΛ^s(f g)k_Lp≤ckfk_L^p1kgkH˙^s,p2 +ckgk_L^p4kfkH˙^s,p3

withp2, p3∈(1,+∞)such that ¹_p =_p¹

1 +_p¹

2 = _p¹

3 +_p¹

4. Proposition 2.3. If 0≤s≤2,f ∈ S(Rⁿ), then

2f(x)Λ^sf(x)≥Λ^sf²(x) for allx∈Rⁿ.

Proposition 2.4. Let α1 and α2 be two real numbers such that α1 < ⁿ₂,α2< ⁿ₂ and α1+α2 >0. Then there exists a constant C =Cα1,α2 ≥0 such that for all f ∈H˙^α1 andg∈H˙^α2,

kf gkH˙^α≤CkfkH˙^α1kgkH˙^α2, whereα=α₁+α₂−ⁿ₂.

3. Main results

Theorem 3.1. If s ∈ [1/2,1], T > 0, α > ⁿ₂ + 1, u0 ∈ H^α(Rⁿ), v ≥ 0 and v∈C([0, T];H^α(Rⁿ)), then the linear initial-value problem

∂tu=∇u· ∇(−∆)^−sv−v(−∆)^1−su,

u(x,0) =u0. (3.1)

has a unique solution u ∈ C¹([0, T];H^α(Rⁿ)). If the initial data u0 ≥ 0, then u≥0,(x, t)∈Rⁿ×[0, T].

Proof. For any >0, we consider the linear problem

∂tu=F(u) =J(∇Ju· ∇(−∆)^−sv)−J(v(−∆)^1−sJu),

u(x,0) =u₀. (3.2)

By Propositions 2.1 and 2.2 and bys≥ ¹₂ we can estimate kF(u₁)−F(u₂)k_Hα

=kJ(∇J(u₁−u₂)· ∇(−∆)^−sv)−J(v(−∆)^1−sJ(u₁−u₂)kH^α

≤C(,kvkH^α)ku₁−u₂kH^α.

Using Picard iterations, for any α > ⁿ₂ + 1, > 0, there exists aT =T(u_∗)>0, problem (3.2) has a unique solution u∈C¹([0, T);H^α). By Propositions 2.1 and 2.3,

1 2

d

dtkuk²_L2 = Z

∇Ju· ∇(−∆)^−svJu− Z

v(−∆)^1−sJuJu

≤1 2

Z

∇|Ju|²· ∇(−∆)^−sv−1 2

Z

v(−∆)^1−s|Ju|²

≤1 2

Z

|Ju|²(−∆)^1−sv−1 2

Z

|Ju|²(−∆)^1−sv= 0.

Moreover, for anyα >0, 1

2 d

dtkΛ^αuk²_L2

= Z

Λ^α(∇Ju· ∇(−∆)^−sv)JΛ^αu− Z

Λ^α(v(−∆)^1−sJu)Λ^αJu

≤Ck[Λ^α,∇(−∆)^−sv]∇JukL²kΛ^αukL²+ Z

∇(−∆)^−svΛ^α∇JuΛ^αJu +Ck[Λ^α, v](−∆)^1−sJuk_L2kΛ^αuk_L2−

Z

vΛ^α(−∆)^1−sJuΛ^αJu. By Proposition 2.2 and Sobolev embeddings,

k[Λ^α,∇(−∆)^−sv]∇Juk_L2

≤Ck(−∆)^1−svk_L^∞kΛ^α−1∇Juk_L2+k∇(−∆)^−svkH˙^αk∇Juk_L^∞

≤CkvkH^αkukH^α, k[Λ^α, v](−∆)^1−sJuk_L2

≤Ck∇vkL^∞k(−∆)^1−sJukH˙^α−1+kvkH˙^αk(−∆)^1−sJukL^∞

≤Ckvk_Hαkuk_Hα.

By Proposition 2.3, Z

∇(−∆)^−svΛ^α∇JuΛ^αJu− Z

vΛ^α(−∆)^1−sJuΛ^αJu

≤ 1 2

Z

∇(−∆)^−sv∇(Λ^αJu)²−1 2

Z

v(−∆)^1−s(Λ^αJu)²

≤CkvkH^αkuk²_Hα. Combining the above estimates,

d

dtku(·, t)kH^α ≤CkvkH^αkukH^α. By Gronwall’s inequality,

ku(·, t)kH^α≤ ku0kH^αexp(C sup

0≤t≤T

kvkH^α).

Such the solutionu exists on [0, T]. Similarly, d

dtku(·, t)k_Hα−1≤Ckvk_Hαkuk_Hα ≤C(kvk_Hα,ku₀k_Hα, T).

By Aubin compactness theorem [11], there is a subsequence of{uⁿ¹}_n≥1 that con- vergence strongly touin C([0, T];H^α). Ifα > ^d₂+ 1, H^α,→C¹, so uis a solution of (3.3).

Ifuand ˜uare two solutions of problem (3.3), thenw=u−˜usatisfies

∂_tw=∇w· ∇(−∆)^−sv−v(−∆)^1−sw, w(x,0) = 0.

Similarly, we get _dt^dkwkL² ≤ 0 and _dt^dkwkH˙^α ≤ kvkH^αkwkH^α, i.e, _dt^dkwkH^α ≤ kvk_Hαkwk_Hα. By Gronwall’s inequality,u(x, t) = 0, (x, t)∈Rⁿ×[0, T].

Since u₀ ≥ 0 then if there exists a first time t₀ where for some point x₀ we haveu(x₀, t₀) = 0, then (x₀, t₀) will correspond to a minimum point and therefore

∇u(x0, t0) = 0, and

(−∆)^1−su(x) =c

Z u(x)−u(y)

|y|^n+2−2s dy≤0.

Henceut|(x₀,t₀)≥0. Sou(x, t)≥0 for all (x, t)∈Rⁿ×[0, T].

Theorem 3.2. Let n≥2,s∈[¹₂,1),α >^d₂+ 1,u₀∈H^α(Rⁿ), andu₀≥0. Then there the linear initial value problem

∂_tu=∇ ·(u∇(−∆)^−su), u(x,0) =u0.

has a unique solution u ∈ C¹([0, T0], H^α(Rⁿ)). If the initial data u0 ≥ 0, then u≥0,(x, t)∈Rⁿ×[0, T0].

Proof. Set u¹ = u0. Note that ∂tu = ∇ ·(u∇(−∆)^−su) = ∇u· ∇(−∆)^−su− u(−∆)^1−su, and let{uⁿ} be the sequence defined by

∂_tuⁿ⁺¹=∇uⁿ⁺¹· ∇(−∆)^−suⁿ−uⁿ(−∆)^1−su⁽ⁿ⁺¹⁾,

uⁿ⁺¹(x,0) =u₀. (3.3)

By Theorem 3.1,u²∈C([0, T);H^α), for allT <∞, satisfiesu²≥0 and sup

0≤t≤T

ku²kH^α ≤ ku0kH^αexp(Cku¹kH^αT).

If exp(2Cku1kH^αT0)≤2, for exampleT0=_2C(1+ku^{ln 2}

0k_Hα), we have sup

0≤t≤T0

ku²kH^α ≤2ku0kH^α.

By the standard induction argument, ifuⁿ∈C([0, T0];H^α),uⁿ≥0 is a solution of (3.3) withkuⁿkH^α ≤2ku0kH^α. By Theorem 3.1 uⁿ⁺¹∈C([0, T0];H^α) , uⁿ⁺¹≥0 and

sup

0≤t≤T0

kuⁿ⁺¹kH^α≤ ku0kH^αexp(CkuⁿkH^αT0)≤2ku0kH^α, d

dtkuⁿ⁺¹k_H^α−1 ≤CkuⁿkH^αkuⁿ⁺¹kH^α≤Cku0k²_Hα.

By Aubin compactness theorem [11], there is a subsequence ofuⁿ that convergence strongly touin C([0, T];H^α). Ifu≥0,u˜≥0 are two solutions of problem (3.3), thenw=u−u˜satisfies

∂tw=∇ ·(w∇(−∆)^−su) +∇ ·(˜u∇(−∆)^−sw), w(x,0) = 0.

By Proposition 2.2, 1

2 d

dtkwk²_L2= Z

w∇ ·(w∇(−∆)^−su) + Z

w∇˜u· ∇(−∆)^−sw− Z

w˜u(−∆)^1−sw

=:I1+I2+I3. Note that

I1= Z

w∇w· ∇(−∆)^−su= 1 2

Z

∇w²· ∇(−∆)^−su

= 1 2

Z

w²(−∆)^1−su≤Ckuk_Hαkwk²_L2, I₃≤ −1

2 Z

u(−∆)˜ ^1−sw²= Z

−1

2(−∆)^1−su˜·w²≤CkukH^αkwk²_L2. Whens >1/2,

I₂≤Ckwk_L2k∇u· ∇(−∆)^−swk_L2

≤Ckwk_L2k∇uk_H˙ⁿ₂+1−2sk∇(−∆)^−swkH˙^2s−1 ≤Ckuk_Hαkwk²_L2.

When s = 1/2, the above estimates are still valid. Combining the above esti- mates we have _dt^dkwk_L2 ≤Ckwk_L2kukH^α. By Gronwall’s inequality we can deduce

w(x, t) = 0 on [0, T0].

4. Correction

In [14], trying to establish the well-posedness of (1.1) in Besov spaces the authors incurred in a mistake in page 9 when estimating the termJ₄⁰ in equation [14, (4.5)].

To correct the mistake, we modify our proof the following way.

[14, Theorem 1.1] Let n ≥ 2, s ∈ [¹₂,1], α > n+ 1. If the initial data u0 ∈ B_1,∞^α , then there existsT =T(ku0kB^α_1,∞) such that (1.1) has a unique solution in

[0, T]×Rⁿ. Such a solution belongs toC¹([0, T];B_1,∞^α+2s−2)∩L^∞([0, T];B_1,∞^β ), with β∈[α+ 2s−2, α].

Proof. First we construct the approximate equation

u⁽ⁿ⁺¹⁾_t =∇u⁽ⁿ⁺¹⁾· ∇(−4)^−su⁽ⁿ⁾ −u⁽ⁿ⁾ (−4)^1−su⁽ⁿ⁺¹⁾; u⁽ⁿ⁺¹⁾(0) =σ∗u0, u⁽¹⁾ =σ∗u0.

(4.1) By the argument in section 2, there exists a sequence u⁽ⁿ⁾ that solves the linear systems (4.1). Assuming thatu0 ≥0, we prove thatu⁽ⁿ⁺¹⁾ ≥0. Inspired by [4], we assume thatx₀ is a point of minimum of u⁽ⁿ⁺¹⁾ at timet=t₀. This indicates that∇u⁽ⁿ⁺¹⁾(x₀) = 0, and

(−4)^1−su⁽ⁿ⁺¹⁾(x₀) =c

Z u(x₀)−u(y)

|y|^n+2(1−s) dy≤0.

Thus we deduce that _∂t^∂u⁽ⁿ⁺¹⁾ _t=t

0 ≥ 0, and by induction we have u⁽ⁿ⁺¹⁾ ≥ 0.

Arguing as in [14], taking4j on (4.1), we obtain

∂t4ju⁽ⁿ⁺¹⁾=X

[4j, ∂i(−4)^−su⁽ⁿ⁾ ]∂iuⁿ⁺¹+X

∂i(−4)^−su⁽ⁿ⁾ 4j(∂iu⁽ⁿ⁺¹⁾)

−[4j, u⁽ⁿ⁾ ](−4)^1−su⁽ⁿ⁺¹⁾−u⁽ⁿ⁾ 4j(−4)^1−su⁽ⁿ⁺¹⁾. Multiplying both sides by ^4ju(n+1)

|4ju⁽ⁿ⁺¹⁾|, and integrating over R^d, then denote the corresponding part in the right side by J₁⁰, J₂⁰, J₃⁰, J₄⁰, respectively. We obtain the estimates,

J₁⁰ ≤C2^−jαku⁽ⁿ⁺¹⁾kB_1,∞^α ku⁽ⁿ⁾k_B^α+1−2s

1,∞

J₂⁰ ≤C2^−jαku⁽ⁿ⁺¹⁾kB_1,∞^α ku⁽ⁿ⁾k_B^α+1−2s

1,∞

J₃⁰ ≤C2^−jαku⁽ⁿ⁾k_B^α

1,∞ku⁽ⁿ⁺¹⁾k_Bα+1−2s

1,∞ .

The estimate for the termJ₄⁰ is now replaced by J₄⁰ =−

Z

u⁽ⁿ⁾4j(−4)^1−su⁽ⁿ⁺¹⁾ 4ju⁽ⁿ⁺¹⁾

|4ju⁽ⁿ⁺¹⁾|

≤ − Z

u⁽ⁿ⁾(−4)^1−s|4ju⁽ⁿ⁺¹⁾|

≤ − Z

(−4)^1−su⁽ⁿ⁾|4u⁽ⁿ⁺¹⁾|

≤2^−jαkuⁿk_B^r+2−2s

1,∞ ku⁽ⁿ⁺¹⁾kB^α_1,∞.

Here r > d is any real number. The first inequality uses the following pointwise estimate.

Proposition 4.1 ([10]). If0≤α≤2,p≥1, then

p|f(x)|^p−2f(x)Λ^αf(x)≥Λ^α|f(x)|^p. for any f ∈ S(R^d)

Takingrsuch thatr+ 2−2s < α, e.g. setr=α−1, we conclude d

dtku⁽ⁿ⁺¹⁾k_B^α_1,,∞ ≤ ku⁽ⁿ⁾k_B1,,∞^α ku⁽ⁿ⁺¹⁾k_B1,,∞^α .

The other parts of the proof need no modification.

Acknowledgement. We are very grateful to Ph. D. Mitia Duerinckx for point- ing out our mistake and share some good views on this problem. This paper is supported by the NNSF of China under grants No. 11601223 and No.11626213.

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Xuhuan Zhou

Department of Information Technology, Nanjing Forest Police College, 210023 Nan- jing, China

E-mail address:[email protected]

Weiliang Xiao

School of Applied Mathematics, Nanjing University of Finance and Economics, 210023 Nanjing, China

E-mail address:[email protected]