Scholarly article on topic 'On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces'

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Academic research paper on topic "On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces"

0 Boundary Value Problems

a SpringerOpen Journal

RESEARCH

Open Access

On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin

spaces

Wenxin Yu1* and Yigang He2

"Correspondence: slowbird@sohu.com 1College of Electricaland Information Engineering, Hunan University, Changsha, 410000, P.R. China

Fulllist of author information is available at the end of the article

Abstract

In this paper, we prove the local well-posedness for the incompressible porous media equation in Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. The main tools we use are the Fourier localization technique and Bony's paraproduct decomposition. MSC: 76S05; 76D03

Keywords: well-posedness; incompressible porous media equation; blow-up criterion; Fourier localization; Bony's paraproduct decomposition; Triebel-Lizorkin space

ft Spri

ringer

1 Introduction

In this paper, we are concerned with the incompressible porous media equation (IPM) in Rd (d = 2 or 3):

(IPM) \dt 9 + U 'V0 = ° 0 (*0) = 00' (1.1)

lu = -k(Vp + gy9), div u = 0,

where x e Rd, t > 0, 0 is the liquid temperature, u is the liquid discharge, p is the scalar pressure, k is the matrix of position-independent medium permeabilities in the different directions, respectively, divided by the viscosity, g is the acceleration due to gravity, and Y e Rd is the last canonical vector ed. For simplicity, we only consider k = g = 1.

By rewriting Darcy's law we obtain the expression of velocity u only in terms of temperature 0 [1, 2]. In the 2D case, thanks to the incompressibility, taking the curl operator first and the Vx := (-3X2, dxi) operator second on both sides of Darcy' law, we have

-Au = vx(dxi 0 ) = (-dxi dx2 0, dxl0), thus the velocity u can be recovered as

if / d20 d20 \

u(x, t) =--ln |x -y|(--(y, t), (y, t)| dy, x e R2.

(,) 2^ Jr2 yiV dy2dyi^ ^ d 2yl(y, 7 y,

©2014 Yu and He; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Through integration by parts we finally get

2 2n Jr2

where the kernel H(■) is defined by

u(x, t) = — (0,0(x, t)) + — PW H(x -y)0(y, t) dy, x e R2,

H (x) =

2x1x2 x2 xi

jxj4 , jxj4

Similarly, in 3D case, applying the curl operator twice to Darcy's law, we get

-Au = (-91930, -92930 , 92 0 + 9220), where d¡ := d, thus

u(x, t) = — Í0,0,0(x, t)) + — PU K(x -y)0(y, t) dy, x e R3,

K (x) =

3x1x3 3x2x3 2x3

jxj5 ' jxj5 ' jxj5

We observe that, in general, each coefficient of u(-, t) (with t as parameter) is only the linear combination of the Calderon-Zygmund singular integral (with the definition see the sequel) of 0 and 0 itself. We write the general version as

where T = (Tk), C = (Ck), S = (Sk), 1 < k < N are all operators mapping scalar functions to vector-valued functions and Ck equals a constant multiplication operator whereas Sk means a Calderón-Zygmund singular integral operator. Especially the corresponding specific forms in 2D or 3D are shown as (1.2) or (1.3).

We observe that the system (IPM) is not more than a transport equation with non-local divergence-free velocity field (the specific relationship between velocity and temperature as (1.4) shows). It shares many similarities with another flow model - the 2D dissipative quasi-geostrophic (QG) equation, which has been intensively studied by many authors [3-8]. From a mathematical point of view, the system (IPM) is somewhat a generalization of the (QG) equation. Very recently, the system (IPM) was introduced and investigated by Córdoba et al. In [2], they treated the (IMP) in 2D case and obtained the local existence and uniqueness in Holder space Cs for 0 < S < 1 by the particle-trajectory method and gave some blow-up criteria of smooth solutions. Recently, they proved non-uniqueness for weak solutions of (IPM) in [9]. For the dissipative system related (IPM), in [1], the authors obtained some results on strong solutions, weak solutions and attractors. For finite energy they obtained global existence and uniqueness in the subcritical and critical cases. In the supercritical case, they obtained local results in Hs, s >(N - a)/2 + 1 and extended to be global under a small condition \\Q°||Hs < cv, for s > N/2 + 1, where c is a small fixed constant.

u := T (0 )=C (0 ) + S (0 ),

Recently, Chae studied the local well-posedness and blow-up criterion for the incompressible Euler equations [10,11], and quasi-geostrophic equations [12] in Triebel-Lizorkin spaces. As is well known, Triebel-Lizorkin spaces are the unification of several classical function spaces such as Lebegue spaces Lp(Rd), Sobolev spaces Hp(Rd), Lipschitz spaces Cs(Rd), and so on. In [10], the author first used the Littlewood-Paley operator to localize the Euler equation to the frequency annulus (|f | ~ 2j}, then obtained an integral representation of the frequency-localized solution on the Lagrangian coordinates by introducing a family of particle-trajectory mappings (Xj(a, t)} defined by

Uja, t) = (Sj-2V)(Xj(a, t), t), |x;(a,0) = a,

where v is a divergence-free velocity field and Sj-2 is a frequency projection to the ball (If I ^ 2j} (see Section 2). He also used the following equivalent relation:

fe 2jsq\Ajv(Xj(a, t))\qV

V'eZ '

to estimate the frequency-localized solutions of the Euler equations or quasi-geostrophic equations in Triebel-Lizorkin spaces. However, it seems difficult to give a strict proof for the above equivalent relation (1.6) and its related counterpart due to the lack of a uniform change of the coordinates independent of j. To avoid this trouble, Chen et al. [13] introduced a particle trajectory mapping X(a, t) independent of j defined by

dtX (a, t) = v(X(a, t), t), X(a, 0) = a,

and then established a new commutator estimate to obtain the local well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces.

In this paper, we will adapt the method of Chen et al. [13] to establish the local well-posedness for the incompressible porous media equation (1.1) and to obtain a blow-up criterion of smooth solutions in the framework of Triebel-Lizorkin spaces. Now we state our result as follows.

JP(-da)

YJ2jsq\^iv(x)\

JP(-dx)

Theorem 1.1 (i) Local-in-time existence. Let s > p + 1, 1 < p, q < to. Assume that d0 e Fp,q(Rd), then there exists T = T(||00||Fs ) such that (1.1) has a unique solution 0 e C([0, T];Fp,q(Rd)) n Lip([0, T];F^q1(Rd))."

(ii) Blow-up criterion. The local-in-time solution 0 e C([0, T];Fpq) constructed in (i) blows up atT* > T in Fsp q, i.e.

limsup||0(t)||Fs =+to, T* < to,

if and only if

/ ||0 (t)|L 1 dt =+to.

J0 to,to

2 Preliminaries

Let B = (f e Rd, |f | < 4} and C = (f e Rd, § <|f | < |}. Choose two nonnegative smooth radial functions x, P supported, respectively, in B and C such that

x(f ) + £p(2-jf) =1, f e Rd, £p(2-jf) = 1, f e Rd\(0}.

j>0 jeZ

We denote pj(f) = p(2-jf), h = F-1p and h = F-ix. Then the dyadic blocks Aj and Sj can be defined as follows:

Ajf = p(2-jD)f = 2'dj h(2jy)f (x -y) dy,

Sjf = £ Akf = x (2-jD)f = 2d f h (2jy)f (x - y) dy.

Formally, Aj = S,+i - Sj is a frequency projection to the annulus (|f | ~ 2j}, and Sj is a frequency projection to the ball (|f | < 2j}. One easily verifies that with our choice of p

AjAkf = 0 if |j - k| > 2 and Aj(Sk-f Akf ) = 0 if |j - k|> 5. (2.1)

With the introduction of Aj and Sj, let us recall the definition of the Triebel-Lizorkin space. Let s e R,(p,q) e [1, to) x [1, to], the homogeneous Triebel-Lizorkin space Fp,q is defined by

FP,q = f e 2'(Rd); \f ||Fp,q < то},

|f \\. = f\\(£jez 2jsq Ajf q q \L, for1 < q < to, f Fp,q l\\ supjez(2js| Ajf |)\L, for q = to,

and 2'(Rd) denotes the dual space of 2(Rd) = f e 5(Rd); daf (0) = 0; Va e Nd multi-index} and can be identified by the quotient space of S'lV with the polynomials space P.

For s >0 and (p, q) e [1, to) x [1, to], we define the inhomogeneous Triebel-Lizorkin space Fp,q as follows:

Fp,q = f e S'(Rd); \f H^ < to}, where

\\f \Fp,q = \f Hip + \\f \\Fp,q.

We refer to [14] for more details.

Lemma 2.1 (Bernstein's inequality) [15] Let k e N. There exists a constant C independent off and j such that, for all 1 < p < q <to, the following inequalities hold:

suppf c {|f | < 2j} ^ sup \\daf \\Lq < C2jk+jd(pp-lq)\f \\lp,

suppfc {|f | ~ 2j} ^ \|fhip < C sup 2-jk\daf \If.

Lemma 2.2 [14] For any k e N, there exists a constant Ck such that the following inequality holds:

C-1||vkf |F <II IUk < Ck IIVkf IF .

11 Fp,q F P,q Fp,q

Proposition 2.1 [10] Let s > 0, (p, q) e (1, to) x (1, to], or p = q = to, then there exists a constant C such that

Fp,q < C(|f IlLTOlIgIIPPM + Hg^If IIPPpq), F*M < C(^if IIlto IIgIFp,q + IIgIILTOIf IIFp,q).

Proposition 2.2 [10] Let s > d/2 with p,q e [1, to]. Suppose / e Fp,q, then there exists a constant C such that the following inequality holds:

If IILTO < C(1+If I^TO,TO N+ If IIFp,q +1)).

Proposition 2.3 [13] Let (p, q) e (1, to) x (1, to], orp = q = to, andf be a solenoidal vector field. Then for s >0

| ||2fa( f, Ak] •Vg) |

¿4(1) |Lp < (IIVf IILTO Ig^F^q + IIVgIILTOIf yFp,q), (2.2)

or for s >-1

| l2^ Ak ] • Vg)||W(Z) Ip < (IIVf IILTO IgI^q + IIgIILTOIIVf IFp,q ). (2.3) The classical Calderon-Zygmund singular integrals are operators of the form

Tcf (x) := PV f ^Nf (x - y) dy = lim0 f ^Nf (x - y) dy, JvN lylN ^0J|y|>6 lylN

where ^ is defined on the unit sphere of RN, SN \ and is integrable with zero average

and where y' := -y-. e SN Clearly, the definition is meaningful for Schwartz functions.

Moreover if ^ e C1(SN-1), Tcz is Lp bounded, 1 <p < to.

The general version (1.4) of the relationship between u and 0 is in fact ensured by the following result (see e.g. [16]).

Lemma 2.3 Let m e Cto (Rn\{0}) be a homogeneous function of degree 0, and Tm be the corresponding multiplier operator defined by (Tmf)A = mf, then there exist a e C and ^ e Cto(Sn-1) with zero average such that for any Schwartz function f,

Tmf = Of + PV -N *f.

Remark 2.1 Since -Av = (3i3N0,..., -3N-13N0,3f0 + ••• + d^^O), the Fourier multiplier of the operator T is rather clear. In fact, each component of its multiplier is the linear

!él i i.

jçj^ г,

and is homogeneous of degree 0.

combination of the term like ||L, i,j e {1, 2, ..., N}, whichofcoursebelongsto Cœ(RN\{0})

3 Proof of Theorem 1.1

We divide the proof of Theorem 1.1 into several steps. Step 1. A priori estimates.

Taking the operation Ak on both sides of the first equation of (1.1), we have

dtAk0 + u • VAk0 = u • VAk0 - Ak(uV0) = [u, Ak] • V0.

Let X(a, t) be the solution of the following ordinary differential equations:

UtX(a, t) = u(X(a, t), t), jx(a, 0) = a.

Then it follows from (3.1) that

d Ak0 (X(a, t), t) = [u, Ak] -V0 (X(a, t), t), which implies that

| Ak 0 (X(a, t), t)|<|Ak0a(a)|+ f |([u, Ak] ^V0 )(X(a, t), t )\dr.

Multiplying 2ks, taking the lq(Z) norm on both sides of (3.4), we get by using the Minkowski inequality

^]|2ksAk0 (X(a, t), t)|

< ^|2ksAk00(a)|^q + ^12ks([u,Ak] ^V0)(X(a,t),t)|^qdT. (3.5)

Next, taking the Lp norm with respect to a e Rd on both sides of (3.5), we get by using the Minkowski inequality that

fRd (E|2feAk0(X(a,t),t)

^|2ks([u, Ak] -V0)(X(a,t),t)|'

< \\00WF' +

da I dT. (3.6)

Using the fact that a ^ X(a, t) is a volume-preserving diffeomorphism due to div u = 0, we get from (3.6) that

\\0 (t)\Fp,q < W00WFp,q +fQ \ \2k^ [U, Ak] • V0) \iq(keZ)\^dT. (3.^)

Thanks to Proposition 2.3, the last term on the right side of (3.7) is dominated by

f (\\vu\\lto\\0IF + \\V0WltotoWuWf' )dT, J0 p'q p'q

and thus

||e (t)||Fp,a <110011^ + j (\\v u\\l« + live Hi« )\\e WpsjT, (3.9)

where we used (1.4) and the boundedness of the Calderon-Zygmund singular integral operator on Fp a.

Now from (1.1) we have immediately

||e (t)|L = \\0ohlp (3.10)

for all 1 < p <«, since div u = 0. Summing up (3.9) and (3.10) yields

||e(i)|FPa < H0oHFp,a + j (HVuhl« + live\\l«)\\eHrp,qdr, (3.11)

which together with the Gronwall inequality gives

||e(t)\pM < H0oHFp,aexp(C^ (HVu\\L« + \\ve\\l«)d^j. (3.12)

Step 2. Approximate solutions and uniform estimates.

We construct the approximate solutions of (1.1). Define the sequence {0(n), u(k)}no=nu{o} by solving the following systems:

dte(n+1) + u(n) • ve(n+1) = o,

u(n) = C(e(n)) + s (e(n)), (313)

div u(n) = 0, (. )

e (n+1)|t=0 = Sn+2e0.

We set (e(0), u(0)) = (0,0) and letX(n)(a, t) be the solutions of the following ordinary differential equations:

Î9tX(n)(a, t) = u(n)(X(n) (a, t), t),

jx(n)(a, 0) = a (. )

for each n e N. Then, following the same procedure of estimate leading to (3.11), we obtain

\\Sn+2eoHFp,a + /(|v u(n)|i« ||e (n+1)|fp,a + ||V e (n+1)|

L«||u n hsp) dx

<\\eo\Fa + i (||vu(n)|^-1 ||e(n+1)|fS + ||ve^H 1 ||u(n)|F )dr

p a Jo p, a Fp,a rp,a p, a

Fp,a 'I, II" Il F

< \\eoHFpa + ft||e(n)||F ||e(n+1)|fS dx,

p a J0n "tpa" nJrp, a

..........(3.15)

p,a ------

where we used the fact that ||Sn+200|| < ||00||, Sobolev embedding theorem Fpq Lto for s - 1 > d/p, (1.4) and the boundedness of the Calderon-Zygmund singular integral operator on F . Equation (3.15) together with the Gronwall inequality implies that

||0(n+1)(t)|F.q < H00Hfsm exp(C^||0(n)|F.qd^, (3.16)

for some C >0 independent of n. Thus, if we choose T0 = T0(||00 ||Fpq) > 0 such that

mm^ q < 2c ln 2,

we have, for any n e N0,

sup ||0(n+1)(t)|Fs < 2CH00HFs , (.17)

0<t<T0 p,q

by the standard induction arguments. Then, 0(n+1) e C([0, T0];Fs (Rd)). Moreover, it follows from Proposition 2.1 that

| dt0(n+1)(t)|Fs-1 = |u(n) ^V0(n+1)|j

< C(|u(n) |lto || v0(n+1) ||F,_1 + |u(n) \Fq || V0(n+1) |lTO)

< C|0(n)|Fs-11|0(n+1)|Fs

by Sobolev embedding and the boundedness of the Calderon-Zygmund singular integral operator on F , and then

sup |9t0(n+1)(t)|Fs < CH00HF , 0!8)

0<t<T0 p,q p,q

which implies that dt0(n+1) e C([0, T0];Fp-?1(Md)). This together with (3.17) gives the uniform estimate of 0 (n)(x, t) in n. Step 3. Existence.

We will show that there exists a positive time T1 (< T0) independent of n such that 0(n) and u(n) are Cauchy sequences in XT-1 — C([0, T1]; Fs—). For this purpose, we set

50(n+1) = 0(n+1) - 0(n), 5u(n+1) = u(n+1) - u(n).

Then, it follows that 50 (n+1) satisfies the equations

I dt50(n+1) + u(n) • V50(n+1) = -5u(n) • V0(n),

150 (n+1)|t=0 = An+100. (3.19)

Applying Ak to the first equation of (3.19), we get

3tAk50(n+1) + u(n) • VAk50(n+1) = [u(n), Ak] • V50(n+1) - Ak(5u(n) • V0(n)). (3.20)

Exactly as in the proof of (3.7), we get

< C|| A„+i0o IIf*-i * «« 2k(s-1) ([u(n), Ak] • V50(n+1))(t) « tq(Z) «p dT + f «su(n) •ve(n)(T)|F_i dT

Jo p,q

< QAnMp- +jQ*(«■vu(n) «iœ «50(«+1) «^ + «50(n+1) «iœ «vu(n) «Fpqi) dT

+ f * («5u(n)«L. «ve (n)«^ + «5u(n)«F»q1 «v 0 (n)«U ) dT

J o p'q p'q

< C|An+10oI^^ * («u(n)«Fpq «50(n+1)«f^q + «Su(*«F-1 «0 (n)«f^q ) dT

< C||An+10oIlF-+/ * d0 (n) «Fpq «50(n+1)«Fp,q + «50 ^ H0 ^ ) ^ ' ^21)

where we used Proposition 2.1, Proposition 2.3, the embedding Fp— Lœ, and the boundedness of the Calderon-Zygmund singular integral operator on Fp,q. Thanks to the Fourier support of An+10o, we have

I|An+10oIF-i < C2-(n+1)|0olFPq. (.22)

p,q p,q

Now, we estimate the Lp norm of 50(n+1). Multiplying 150(n+1)|p-250(n+1) on both sides of the first equation of (3.19), and integrating the resulting equations over Rd, we obtain

«50(n+1)(t)«Lp < IIAn+1 0oIlp + T«5u(n) ^V0(n)(T)«LpdT

< 2-(n+1)«^+ H^ + Cf « 5u(n) « LP «V 0 (n)«U dT

p,q Jo

< 2-(n+1)«.+ «FP,q + ^ « 50(n) «LP «0 (n)«FP,qdT, which together with (3.21) and (3.22) gives

«50 (n+1)«Fp-q1 < C2-(n+1) |0o|Fp,q + c£ («0 (n)«FP,q «50 (n+1) « ^ + «50(n)«Fpq1 «0 (n)«Fp,q) dT

< C2-(n+1) I0o|Fp,q + CTSUPJ0 (n)«Fpq «50 (n+1)«Fp7q1

+ CT sup «0(n)«Fpq«50(n)«Fp- (3.23)

te[o,T] p,q p,q

Equation (3.23) together with (3.17) yields

« 50(n+1) «xs_1 < C12-(n+1) + C1T « 50(n+1) «xs_1

+ C1T «50 (n)«xs_1, (.4)

-s-1. T

where C = C1(H0o Hr^a). Thus, if QT < 4, then

| se (n+u |Xs-1 < C12-n + 2C1T| se(n) |

This implies that

|se(n+u X-1 < 2C:2-(n+:).

Thus, [0(K)}„eN0 is a Cauchy sequence in XT-1. By the standard argument, for T < min[T0, 4C}, the limit 0 e XsTl solves (1.1) with the initial data 00. The fact that 0 e Lip([0, T]; Fs ) follows from the uniform estimate (3.18). Step 4. Uniqueness.

Consider 0 e C([0, T1]; (Fpq)) is another solution to (1.1) with the same initial data. Let SO = 0 - 0 and Su = u - u. Then SO satisfies the following equation:

jdtS0 + u -VS0 = -Su -V0, [S0|t=o = 0.

In the same way as the derivation in (3.24), we obtain

X- < C2T||S0Hxi-1

for sufficiently small T. This implies that se = 0, i.e., e = e. Blow-up criterion.

For the a priori estimate (3.12), we only need to dominate \\Vu\\L« and \\Ve \\L«. From Proposition 2.2 and the boundedness of the Calderon-Zygmund operator from F«,« into itself, we have

\\VuHl« < C(1 + HVuHjr«,«(log+ \\Vu\\fp-ai + 1)) < C(1 + HeHf«,«(log+ HeHFp,a + 1)).

Similarly,

\\ve Hl«< c(1+ \eHf«,« (iog+ \\eh^ + 1)).

Thus, the a priori estimate (3.12) gives

\\e HFp,a < C\\e0 Hfp,a exp( |e (T )|f«,« (l0g+|e (t ^ + 1) dt). By the Gronwall inequality

\\e HFp,a < CHe0 Hfp,a exp(C exp(^t |e(T)||f«,« .

Therefore,iflimsupt^T* \\e(t)HFpa = «,then fQ \\e(t)\\j-««dt = «.

On the other hand, it follows from the Sobolev embedding F ^ for

s > d/p + 1 that

/. t *

/ |9(t)|^ dt < T* sup |9(T)|M

Jo 0<T <T*

n ™ 11 "^(XyX)

< t* sup ||ve(T)|

0<T <T*

< T* sup ||e(T)| .

0<T <T*

Then/0 (t)^^ dt = to implies limsup^T* II9(t)\\i*M = to.

Competing interests

The authors declare that they have no competing Interests. Authors' contributions

Allauthors contributed equally to the writing of this paper. Allauthors read and approved the finalmanuscript. Author details

'College of Electricaland Information Engineering, Hunan University, Changsha,410000, P.R. China. 2Schoolof Electrical and Automation Engineering, Hefei University of Technology, Anhui Pro., Hefei, 230009, P.R. China.

Acknowledgements

This work was supported by the NationalNaturalScience Funds of China for Distinguished Young Scholar under Grant No. 50925727, The NationalDefense Advanced Research Project Grant Nos. C1120110004, 9140A27020211DZ5102, the Key Grant Project of Chinese Ministry of Education under Grant No. 313018, and the FundamentalResearch Funds for the CentralUniversities (2012HGCX0003).

Received: 22 January 2014 Accepted: 24 March 2014 Published: 06 May 2014 References

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10.1186/1687-2770-2014-95

Cite this article as: Yu and He: On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces. Boundary Value Problems 2014, 2014:95