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

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1 Electronic Journal of Differential Equations, Vol. 017 (017), No. 38, pp ISSN: URL: or 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 egenerate iffusion transport equation from which we erive the existence an uniqueness of the solution for the fractional porous meium equation in Sobolev spaces H α with nonnegative initial ata, α > + 1. We also correct a mistake in our previous paper [14]. 1. Introuction We consier the porous meium type equation t u = (u p), p = ( ) s u, 0 < s < 1, u(x, 0) 0, (1.1) where x R n, n, an t > 0 an the fractional Laplacian ( ) s/ := Λ s is given by the psueo ifferential operator with symbol ξ s, that is: ( ) s/ f = Λ s f = F 1 ξ s Ff. Using the Riesz potential, one can also efine this operator as ( ) s/ f(x) = Λ s f(x) f(y) f(x) = c n,s y n+s y. This moel is base on Darcy s law with pressure, p, is given by an inverse fractional Laplacian operator. It was first introuce by Caffarelli an Vázquez [4], in which they prove the existence of a weak solution when u 0 is a boune function R n n n+ s with exponential ecay at infinity. For α =, Caffarelli, Soria an Vázquez [3] prove that the boune nonnegative solutions are C α continuous in a strip of space-time for s 1/. An same conclusion for the inex s = 1/ was prove by Caffarelli an Vázquez in [5]. [7, 6, 13] give a etaile escription of the large-time asymptotic behaviour of the solutions of (1.1). [, 1] consier egenerate cases an show the existence an properties of self-similar solutions. Allen, Caffarelli an Vasseur [1] stuy the equation with fractional time erivative, an prove the Höler continuity for its weak solutions. In this paper, we stuy the existence an uniqueness of solutions of (1.1) in Sobolev spaces. Unlike consiering the existence of weak solution in L or constructing approximate solutions of linear transport systems, we solve equation (1.1) by constructing solutions to a linear egenerate iffusion transport systems. The 010 Mathematics Subject Classification. 35K55, 35K65, 76S05. Key wors an phrases. Fractional porous meium equation; Sobolev space; egenerate iffusion transport equation. c 017 Texas State University. Submitte December, 016. Publishe October 3,

2 X. ZHOU, W. XIAO EJDE-017/38 well-poseness an properties of the linear egenerate iffusion transport are interesting results by themselves an lea us to proving that for s [ 1, 1), α > + 1, u 0 H α (R n ) nonnegative, an some T 0 > 0, the equation (1.1) in R n [0, T 0 ] has a unique solutions. Besies, using the methos an results in this paper, we correct a mistake in our previous paper [14]. Define ρ C c (R n ) by. Preliminaries ρ(x) = { c0 exp( 1 1 x ), x < 1, 0, x 1, where c 0 is selecte such that ρ(x)x = 1. Let J ɛ be efine by J ɛ u = ρ ɛ u = ɛ n ρ( ɛ ) u. This operator satisfies the following properties. Proposition.1. (1) Λ s J ɛ u = J ɛ Λ s u, s R. () For all u L p (R n ), v H α (R n ), with 1 p + 1 q = 1, (J ɛ f)g = f(j ɛ g). (3) For all u H α (R n ), lim J ɛu u H α = 0, lim J ɛ u u H α 1 C u H α. ɛ 0 ɛ 0 (4) For all u H α (R n ), s R, k Z {0}, then J ɛ u H α+k C αk ɛ k u H α, J ɛd k u L C k ɛ n +k u H α The following propositions can be foun in [8, 9]. Proposition.. Suppose that s > 0 an 1 < p <. If f, g S, the Schwartz class, then we have Λ s (fg) fλ s g L p c f L p 1 g Ḣs 1,p + c g L p 4 f Ḣs,p 3, Λ s (fg) L p c f L p 1 g Ḣs,p + c g L p 4 f Ḣs,p 3 with p, p 3 (1, + ) such that 1 p = 1 p p = 1 p p 4. Proposition.3. If 0 s, f S(R n ), then f(x)λ s f(x) Λ s f (x) for all x R n. Proposition.4. Let α 1 an α be two real numbers such that α 1 < n, α < n an α 1 + α > 0. Then there exists a constant C = C α1,α 0 such that for all f Ḣα1 an g Ḣα, fg Ḣα C f Ḣα 1 g Ḣα, where α = α 1 + α n.

3 EJDE-017/38 WELL-POSEDNESS OF A POROUS MEDIUM FLOW 3 3. Main results Theorem 3.1. If s [1/, 1], T > 0, α > n + 1, u 0 H α (R n ), v 0 an v C([0, T ]; H α (R n )), then the linear initial-value problem t u = u ( ) s v v( ) 1 s u, u(x, 0) = u 0. (3.1) has a unique solution u C 1 ([0, T ]; H α (R n )). If the initial ata u 0 0, then u 0, (x, t) R n [0, T ]. Proof. For any ɛ > 0, we consier the linear problem t u ɛ = F ɛ (u ɛ ) = J ɛ ( J ɛ u ɛ ( ) s v) J ɛ (v( ) 1 s J ɛ u ɛ ), u ɛ (x, 0) = u 0. By Propositions.1 an. an by s 1 F ɛ (u ɛ 1) F ɛ (u ɛ ) H α we can estimate = J ɛ ( J ɛ (u ɛ 1 u ɛ ) ( ) s v) J ɛ (v( ) 1 s J ɛ (u ɛ 1 u ɛ ) H α C(ɛ, v H α) u ɛ 1 u ɛ H α. (3.) Using Picar iterations, for any α > n + 1, ɛ > 0, there exists a T ɛ = T ɛ (u ) > 0, problem (3.) has a unique solution u ɛ C 1 ([0, T ɛ ); H α ). By Propositions.1 an.3, 1 t uɛ L = J ɛ u ɛ ( ) s vj ɛ u ɛ v( ) 1 s J ɛ u ɛ J ɛ u ɛ 1 J ɛ u ɛ ( ) s v 1 v( ) 1 s J ɛ u ɛ 1 J ɛ u ɛ ( ) 1 s v 1 J ɛ u ɛ ( ) 1 s v = 0. Moreover, for any α > 0, 1 t Λα u ɛ L = Λ α ( J ɛ u ɛ ( ) s v)j ɛ Λ α u ɛ Λ α (v( ) 1 s J ɛ u ɛ )Λ α J ɛ u ɛ C [Λ α, ( ) s v] J ɛ u ɛ L Λ α u ɛ L + ( ) s vλ α J ɛ u ɛ Λ α J ɛ u ɛ + C [Λ α, v]( ) 1 s J ɛ u ɛ L Λ α u ɛ L vλ α ( ) 1 s J ɛ u ɛ Λ α J ɛ u ɛ. By Proposition. an Sobolev embeings, [Λ α, ( ) s v] J ɛ u ɛ L C ( ) 1 s v L Λ α 1 J ɛ u ɛ L + ( ) s v Ḣα J ɛ u ɛ L C v H α u ɛ H α, [Λ α, v]( ) 1 s J ɛ u ɛ L C v L ( ) 1 s J ɛ u ɛ Ḣα 1 + v Ḣα ( ) 1 s J ɛ u ɛ L C v H α u ɛ H α.

4 4 X. ZHOU, W. XIAO EJDE-017/38 By Proposition.3, ( ) s vλ α J ɛ u ɛ Λ α J ɛ u ɛ 1 ( ) s v (Λ α J ɛ u ɛ ) 1 C v H α u ɛ H α. Combining the above estimates, t uɛ (, t) H α C v H α u ɛ H α. By Gronwall s inequality, vλ α ( ) 1 s J ɛ u ɛ Λ α J ɛ u ɛ v( ) 1 s (Λ α J ɛ u ɛ ) u ɛ (, t) H α u 0 H α exp(c sup v H α). 0 t T Such the solution u ɛ exists on [0, T ]. Similarly, t uɛ (, t) H α 1 C v H α u ɛ H α C( v H α, u 0 H α, T ). By Aubin compactness theorem [11], there is a subsequence of {u 1 n } n 1 that convergence strongly to u in C([0, T ]; H α ). If α > + 1, Hα C 1, so u is a solution of (3.3). If u an ũ are two solutions of problem (3.3), then w = u ũ satisfies t w = w ( ) s v v( ) 1 s w, w(x, 0) = 0. Similarly, we get t w L 0 an t w Ḣ v α H α w H α, i.e, t w H α v H α w H α. By Gronwall s inequality, u(x, t) = 0, (x, t) R n [0, T ]. Since u 0 0 then if there exists a first time t 0 where for some point x 0 we have u(x 0, t 0 ) = 0, then (x 0, t 0 ) will correspon to a minimum point an therefore u(x 0, t 0 ) = 0, an u(x) u(y) ( ) 1 s u(x) = c y 0. y n+ s Hence u t (x0,t 0) 0. So u(x, t) 0 for all (x, t) R n [0, T ]. Theorem 3.. Let n, s [ 1, 1), α > + 1, u 0 H α (R n ), an u 0 0. Then there the linear initial value problem t u = (u ( ) s u), u(x, 0) = u 0. has a unique solution u C 1 ([0, T 0 ], H α (R n )). u 0, (x, t) R n [0, T 0 ]. If the initial ata u 0 0, then Proof. Set u 1 = u 0. Note that t u = (u ( ) s u) = u ( ) s u u( ) 1 s u, an let {u n } be the sequence efine by t u n+1 = u n+1 ( ) s u n u n ( ) 1 s u (n+1), u n+1 (x, 0) = u 0. (3.3)

5 EJDE-017/38 WELL-POSEDNESS OF A POROUS MEDIUM FLOW 5 By Theorem 3.1, u C([0, T ); H α ), for all T <, satisfies u 0 an sup u H α u 0 H α exp(c u 1 H αt ). 0 t T ln If exp(c u 1 H αt 0 ), for example T 0 = C(1+ u 0 H α ), we have sup 0 t T 0 u H α u 0 H α. By the stanar inuction argument, if u n C([0, T 0 ]; H α ), u n 0 is a solution of (3.3) with u n H α u 0 H α. By Theorem 3.1 u n+1 C([0, T 0 ]; H α ), u n+1 0 an sup u n+1 H α u 0 H α exp(c u n H αt 0 ) u 0 H α, 0 t T 0 t un+1 H α 1 C u n H α u n+1 H α C u 0 H α. By Aubin compactness theorem [11], there is a subsequence of u n that convergence strongly to u in C([0, T ]; H α ). If u 0, ũ 0 are two solutions of problem (3.3), then w = u ũ satisfies t w = (w ( ) s u) + (ũ ( ) s w), w(x, 0) = 0. By Proposition., 1 t w L = w (w ( ) s u) + =: I 1 + I + I 3. w ũ ( ) s w wũ( ) 1 s w Note that I 1 = w w ( ) s u = 1 w ( ) s u = 1 w ( ) 1 s u C u H α w L, I 3 1 ũ( ) 1 s w = 1 ( )1 s ũ w C u H α w L. When s > 1/, I C w L u ( ) s w L C w L u Ḣ n +1 s ( ) s w Ḣs 1 C u H α w L. When s = 1/, the above estimates are still vali. Combining the above estimates we have t w L C w L u Hα. By Gronwall s inequality we can euce w(x, t) = 0 on [0, T 0 ]. 4. Correction In [14], trying to establish the well-poseness of (1.1) in Besov spaces the authors incurre in a mistake in page 9 when estimating the term J 4 in equation [14, (4.5)]. To correct the mistake, we moify our proof the following way. [14, Theorem 1.1] Let n, s [ 1, 1], α > n + 1. If the initial ata u 0 B α, then there exists T = T ( u 0 B α ) such that (1.1) has a unique solution in

6 6 X. ZHOU, W. XIAO EJDE-017/38 [0, T ] R n. Such a solution belongs to C 1 ([0, T ]; B α+s ) L ([0, T ]; B β ), with β [α + s, α]. Proof. First we construct the approximate equation u (n+1) t = u (n+1) ( ) s u (n) ɛ u (n) ɛ ( ) 1 s u (n+1) ; u (n+1) (0) = σ ɛ u 0, u (1) = σ ɛ u 0. (4.1) By the argument in section, there exists a sequence u (n) that solves the linear systems (4.1). Assuming that u 0 0, we prove that u (n+1) 0. Inspire by [4], we assume that x 0 is a point of minimum of u (n+1) at time t = t 0. This inicates that u (n+1) (x 0 ) = 0, an ( ) 1 s u (n+1) u(x0 ) u(y) (x 0 ) = c y 0. y n+(1 s) Thus we euce that t u(n+1) t=t0 0, an by inuction we have u (n+1) 0. Arguing as in [14], taking j on (4.1), we obtain t j u (n+1) = [ j, i ( ) s u (n) ɛ ] i u n+1 + i ( ) s u (n) ɛ j ( i u (n+1) ) [ j, u (n) ɛ ]( ) 1 s u (n+1) u (n) ɛ j ( ) 1 s u (n+1). Multiplying both sies by ju(n+1) ju (n+1), an integrating over R, then enote the corresponing part in the right sie by J 1, J, J 3, J 4, respectively. We obtain the estimates, J 1 C jα u (n+1) B α u (n) B α+1 s J C jα u (n+1) B α u (n) B α+1 s J 3 C jα u (n) B α u (n+1) B α+1 s. The estimate for the term J 4 is now replace by J 4 = u (n) j ( ) 1 s u (n+1) ju (n+1) j u (n+1) u (n) ( ) 1 s j u (n+1) ( ) 1 s u (n) u (n+1) jα u n B r+ s u (n+1) B α. Here r > is any real number. The first inequality uses the following pointwise estimate. Proposition 4.1 ([10]). If 0 α, p 1, then for any f S(R ) p f(x) p f(x)λ α f(x) Λ α f(x) p. Taking r such that r + s < α, e.g. set r = α 1, we conclue t u(n+1) B α 1,, u (n) B α 1,, u (n+1) B α 1,,.

7 EJDE-017/38 WELL-POSEDNESS OF A POROUS MEDIUM FLOW 7 The other parts of the proof nee no moification. Acknowlegement. We are very grateful to Ph. D. Mitia Duerinckx for pointing out our mistake an share some goo views on this problem. This paper is supporte by the NNSF of China uner grants No an No References [1] M. Allen, L. Caffarelli, A. Vasseur; Porous meium flow with both a fractional potential pressure an fractional time erivative. arxiv preprint arxiv: , 015. [] P. Biler, C. Imbert, G. Karch; The nonlocal porous meium equation: Barenblatt profiles an other weak solutions. Archive for Rational Mechanics an Analysis, 015, 15(): [3] L. Caffarelli, F. Soria, J. L. Vázquez; Regularity of solutions of the fractional porous meium flow. arxiv preprint arxiv: , 01. [4] L. Caffarelli, J. L. Vázquez; Nonlinear porous meium flow with fractional potential pressure. Archive for rational mechanics an analysis, 011, 0(): [5] L. Caffarelli, J. L. Vázquez; Regularity of solutions of the fractional porous meium flow with exponent 1/. arxiv preprint arxiv: , 014. [6] L. Caffarelli, J. L. Vázquez; Asymptotic behaviour of a porous meium equation with fractional iffusion. arxiv preprint arxiv: , 010. [7] J. A. Carrillo, Y. Huang, M. C. Santos, J. L. Vázquez; Exponential convergence towars stationary states for the 1D porous meium equation with fractional pressure. Journal of Differential Equations, 015, 58(3): [8] A. Córoba, D. Córoba; A maximum principle applie to quasi-geostrophic equations. Communications in mathematical physics, 004, 49(3): [9] N. Ju; Existence an uniqueness of the solution to the issipative D quasi-geostrophic equations in the Sobolev space. Communications in Mathematical Physics, 004, 51(): [10] C. Miao, J. Wu, Z. Zhang; Littlewoo-Paley Theory an Applications to Flui Dynamics Equations. Monographson Moern Pure Mathematics, No. 14 (Science Press, Beijing, 01). [11] J. Simon. Compact sets in the space L p (0, T ; B) Ann. Mat. Pura. Appl., (4) 146, 1987, [1] D. Stan, F. el Teso, J. L. Vázquez; Transformations of self-similar solutions for porous meium equations of fractional type. Nonlinear Analysis: Theory, Methos Applications, 015, 119: [13] J. L. Vázquez; Nonlinear iffusion with fractional Laplacian operators. Nonlinear Partial Differential Equations. Springer Berlin Heielberg, 01: [14] X. Zhou, W. Xiao, J. Chen; Fractional porous meium an mean fiel equations in Besov spaces, Electron. J. Differential Equations, 014 (014), no. 199, Xuhuan Zhou Department of Information Technology, Nanjing Forest Police College, 1003 Nanjing, China aress: zhouxuhuan@163.com Weiliang Xiao School of Applie Mathematics, Nanjing University of Finance an Economics, 1003 Nanjing, China aress: xwltc13@163.com