Recent result on porous medium equations with nonlocal pressure

Transcription

1 Recent result on porous medium equations with nonlocal pressure Diana Stan Basque Center of Applied Mathematics joint work with Félix del Teso and Juan Luis Vázquez November th workshop on Fractional Calculus, Probability and Non-local Operators BCAM 1 / 21

2 Preliminaries Physical Model: a continuum (fluid or population) with density distribution u(x, t) 0 and velocity field v(x, t). Continuity equation u t = (u v). Darcy s law: v derives from a potential (fluids in porous media): v = P. The relation between P and u: for gasses in porous media, Leibenzon and Muskat (1930) derived a relation in the form of the state law P = f (u), where f is a nondecreasing scalar function. f (u) is linear when the flow is isothermal and is a higher power of u when the flow is adiabatic, i.e. f (u) = cu m 1 with c > 0 and m > 1. The linear dependence f (u) = cu Boussinesq (1903) modelling water infiltration in an almost horizontal soil layer u t = c u 2. The model u t = (c/m) u m. The Porous Medium Equation u t = u m. 2 / 21

3 Porous Medium Equation / Fast Diffusion Equation PME/FDE u t(x, t) = u m (x, t) x R N, t > 0 Self Similar solutions: U(x, t) = t N N(m 1)+2 F ( x t 1 N(m 1)+2 ) Slow Diffusion m > 1, Profile (R 2 y 2 ) 1/(m 1) + Fast Diffusion m < 1, Profile (R 2 + y 2 ) 1/(1 m) 3 / 21

4 Definition of the Fractional Laplacian Several equivalent definitions of the nonlocal operator ( ) s (Laplacian of order 2s): 1 Fourier transform ( ) s g(ξ) = (2π ξ ) 2s ĝ(ξ). [ can be used for positive and negative values of s ] 2 Singular Kernel ( ) s g(x) = c N,s P.V. R N 3 Heat semigroup g(x) g(z) x z N+2s [ can be used for 0 < s < 1, where c N,s is a normalization constant. ] ( ) s g(x) = 4 Generator of the 2s-stable Levy process: dz 1 Γ( s) (e t g(x) g(x)) dt 0 t. 1+s ( ) s 1 g(x) = lim h 0 h E[g(x) g(x + X h)]. 4 / 21

5 Porous medium with nonlocal pressure The pressure p = ( ) s (u), 0 < s < 1: The model: ( ) s (u) = K s u = R N u(y) x y N 2s dy, Ks(x) = C N,s x (N 2s). tu = (u p), p = ( ) s (u). Difficulties: no maximum principle, no uniqueness. References: 1D: crystal dislocations model, Biler, Karch and Monneau, Comm.Math.Phys Existence and finite speed of propagation: Caffarelli and Vázquez, ARMA Asymptotic behavior: Caffarelli and Vázquez, DCDS Regularity: Caffarelli, Soria and Vázquez, JEMS Exponential convergence towards stationary states in 1D: Carrillo, Huang, Santos and Vázquez, JDE / 21

6 Porous Medium with nonlocal pressure t u = (u m 1 p), p = ( ) s (u). (P) for x R N, t > 0, N 1. We take m > 1, 0 < s < 1 and u(x, t) 0. The initial data u(x, 0) = u 0(x) for x R N, u 0 R N [0, ) is assumed to be a bounded integrable function. D. Stan, F. del Teso and J.L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure, Comptes Rendus Mathematique, 352 (2014), Issue 2, D. Stan, F. del Teso and J.L. Vázquez, Transformations of Self-Similar Solutions for porous medium equations of fractional type, Nonlinear Analysis Theory Meth.Appl., 119 (2015) D. Stan, F. del Teso and J.L. Vázquez, Finite and infinite speed of propagation for porous medium equations with nonlocal pressure, Journal of Differential Equations 260 (2016), Issue 2, D. Stan, F. del Teso and J.L. Vázquez, Existence of weak solutions for a general porous medium equation with nonlocal pressure, arxiv: / 21

7 Figure : m = 1.5, s = 0.25 Figure : m = 2, s = 0.25 Figure : m = 1.5, s = 0.75 Figure : m = 2, s = 0.75 Infinite vs. finite speed of propagation 7 / 21

8 New idea: existence for all m > 1 when u 0 L 1 (R N ) L (R N ) We write the problem in the form Then, formally: u t = (u m 1 ( ) 1 ( ) 1 s u) up 0 (x)dx u(x, t t)p dx = C 1 R N R N um+p 2 ( ) 1 s u dx dt R N C 2 t 0 R N by the Stroock-Varoupolos Inequality. Here C 1 = (p 1)/(m + p 2). 0 1 s ( ) 2 u m+p dxdt D. Stan, F. del Teso and J.L. Vázquez, Existence of weak solutions for a general porous medium equation with nonlocal pressure, arxiv: / 21

9 New approximation method u t = (u m 1 ( ) 1 ( ) 1 s u) (P) Then we approximate the operator L = ( ) 1 s by L 1 s ɛ (u)(x) = C N,1 s R N u(x) u(y) ( x y 2 + ɛ 2 ) N+2 2s 2 Convergence: L 1 s ɛ [u] ( ) 1 s u pointwise in R N as ɛ 0 Generalized Stroock-Varopoulos Inequality for L s ɛ : Let u H s ɛ(r N ). Let ψ R R such that ψ C 1 (R) and ψ 0. Then dy. where ψ = (Ψ ) 2. R N ψ(u)ls ɛ[u]dx R N (Ls ɛ) 1 2 [Ψ(u)] 2 dx, 9 / 21

10 Approximating problem We consider the approximating problem (P ɛδµr ) (U 1) t = δ U 1 + ((U 1 + µ) m 1 ( ) 1 Lɛ 1 s [U 1]) for (x, t) B R (0, T ), U 1(x, 0) = û 0(x) for x B R, U 1(x, t) = 0 for x B R, t (0, T ), with parameters ɛ, δ, µ, R > 0. Existence of solutions of (P ɛδµr ) fixed points of the following map given by the Duhamel s formula T (v)(x, t) = e δt u 0(x) + 0 t e δ(t τ) G(v)(x, τ)dτ, where G(v) = (v + µ) m 1 ( ) 1 L s ɛ[v] and e t is the Heat Semigroup. Existence of solutions of (P) (P ɛδµr ) ɛ 0 (P δµr ) R (P δµ ) µ 0 (P δ ) ɛ 0 (P). 10 / 21

11 Existence of weak solutions for m > 1 Theorem. Let 1 < m <, N 1, and let u 0 L 1 (R N ) L (R N ) and nonnegative. Main results: Existence of a weak solution u 0 of Problem (PMFP) with initial data u 0. Conservation of mass: For all 0 < t < T we have R N u(x, t)dx = R N u0(x)dx. L estimate: u(, t) u 0, 0 < t < T L p energy estimate: For all 1 < p < and 0 < t < T we have 4p(p 1) up (x, t)dx + R N (m + p 1) 2 t 0 R N 1 s ( ) 2 u m+p dxdt up 0 (x)dx. R N Second energy estimate: For all 0 < t < T we have 1 2 s ( ) 2 u(t) 2 dx+ R N 0 t R N um 1 ( ) s u(t) 2 dxdt 1 2 R N ( ) s 2 u 0 2 dx. 11 / 21

12 Smoothing effect Theorem Let u 0 be a weak solution of Problem (PMFP) with u 0 L 1 (R N ) L (R N ), u 0 0, as constructed before. Then where γ p = u(, t) L (R N ) C N,s,m,p t γp u 0 δp L p (R N ) for all t > 0, N (m 1)N+2p(1 s), δ p = 2p(1 s) (m 1)N+2p(1 s). Existence of weak solutions for only u 0 M + (R N ). Existence of weak solutions for only u 0 L 1 (R N ). 12 / 21

13 Existence for measure data Let 1 < m <, N 1 and µ M + (R N ). Then there exists a weak solution u 0 of Problem (P) s.t. the smoothing effect holds for p = 1 in the following sense: u(, t) L (R N ) C N,s,m t γ µ(r N ) δ for all t > 0, (1) N where γ = 2(1 s) (m 1)N+2(1 s) (m 1)N+2(1 s) Regularity: u L ((τ, ) L 1 (R N )) L (R N (τ, )) L ((0, ) M + (R N )) for all τ > 0 Conservation of mass: For all 0 < t < T we have R N u(x, t)dx = R N dµ(x). L p energy estimate: For all 1 < p < and 0 < τ < t < T we have 4p(p 1) up (x, t)dx + R N (m + p 1) 2 Second energy estimate: For all 0 < τ < t < T we have 1 2 s ( ) 2 u(t) 2 dx+ R N τ t τ t 1 s ( ) 2 u m+p dxdt up (x, τ)dx. RN R N R N um 1 ( ) s u(t) 2 dxdt 1 2 R N ( ) s 2 u(τ) 2 dx 13 / 21

14 Finite speed of propagation for m 2 Theorem Assume that u 0 has compact support and u(x, t) is bounded for all x, t. Then u(, t) is compactly supported for all t > 0. If 0 < s < 1/2 and u 0(x) U 0(x) = a( x b) 2, then there is a constant C large enough s.t. u(x, t) U(x, t) = a(ct ( x b)) 2. For 1/2 s < 2 C = C(t) is an increasing function of t. Consequence: Free Boundaries! Figure : u 0 U 0 Figure : u(x, t) U(x, t) 14 / 21

15 Infinite speed of propagation for m (1, 2) and N = 1 Theorem. Let m (1, 2), s (0, 1) and N = 1. Let u be the solution of Problem (PMFP) with initial data u 0 0 radially symmetric and monotone decreasing in x. Then u(x, t) > 0 for all t > 0, x R. Idea of the proof: Prove that v(x, t) = x u(y, t)dy > 0 for t > 0, x R. The integrated problem tv = v x m 1 ( ) 1 s v The initial data is given by v 0(x) = x u0(y)dy. (IP) Initial data v 0(x) satisfies: v 0(x) = 0 for x < η, v 0(x) = M for x > η, v 0(x) 0 for x ( η, η). Figure : Typical initial data for models (P) and (IP). 15 / 21

16 Fractional Porous Medium Equation U t + ( ) s U m = 0 (FPME) Porous Medium with Fractional Pressure V t = (V m 1 ( ) s V ) (PMFP) Self Similar Solutions U(x, t) = t α 1 F 1(t β 1 x) α 1 = Nβ 1, β 1 = with 1 N(m 1)+2s, ( ) s F m 1 = β 1 (y F 1). (P1) Self Similar Solutions V (x, t) = t α 2 F 2(t β 2 x) α 2 = Nβ 2, β 2 = with 1 N( m 1)+2 2 s, (F m 1 2 ( ) s F 2) = β 2 (y F 2). (P2) J. L. Vázquez. Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. JEMS D. Stan, F. del Teso and J.L. Vázquez, Transformations of Self-Similar Solutions for porous medium equations of fractional type, Non.Analysis, arxiv: Theorem. Transformation of self similar solutions If m > N/(N + 2s), s (0, 1) and F 1 is a solution to the profile equation (P1) then F 2(x) = (β 1/β 2) m 1 m (F 1(x)) m is a solution to the profile equation (P2) if we put m = (2m 1)/m and s = 1 s. 16 / 21

17 FPME: The profile F 1(y) is a smooth and positive function in R N, it is a radial function, it is monotone decreasing in r = y and has a definite decay rate as y, that depends on m. For m > N/(N + 2s), F 1(y) y (N+2s) for large y. [Vázquez, JEMS 2014] Consequence: PMFP: F 2 > 0 and F 2(x) C x (N+2 2 s)/(2 m) if m ((N s)/n, 2). Infinite Propagation for Self-Similar Solutions of the PMFP in R N, N 1, m < 2. Similar results are proved for smaller values of m. 17 / 21

18 Explicit Solutions Fractional Porous Medium Equation U t + ( ) s U m = 0 s (0, 1) m = N+2 2s N+2s (FPME) > m c = N 2s N Porous Medium with Fractional Pressure (II ) V t = (V ( ) s V m 1 ), m > 1 Biler, Imbert, Karch [2013]: v(x, t) = at Nβ 2 (R 2 xt β 2 2 ) (1 s)/( m 1) u(x, t) = at Nβ 1 (R 2 + xt β 1 2 ) (N+2s)/2 Huang, Bul.Lon.Math.Soc (I ) V t = (V m 1 ( ) s V ), m < 2 Huang [2014]: s (0, 1) m = N+6 s 2 N+2 s < 2 v(x, t) = at Nβ 2 (R 2 + xt β 2 2 ) (N+2 s)/2 18 / 21

19 Explicit Solutions Fractional Porous Medium Equation U t + ( ) s U m = 0 s (0, 1) m = N+2 2s N+2s (FPME) > m c = N 2s N Porous Medium with Fractional Pressure (II ) V t = (V ( ) s V m 1 ), m > 1 Biler, Imbert, Karch [2013]: v(x, t) = at Nβ 2 (R 2 xt β 2 2 ) (1 s)/( m 1) u(x, t) = at Nβ 1 (R 2 + xt β 1 2 ) (N+2s)/2 Huang, Bul.Lon.Math.Soc (I ) V t = (V m 1 ( ) s V ), m < 2 Huang [2014]: s (0, 1) m = N+6 s 2 N+2 s < 2 v(x, t) = at Nβ 2 (R 2 + xt β 2 2 ) (N+2 s)/2 Transformation Formula 19 / 21

20 Nonlocal/Local. Self-Similar Solutions V t = (V ( ) s V m 1 ), m > 1 V t = (V m 1 ( ) s V ), m 2 PME/FDE u t = u m m > 1, Profile (R 2 y 2 ) 1/(m 1) + U t + ( ) s U m = 0 V t = (V m 1 ( ) s V ), m < 2 m < 1, Profile (R 2 + y 2 ) 1/(1 m) 20 / 21

21 Gracias! Eskerrik asko! 21 / 21