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Weak solution of the equation for a fractional porous medium with a forcing term
Computers and Mathematics with Applications 67 (2014) 145–150
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Weak solution of the equation for a fractional porous medium with a forcing term✩ Mingshu Fan a,b , Shan Li c , Lei Zhang d,∗ a
College of Mathematics, Southwest Jiaotong University, Chengdu, 610031, PR China
b
Department of Mathematics, Jingcheng College of Sichuan University, Chengdu 611731, PR China
c
Business School, Sichuan University, Chengdu 610064, PR China
d
College of Computer Science, Sichuan University, Chengdu 610064, PR China
article
abstract
info
Article history: Received 18 March 2013 Received in revised form 27 July 2013 Accepted 30 September 2013
We consider the Cauchy problem with a forcing term for a fractional porous medium, which arises in statistical mechanics and heat control. The existence and uniqueness of a weak energy solution are established using implicit time discretization and L1 contraction semigroup arguments. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Fractional diffusion Porous medium equation Weak solution Forcing term
1. Introduction We consider the following Cauchy problem involving a forcing term for a fractional porous medium (FPM): ut + (−△)1/2 |u|m−1 u = f (x, t ), u(x, 0) = u0 (x),
(x, t ) ∈ RN × (0, +∞), (1.1) x ∈ RN , where m > 0, the forcing term is f (x, t ) ∈ C 0, ∞; L1 (RN ) , and the initial data are u0 (x) ∈ L1 (RN ) ∩ L∞ (RN ). We look for
a suitable class of weak solutions for (1.1). The fractional operator, known as the square root of the Laplacian, is defined as a pseudo-differential operator by Fourier transformation
1/2 g (ξ ) = |ξ |ˆ (−△) g (ξ )
(1.2)
for any smooth function g in the Schwartz class. The nonlocal operator can also be expressed using the Riesz potential approach [1,2]: 1/2
(−△)
g (x) = C (N ) P.V.
g (x) − g (y)
RN
|x − y|N +1
dy,
(1.3)
✩ M.S.F. is supported by an NSFC grant. S.L. is supported in part by SRFDP (No. 20100181120031), Fundamental Research Funds for the Central Universities (skqy201224) and China Postdoctoral Funds (2013M542285). ∗ Corresponding author. E-mail addresses: [email protected] (M. Fan), [email protected] (S. Li), [email protected] (L. Zhang).
0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.camwa.2013.09.025
146
M. Fan et al. / Computers and Mathematics with Applications 67 (2014) 145–150
which is an operator of integral type defined by convolution on the whole domain RN , where C (N ) is some normalization constant. There is wide interest in studying fractional diffusion in model diffusive processes, especially for propagation of longrange diffusive interactions in porous media and infinitesimal generators of stable Lévy processes [3,4]. Fractional operators [5,6], fractional partial differential equations [7–11], and equations for porous media [12–18] have been reviewed elsewhere. Pablo et al. considered the existence, uniqueness, and properties of Cauchy problem (1.1) without a forcing term f ≡ 0, and established a satisfactory theory not only for a weak solution but also for a strong solution [19]. To deal with the nonlocal operator (−△)1/2 , they applied a third way to express the half Laplacian, which was introduced by Caffarelli and Silvestre [20] through the so-called Dirichlet–Neumann operator. The operator can be obtained from harmonic extension of the problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann one. Suppose that v(x, y) is the harmonic extension of the bounded smooth function g (x), written as v = E (g ), as the solution to
△x,y v = 0, v(x, 0) = g (x),
(x, y) ∈ RN × R+ , x ∈ RN ,
(1.4)
where △x,y is the Laplacian in the variables (x, y) ∈ RN × R+ . We define the operator T as g → − ∂v (x, 0). Applying the ∂y operator T twice to g yields T 2 g (x) = T
∂v − (x, 0) = uyy (x, 0) = −△g (x). ∂y
Using the fact that T is a positive operator, we have
−
∂v (x, 0) = (−△)1/2 g (x). ∂y
(1.5)
Applying the third expression for fractional diffusion by harmonic extension to (1.1), we formulate the following problem for the extension function w = E |u|m−1 u :
△w = 0, 1 m −1 w ∂ |w| ∂w − = −f (x, t ), ∂y ∂t w(x, 0, 0) = um 0 (x),
N +1 for (x, y) ∈ R+ , t > 0,
(1.6)
for x ∈ RN , y = 0, t > 0, for x ∈ RN .
1
The solution u of the original (1.1) can be understood as the trace of E |w| m −1 w
for y = 0. This modified problem is
a quasi-stationary problem with a dynamic boundary condition. Compared to the original problem, the advantage of the modified (1.6) is that there is no nonlocal operator. For the case f ≡ 0, Athanasopoulos and Caffarelli [21]showed the continuity of the bounded weak energy solution of (1.6) for m > 1. Pablo et al. established a systematic theory for the FPM equation without a forcing term [19]. They obtained the existence, uniqueness, and properties such as regularity, positivity, and comparison for a suitable weak solution of (1.1) for f ≡ 0. The existence of their solution was based on solving the modified (1.6) (for f ≡ 0) and an L1 -contraction semigroup. The results were then extended to a general FPM equation [22]. Following the ideas of Pablo et al., the main purpose of this paper is to show the existence and uniqueness of a suitable weak solution of (1.1) for a complete FPM equation. We start from the definition of a suitable weak solution of (1.1) according to the quasi-stationary problem (1.6). For convenience, we adopt the same notation as in [19], that is, x¯ = (x, y), Ω = RN++1 , Γ = RN × {0}, throughout the paper. ¯ × [0, T )) and integrating by parts yields Multiplying both sides of the first equation in (1.6) by a test function ϕ ∈ C01 (Ω
∂ϕ dxds + ∂t 0 Ω 0 Γ 1 where u is understood as the trace Tr |w| m −1 w .
−
T
⟨∇w, ∇ϕ⟩dx¯ ds +
T
T
u
0
Γ
f (x, s)ϕ dxds = 0,
1,1
(1.7)
1
Definition 1.1 (Weak Solution). Assume that w ∈ L1 (0, T ); Wloc (Ω ) , u = Tr |w| m −1 w
∈ L1 (Γ × (0, T )) and the
¯ × [0, T )). Then the pair of functions (u, w) is called a weak solution of (1.6) provided identity (1.7) holds for any ϕ ∈ C01 (Ω that u(·, t ) ∈ L1 (Γ ) for any t > 0 and limt →0 u(·, t ) = u0 in L1 (Γ ). For weak solutions the restriction conditions are relaxed and the solutions are obtained in Sobolev classes of weakly differentiable functions. However, we have to give a suitable condition to ensure the uniqueness of the problem. According to the notion for classical porous-medium equations, the concept of a weak energy solution is acceptable.
M. Fan et al. / Computers and Mathematics with Applications 67 (2014) 145–150
147
Definition 1.2 (Weak Energy Solution). A weak solution pair is called a weak energy solution provided w ∈ L2 [0, T ];
H 1 (Ω ) .
Our main result is the existence and uniqueness of the weak energy solution, as follows. Theorem 1.1. Assume that m > 0, f ∈ C 0, ∞; L1 (RN ) and the data u0 ∈ L1 (RN ) ∩ L∞ (RN ). Then there exists a pair of
unique weak energy solution (w, u) to (1.6), and u ∈ C [0, ∞); L1 (RN ) ∩ L∞ (RN × [0, ∞)). 2. Proof of the main result
In this section, we construct weak solutions for the complete FPM equation (1.1) by viewing it as an abstract evolutionary equation. More precisely, we consider an ordinary differential equation with data in a Banach space and write the solution as u(·, t ) instead of u(t ) for convenience. To solve the abstract differential equations, we use the implicit time discretization method to construct the approximate solution, which exists uniquely according to elliptic theory in a half plane. The convergence of discrete approximated solutions comes from classical work by Crandall and Liggett [23] when the abstract operator has some contraction properties, called an acrretive operator. Finally, we show that the mild solution (the limit of the discrete approximated solution) is in fact the weak solution of the modified Eq. (1.6). Consider the abstract Cauchy problem du
+ A(u) = f , u(0) = u0 , where f ∈ C 0, T ; L1 (RN ) and u0 ∈ L1 (Γ ).
(2.1)
dt
Now we formulate the approximated problems using the discretization process as follows. Taking a partition P = {0 = t0 < t1 < · · · < tk−1 < tk < · · · < tn = T }, tk = kε = kT /n for k = 0, 1, 2, . . . , n, we solve the abstract problem (2.1) as the system of difference relations uε,k − uε,k−1
+ A(uε,k ) = fk , for k = 1, 2, . . . , n, (2.2) ε where {f1 , f2 , . . . , fn } is a discretization of f adapted to the partition P , fk = f (·, kε) for k = 0, 1, . . . , n. Eq. (2.2) can be also written as uε,k + ε A(uε,k ) = uε,k−1 + ε fk .
(2.3)
In fact, the square root of the Laplacian operator A : D(A) → L1 (Γ ) can be defined as A(v) = −Tr
∂ E (|v|m−1 v) ∂y
(2.4)
and D(A) = v ∈ L1 (Γ ) ∩ L∞ (Γ ) | A(v) ∈ L1 (Γ ), ∥v∥L∞ (Γ ) ≤ ∥u0 ∥L∞ (Γ ) + T ∥f ∥L∞ (Γ ) .
(2.5)
Applying the discretization process to the modified (1.6), we obtain the discretized problems for k = 1, 2, . . . , n:
△wε,k = 0, ∂wε,k = uε,k − uε,k−1 − ε fk , ε ∂y
in Ω , (2.6)
on Γ ,
1
with the first step uε,0 = u0 on Γ , and uε,k can be understood as Tr |wε,k | m −1 wε,k . This is an elliptic equation with a nonlinear Neumann boundary condition in the half plane. The solvability of the elliptic problem has been proved elsewhere [19] and is stated as a lemma here. Lemma 2.1. For any ε > 0, consider the following Neumann problem for an elliptic equation:
△w = 0, ∂w 1 = |w| m −1 w − g , ε ∂y
in Ω , (2.7)
on Γ .
1
For g ∈ L1 (Γ ) ∩ L∞ (Γ ), there exists a unique weak solution w ∈ H 1 (Ω ) such that u = Tr (|w| m −1 w)
∈ L1 (Γ ) ∩ L∞ (Γ )
and ∥w∥L∞ (Γ ) ≤ ∥g ∥L∞ (Γ ) . Moreover, if w and w ˜ are the solutions corresponding to data g and g˜ , then we have
Γ
1
1
|w| m −1 w − |w| ˜ m −1 w ˜
+
g − g˜ + dx,
dx ≤ Γ
¯ provided that g ≥ 0 in Γ . which implies that w ≥ 0 in Ω
148
M. Fan et al. / Computers and Mathematics with Applications 67 (2014) 145–150
Fig. 1. Construction of the approximate solution wε .
As a direct consequence, we have the following results for the solvability and contraction properties of the discretized problems. Proposition 2.2. For any ε > 0, consider the following discretized problems:
△wε,k = 0, ∂wε,k 1 1 − |wε,k | m −1 wε,k = −|wε,k−1 | m −1 wε,k−1 − ε fk , ε ∂y
in Ω , (2.8)
on Γ .
For any given wε,k−1 ∈ H 1 (Ω ) and fk ∈ L1 (Γ ) ∩ L∞ (Γ ), there exists a unique weak solution wε,k ∈ H 1 (Ω ), such that 1
uε,k = Tr(|wε,k | m −1 wε,k ) ∈ L1 (Γ ) ∩ L∞ (Γ ) and
∥uε,k ∥L∞ (Γ ) ≤ ∥uε,k−1 ∥L∞ (Γ ) + ε∥fk ∥L∞ (Γ ) .
(2.9)
Moreover, uε,k and u˜ ε,k are solutions corresponding to the data uε,0 = u0 and u˜ ε,0 = u˜ 0 . Then we have
uε,k − u˜ ε,k + dx ≤
Γ
u0 − u˜ 0 + dx
Γ
for any k = 0, 1, 2, . . . , N. Next, we establish a weak solution for the evolutionary problem using the process mentioned above. Theorem 2.3. There exists a weak solution (u, w) to (1.6) and u(·, t ) ∈ L1 (Γ ) ∩ L∞ (Γ ) for every t > 0 and w ∈ L2 [0, T ];
H 1 (Ω ) . Moreover, there exists almost one weak energy solution w ∈ L2 [0, T ]; H 1 (Ω ) to (1.6) (Fig. 1). Proof. Dividing the interval [0, T ] into n subintervals, the length of each subinterval is T /n. We construct the approximate solution wε (x), which is a piecewise constant function in each interval (tk−1 , tk ] for tk = kε, k = 1, 2, . . . , n. For each k > 0, wε,k satisfies
1wε,k = 0 ∂wε,k = uε,k − uε,k−1 + ε fk ε ∂y
in Ω , (2.10)
on Γ ,
where uε,0 = u0 (x), fk = f (x, tk−1 ). Consider the limit (uε , wε ) → (u, w) as ε → 0 in L1loc (Ω ), which is a so-called mild solution. It follows from classical semigroup theory that u ∈ C [0, ∞); L1 (Γ ) . Furthermore, multiplying (2.10) by wε,k and integrating by parts yields
ε
Ω
∇wε,k 2 dx¯ ≤
Γ
|uϵ,k |m−1 uϵ,k | uε,k−1 − uε,k + εfk dx.
(2.11)
It follows from Young’s inequality that
ε
Ω
∇wε,k 2 dx¯ ≤
1
m+1
Γ
|uε,k−1 |m+1 dx −
Γ
m+1 uε,k |uϵ,k |m−1 uϵ,k |dx dx + ε Γ
for k = 1, 2, . . . , n. Then adding from k = 1 to k = n gives T
0
Ω
|∇wε (¯x, t )|2 dx¯ dt ≤
1 m+1
Γ
|u0 (x)|m+1 dx +
T
0
Γ
|uε (x, t )|m f (x, t )dxdt .
(2.12)
M. Fan et al. / Computers and Mathematics with Applications 67 (2014) 145–150
149
Taking the limit ε → 0 implies that w ∈ L2 [0, T ]; H 1 (Ω ) . Moreover, choosing some appropriate test function gives the following weak form:
Ω
⟨∇wε,k , ∇ϕ⟩dx¯ =
Γ
1 uε,k−1 − uue ,k ϕ + fk ϕ dx. ε
Integrating over (tk−1 , tk ) and adding these together leads to n
tk
T
Γ
0
−
ε
((uε,k−1 (x, t ) − uε,k )) + fk ϕ(x, 0, t )dxdt
ϕ(x, 0, t + ε) − ϕ(x, 0, t ) 1 ε u0 (x)ϕ(x, 0, t )dxdt dxdt + ε ε 0 Γ T f (x, t )ϕ(x, 0, t )dxdt . uε (x, T )ϕ(x, 0, t )dxdt +
uε (x, t )
= 1
ε
Γ
tk−1
k =1
1
T
Γ
T −ε
(2.13)
Γ
0
Taking the limit ε → 0, the second and third integral terms on the right-hand side vanish and we obtain the weak formulation (1.7). Thus, we can conclude that the function (u, w) is indeed a weak solution to (1.6). Next we show the uniqueness of the weak energy solution to (1.6) using the standard energy method. Let (ui , wi ) be the weak energy solutions to (1.6) corresponding to the same initial data g (x) for i = 1, 2. Consider the deviation of the two solutions w(¯x, t ) = w1 (¯x, t ) − w2 (¯x, t ), which satisfy wi = E |ui |m−1 ui for i = 1.2,
△w = 0, ∂w ∂(u1 − u2 ) − = 0, ∂y ∂t w(x, 0, 0) = 0,
for (x, y) ∈ Ω , t > 0, for x ∈ Γ , t > 0,
(2.14)
for x ∈ Γ .
Choosing an appropriate test function
ϕ(¯x, t ) =
T
w(¯x, s)ds
for 0 ≤ t ≤ T ,
t
0,
for t ≥ T ,
and multiplying it by (2.14), integration by parts yields T
Ω
0
∇w(¯x, t ),
T
T
∇w(¯x, s)ds dx¯ dt +
t
0
Γ
m (u1 − u2 )(um 1 − u2 )dxdt = 0.
That is,
2
T
1
Ω
0
2 ∇w(¯x, s)ds dx¯ +
T 0
Γ
m (u1 − u2 )(um 1 − u2 )dxdt = 0,
which implies that u1 = u2 in Γ and w1 = w2 in Ω .
Acknowledgments We would like to thank the anonymous referees for suggestions that helped to improve our paper. References [1] N.S. Landkof, Foundations of Modern Potential Theory, in: Grundlehren der mathematischen Wissenschaftern, vol. 180, Springer-Verlag, New York, Heidelberg, 1972. [2] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, in: Princeton Mathematics Series, vol. 30, Princeton University Press, Princeton, NJ, 1970. [3] D. Applebaum, Lévy Processes and Stochastic Calculus, second ed., in: Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge, 2009. [4] J. Bertoin, Lévy Processes, in: Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. [5] J.T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1140–1153. [6] J.L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in: Nonlinear Partial Differential Equations, The Abel Symposium 2010, pp. 271–298. Abel Symposia, 7. Berlin-Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-25361-4_15. [7] B.L. Guo, D.W. Huang, Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys. 53 (2012) 083702. [8] F.H. Huang, B.L. Guo, General solutions to a class of time fractional partial differential equations, Appl. Math. Mech. (English Ed.) 31 (2010) 815–826. [9] X.K. Pu, B.L. Guo, J.J. Zhang, Global weak solutions to the 1-D fractional Landau–Lifshitz equation, Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 199–207. [10] X.K. Pu, On the local well-posedness for the fractional Landau–Lifshitz–Gilbert equation, Comput. Math. Appl. 64 (2012) 3276–3280. [11] X.K. Pu, B.L. Guo, Weak attractors for the dissipative fractional Heisenberg equation, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 764–768.
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[12] J.L. Vázquez, The Porous Medium Equation, Mathematical Theory, in: Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [13] L.L. Du, X.Y. Xiang, A further blow-up analysis for a localized porous medium equation, Appl. Math. Comput. 179 (2006) 200–208. [14] L.L. Du, Z.A. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation, Commun. Pure Appl. Anal. 6 (2007) 183–190. [15] L.L. Du, Blow-up for a degenerate reaction–diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math. 202 (2007) 237–247. [16] L.L. Du, Blow-up for a degenerate reaction–diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304–320. [17] L.L. Du, C.L. Mu, M.S. Fan, Global existence and non-existence for a quasilinear degenerate parabolic system with non-local source, Dyn. Syst. 20 (2005) 401–412. [18] M.S. Fan, C.L. Mu, L.L. Du, Uniform blow-up profiles for a nonlocal degenerate parabolic system, Appl. Math. Sci. 1 (2007) 13–23. [19] A. Pablo, F. Quirós, A. Rodríguez, J.L. Vázquez, A fractional porous medium equation, Adv. Math. 226 (2011) 1378–1409. [20] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differential Equations 32 (2007) 1245–1260. [21] I. Athanasopoulos, L.A. Caffarelli, Continuity of the temperature in boundary heat control problems, Adv. Math. 224 (1) (2010) 293–315. [22] A. Pablo, F. Quirós, A. Rodríguez, J.L. Vázquez, A fractional porous medium equation, Comm. Pure Appl. Math. 65 (2012) 1242–1284. [23] M.G. Crandall, T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Am. J. Math. 93 (1971) 265–298.
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Weak solution of the equation for a fractional porous medium with a forcing term✩ Mingshu Fan a,b , Shan Li c , Lei Zhang d,∗ a
College of Mathematics, Southwest Jiaotong University, Chengdu, 610031, PR China
b
Department of Mathematics, Jingcheng College of Sichuan University, Chengdu 611731, PR China
c
Business School, Sichuan University, Chengdu 610064, PR China
d
College of Computer Science, Sichuan University, Chengdu 610064, PR China
article
abstract
info
Article history: Received 18 March 2013 Received in revised form 27 July 2013 Accepted 30 September 2013
We consider the Cauchy problem with a forcing term for a fractional porous medium, which arises in statistical mechanics and heat control. The existence and uniqueness of a weak energy solution are established using implicit time discretization and L1 contraction semigroup arguments. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Fractional diffusion Porous medium equation Weak solution Forcing term
1. Introduction We consider the following Cauchy problem involving a forcing term for a fractional porous medium (FPM): ut + (−△)1/2 |u|m−1 u = f (x, t ), u(x, 0) = u0 (x),
(x, t ) ∈ RN × (0, +∞), (1.1) x ∈ RN , where m > 0, the forcing term is f (x, t ) ∈ C 0, ∞; L1 (RN ) , and the initial data are u0 (x) ∈ L1 (RN ) ∩ L∞ (RN ). We look for
a suitable class of weak solutions for (1.1). The fractional operator, known as the square root of the Laplacian, is defined as a pseudo-differential operator by Fourier transformation
1/2 g (ξ ) = |ξ |ˆ (−△) g (ξ )
(1.2)
for any smooth function g in the Schwartz class. The nonlocal operator can also be expressed using the Riesz potential approach [1,2]: 1/2
(−△)
g (x) = C (N ) P.V.
g (x) − g (y)
RN
|x − y|N +1
dy,
(1.3)
✩ M.S.F. is supported by an NSFC grant. S.L. is supported in part by SRFDP (No. 20100181120031), Fundamental Research Funds for the Central Universities (skqy201224) and China Postdoctoral Funds (2013M542285). ∗ Corresponding author. E-mail addresses: [email protected] (M. Fan), [email protected] (S. Li), [email protected] (L. Zhang).
0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.camwa.2013.09.025
146
M. Fan et al. / Computers and Mathematics with Applications 67 (2014) 145–150
which is an operator of integral type defined by convolution on the whole domain RN , where C (N ) is some normalization constant. There is wide interest in studying fractional diffusion in model diffusive processes, especially for propagation of longrange diffusive interactions in porous media and infinitesimal generators of stable Lévy processes [3,4]. Fractional operators [5,6], fractional partial differential equations [7–11], and equations for porous media [12–18] have been reviewed elsewhere. Pablo et al. considered the existence, uniqueness, and properties of Cauchy problem (1.1) without a forcing term f ≡ 0, and established a satisfactory theory not only for a weak solution but also for a strong solution [19]. To deal with the nonlocal operator (−△)1/2 , they applied a third way to express the half Laplacian, which was introduced by Caffarelli and Silvestre [20] through the so-called Dirichlet–Neumann operator. The operator can be obtained from harmonic extension of the problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann one. Suppose that v(x, y) is the harmonic extension of the bounded smooth function g (x), written as v = E (g ), as the solution to
△x,y v = 0, v(x, 0) = g (x),
(x, y) ∈ RN × R+ , x ∈ RN ,
(1.4)
where △x,y is the Laplacian in the variables (x, y) ∈ RN × R+ . We define the operator T as g → − ∂v (x, 0). Applying the ∂y operator T twice to g yields T 2 g (x) = T
∂v − (x, 0) = uyy (x, 0) = −△g (x). ∂y
Using the fact that T is a positive operator, we have
−
∂v (x, 0) = (−△)1/2 g (x). ∂y
(1.5)
Applying the third expression for fractional diffusion by harmonic extension to (1.1), we formulate the following problem for the extension function w = E |u|m−1 u :
△w = 0, 1 m −1 w ∂ |w| ∂w − = −f (x, t ), ∂y ∂t w(x, 0, 0) = um 0 (x),
N +1 for (x, y) ∈ R+ , t > 0,
(1.6)
for x ∈ RN , y = 0, t > 0, for x ∈ RN .
1
The solution u of the original (1.1) can be understood as the trace of E |w| m −1 w
for y = 0. This modified problem is
a quasi-stationary problem with a dynamic boundary condition. Compared to the original problem, the advantage of the modified (1.6) is that there is no nonlocal operator. For the case f ≡ 0, Athanasopoulos and Caffarelli [21]showed the continuity of the bounded weak energy solution of (1.6) for m > 1. Pablo et al. established a systematic theory for the FPM equation without a forcing term [19]. They obtained the existence, uniqueness, and properties such as regularity, positivity, and comparison for a suitable weak solution of (1.1) for f ≡ 0. The existence of their solution was based on solving the modified (1.6) (for f ≡ 0) and an L1 -contraction semigroup. The results were then extended to a general FPM equation [22]. Following the ideas of Pablo et al., the main purpose of this paper is to show the existence and uniqueness of a suitable weak solution of (1.1) for a complete FPM equation. We start from the definition of a suitable weak solution of (1.1) according to the quasi-stationary problem (1.6). For convenience, we adopt the same notation as in [19], that is, x¯ = (x, y), Ω = RN++1 , Γ = RN × {0}, throughout the paper. ¯ × [0, T )) and integrating by parts yields Multiplying both sides of the first equation in (1.6) by a test function ϕ ∈ C01 (Ω
∂ϕ dxds + ∂t 0 Ω 0 Γ 1 where u is understood as the trace Tr |w| m −1 w .
−
T
⟨∇w, ∇ϕ⟩dx¯ ds +
T
T
u
0
Γ
f (x, s)ϕ dxds = 0,
1,1
(1.7)
1
Definition 1.1 (Weak Solution). Assume that w ∈ L1 (0, T ); Wloc (Ω ) , u = Tr |w| m −1 w
∈ L1 (Γ × (0, T )) and the
¯ × [0, T )). Then the pair of functions (u, w) is called a weak solution of (1.6) provided identity (1.7) holds for any ϕ ∈ C01 (Ω that u(·, t ) ∈ L1 (Γ ) for any t > 0 and limt →0 u(·, t ) = u0 in L1 (Γ ). For weak solutions the restriction conditions are relaxed and the solutions are obtained in Sobolev classes of weakly differentiable functions. However, we have to give a suitable condition to ensure the uniqueness of the problem. According to the notion for classical porous-medium equations, the concept of a weak energy solution is acceptable.
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Definition 1.2 (Weak Energy Solution). A weak solution pair is called a weak energy solution provided w ∈ L2 [0, T ];
H 1 (Ω ) .
Our main result is the existence and uniqueness of the weak energy solution, as follows. Theorem 1.1. Assume that m > 0, f ∈ C 0, ∞; L1 (RN ) and the data u0 ∈ L1 (RN ) ∩ L∞ (RN ). Then there exists a pair of
unique weak energy solution (w, u) to (1.6), and u ∈ C [0, ∞); L1 (RN ) ∩ L∞ (RN × [0, ∞)). 2. Proof of the main result
In this section, we construct weak solutions for the complete FPM equation (1.1) by viewing it as an abstract evolutionary equation. More precisely, we consider an ordinary differential equation with data in a Banach space and write the solution as u(·, t ) instead of u(t ) for convenience. To solve the abstract differential equations, we use the implicit time discretization method to construct the approximate solution, which exists uniquely according to elliptic theory in a half plane. The convergence of discrete approximated solutions comes from classical work by Crandall and Liggett [23] when the abstract operator has some contraction properties, called an acrretive operator. Finally, we show that the mild solution (the limit of the discrete approximated solution) is in fact the weak solution of the modified Eq. (1.6). Consider the abstract Cauchy problem du
+ A(u) = f , u(0) = u0 , where f ∈ C 0, T ; L1 (RN ) and u0 ∈ L1 (Γ ).
(2.1)
dt
Now we formulate the approximated problems using the discretization process as follows. Taking a partition P = {0 = t0 < t1 < · · · < tk−1 < tk < · · · < tn = T }, tk = kε = kT /n for k = 0, 1, 2, . . . , n, we solve the abstract problem (2.1) as the system of difference relations uε,k − uε,k−1
+ A(uε,k ) = fk , for k = 1, 2, . . . , n, (2.2) ε where {f1 , f2 , . . . , fn } is a discretization of f adapted to the partition P , fk = f (·, kε) for k = 0, 1, . . . , n. Eq. (2.2) can be also written as uε,k + ε A(uε,k ) = uε,k−1 + ε fk .
(2.3)
In fact, the square root of the Laplacian operator A : D(A) → L1 (Γ ) can be defined as A(v) = −Tr
∂ E (|v|m−1 v) ∂y
(2.4)
and D(A) = v ∈ L1 (Γ ) ∩ L∞ (Γ ) | A(v) ∈ L1 (Γ ), ∥v∥L∞ (Γ ) ≤ ∥u0 ∥L∞ (Γ ) + T ∥f ∥L∞ (Γ ) .
(2.5)
Applying the discretization process to the modified (1.6), we obtain the discretized problems for k = 1, 2, . . . , n:
△wε,k = 0, ∂wε,k = uε,k − uε,k−1 − ε fk , ε ∂y
in Ω , (2.6)
on Γ ,
1
with the first step uε,0 = u0 on Γ , and uε,k can be understood as Tr |wε,k | m −1 wε,k . This is an elliptic equation with a nonlinear Neumann boundary condition in the half plane. The solvability of the elliptic problem has been proved elsewhere [19] and is stated as a lemma here. Lemma 2.1. For any ε > 0, consider the following Neumann problem for an elliptic equation:
△w = 0, ∂w 1 = |w| m −1 w − g , ε ∂y
in Ω , (2.7)
on Γ .
1
For g ∈ L1 (Γ ) ∩ L∞ (Γ ), there exists a unique weak solution w ∈ H 1 (Ω ) such that u = Tr (|w| m −1 w)
∈ L1 (Γ ) ∩ L∞ (Γ )
and ∥w∥L∞ (Γ ) ≤ ∥g ∥L∞ (Γ ) . Moreover, if w and w ˜ are the solutions corresponding to data g and g˜ , then we have
Γ
1
1
|w| m −1 w − |w| ˜ m −1 w ˜
+
g − g˜ + dx,
dx ≤ Γ
¯ provided that g ≥ 0 in Γ . which implies that w ≥ 0 in Ω
148
M. Fan et al. / Computers and Mathematics with Applications 67 (2014) 145–150
Fig. 1. Construction of the approximate solution wε .
As a direct consequence, we have the following results for the solvability and contraction properties of the discretized problems. Proposition 2.2. For any ε > 0, consider the following discretized problems:
△wε,k = 0, ∂wε,k 1 1 − |wε,k | m −1 wε,k = −|wε,k−1 | m −1 wε,k−1 − ε fk , ε ∂y
in Ω , (2.8)
on Γ .
For any given wε,k−1 ∈ H 1 (Ω ) and fk ∈ L1 (Γ ) ∩ L∞ (Γ ), there exists a unique weak solution wε,k ∈ H 1 (Ω ), such that 1
uε,k = Tr(|wε,k | m −1 wε,k ) ∈ L1 (Γ ) ∩ L∞ (Γ ) and
∥uε,k ∥L∞ (Γ ) ≤ ∥uε,k−1 ∥L∞ (Γ ) + ε∥fk ∥L∞ (Γ ) .
(2.9)
Moreover, uε,k and u˜ ε,k are solutions corresponding to the data uε,0 = u0 and u˜ ε,0 = u˜ 0 . Then we have
uε,k − u˜ ε,k + dx ≤
Γ
u0 − u˜ 0 + dx
Γ
for any k = 0, 1, 2, . . . , N. Next, we establish a weak solution for the evolutionary problem using the process mentioned above. Theorem 2.3. There exists a weak solution (u, w) to (1.6) and u(·, t ) ∈ L1 (Γ ) ∩ L∞ (Γ ) for every t > 0 and w ∈ L2 [0, T ];
H 1 (Ω ) . Moreover, there exists almost one weak energy solution w ∈ L2 [0, T ]; H 1 (Ω ) to (1.6) (Fig. 1). Proof. Dividing the interval [0, T ] into n subintervals, the length of each subinterval is T /n. We construct the approximate solution wε (x), which is a piecewise constant function in each interval (tk−1 , tk ] for tk = kε, k = 1, 2, . . . , n. For each k > 0, wε,k satisfies
1wε,k = 0 ∂wε,k = uε,k − uε,k−1 + ε fk ε ∂y
in Ω , (2.10)
on Γ ,
where uε,0 = u0 (x), fk = f (x, tk−1 ). Consider the limit (uε , wε ) → (u, w) as ε → 0 in L1loc (Ω ), which is a so-called mild solution. It follows from classical semigroup theory that u ∈ C [0, ∞); L1 (Γ ) . Furthermore, multiplying (2.10) by wε,k and integrating by parts yields
ε
Ω
∇wε,k 2 dx¯ ≤
Γ
|uϵ,k |m−1 uϵ,k | uε,k−1 − uε,k + εfk dx.
(2.11)
It follows from Young’s inequality that
ε
Ω
∇wε,k 2 dx¯ ≤
1
m+1
Γ
|uε,k−1 |m+1 dx −
Γ
m+1 uε,k |uϵ,k |m−1 uϵ,k |dx dx + ε Γ
for k = 1, 2, . . . , n. Then adding from k = 1 to k = n gives T
0
Ω
|∇wε (¯x, t )|2 dx¯ dt ≤
1 m+1
Γ
|u0 (x)|m+1 dx +
T
0
Γ
|uε (x, t )|m f (x, t )dxdt .
(2.12)
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Taking the limit ε → 0 implies that w ∈ L2 [0, T ]; H 1 (Ω ) . Moreover, choosing some appropriate test function gives the following weak form:
Ω
⟨∇wε,k , ∇ϕ⟩dx¯ =
Γ
1 uε,k−1 − uue ,k ϕ + fk ϕ dx. ε
Integrating over (tk−1 , tk ) and adding these together leads to n
tk
T
Γ
0
−
ε
((uε,k−1 (x, t ) − uε,k )) + fk ϕ(x, 0, t )dxdt
ϕ(x, 0, t + ε) − ϕ(x, 0, t ) 1 ε u0 (x)ϕ(x, 0, t )dxdt dxdt + ε ε 0 Γ T f (x, t )ϕ(x, 0, t )dxdt . uε (x, T )ϕ(x, 0, t )dxdt +
uε (x, t )
= 1
ε
Γ
tk−1
k =1
1
T
Γ
T −ε
(2.13)
Γ
0
Taking the limit ε → 0, the second and third integral terms on the right-hand side vanish and we obtain the weak formulation (1.7). Thus, we can conclude that the function (u, w) is indeed a weak solution to (1.6). Next we show the uniqueness of the weak energy solution to (1.6) using the standard energy method. Let (ui , wi ) be the weak energy solutions to (1.6) corresponding to the same initial data g (x) for i = 1, 2. Consider the deviation of the two solutions w(¯x, t ) = w1 (¯x, t ) − w2 (¯x, t ), which satisfy wi = E |ui |m−1 ui for i = 1.2,
△w = 0, ∂w ∂(u1 − u2 ) − = 0, ∂y ∂t w(x, 0, 0) = 0,
for (x, y) ∈ Ω , t > 0, for x ∈ Γ , t > 0,
(2.14)
for x ∈ Γ .
Choosing an appropriate test function
ϕ(¯x, t ) =
T
w(¯x, s)ds
for 0 ≤ t ≤ T ,
t
0,
for t ≥ T ,
and multiplying it by (2.14), integration by parts yields T
Ω
0
∇w(¯x, t ),
T
T
∇w(¯x, s)ds dx¯ dt +
t
0
Γ
m (u1 − u2 )(um 1 − u2 )dxdt = 0.
That is,
2
T
1
Ω
0
2 ∇w(¯x, s)ds dx¯ +
T 0
Γ
m (u1 − u2 )(um 1 − u2 )dxdt = 0,
which implies that u1 = u2 in Γ and w1 = w2 in Ω .
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