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Regularity of weak solutions of the Cauchy problem to a fractional porous medium equation
- Lei Zhang^{1, 2} and
- Shan Li^{3}Email author
https://doi.org/10.1186/s13661-015-0286-8
© Zhang and Li; licensee Springer. 2015
- Received: 21 August 2014
- Accepted: 12 January 2015
- Published: 11 February 2015
Abstract
This paper concerns the regularity of the weak solutions of the Cauchy problem to a fractional porous medium equation with a forcing term. In the recent work (Fan et al. in Comput. Math. Appl. 67:145-150, 2014), the authors established the existence of the weak solution and the uniqueness of the weak energy solution. In this paper, we show that the every nonnegative bounded weak energy solution is indeed a strong solution.
Keywords
- fractional diffusion
- porous medium equation
- weak solution
- strong solution
MSC
- 26A33
- 35K57
1 Introduction
Recently, the fractional porous medium equation \(u_{t}+(-\Delta)^{\frac{\alpha}{2}}u^{m}= 0\) (\(m>0\), \(\alpha>0\)) has been investigated by Pablo et al. in [2] for \(\alpha =1\) and in [3] for general case \(0 < \alpha< 2\). Systematic and satisfactory results on the weak solutions to the Cauchy problem to the fractional porous medium problem have been obtained, including the existence, uniqueness, comparison principle, and regularity to the suitable weak solutions.
The interest in studying the fractional diffusion in modeling diffusive processes has a wide literature, especially studying the long-range diffusive interaction in porous medium type propagation and infinitesimal generators of stable Lévy processes [4, 5]. We would like to refer to the survey papers [6] and [7] for the fractional operators and to [8–10] for fractional partial differential equations. For instance, a method based on the Jacobi-tau approximation for solving multi-term time-space-fractional partial differential equations. A spectral-tau algorithm is based on the Jacobi operational matrix for a numerical solution of time fractional diffusion-wave equations. (See [11] for details.) On the other side, there is much literature on the porous medium equations (see the classical book [12] and references [1, 10, 13–18] and references therein).
Before we state the main results in this paper, we present some definitions as regards the fractional operator and the weak solutions to the problem (1.1). The nonlocal operator square root Laplacian operator can be illustrated in the following three items.
We focus on the quasi-stationary problem (1.5) with a dynamical boundary condition. Once w is solved, the solution u to the problem (1.1) can be understand as the trace of \(|w|^{\frac{1}{m}-1}w\).
Fan et al. [1] established the existence of the weak solution and the uniqueness of the weak energy solution to the problem (1.1). The weak solution and the weak energy solution of the Cauchy problem are defined as follows.
Then a pair \((u,w)\) of functions is called a weak solution to the problem (1.5) if \(w\in L^{1} ((0,T);W^{1,1}_{\mathrm{loc}}(\Omega) )\), \(u=\operatorname{Tr} (|w|^{\frac{1}{m}-1}w )\in L^{1}(\Gamma\times(0,T))\), and \(u_{0}(x)\in L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})\), the equality (1.6) holds for any \(\varphi\in C_{0}^{1}(\bar{\Omega}\times[0,T))\).
Furthermore, if a pair weak solution pair satisfies \(w\in L^{2} ([0,T];H^{1}(\Omega ) )\), we call the weak solution a weak energy solution.
In a previous work [1], Fan et al. obtained the following existence and uniqueness results.
Theorem 1.1
Assume that \(m>0\), \(f\in C (0,\infty; L^{1}({\mathbb{R}}^{N}) )\) and the data \(u_{0}\in L^{1}({\mathbb{R}}^{N})\cap L^{\infty}({\mathbb{R}}^{N})\), then there exists a unique weak energy solution \((w,u)\) to the problem (1.4), and \(u\in C ([0,\infty);L^{1}({\mathbb{R}}^{N}) )\cap L^{\infty}({\mathbb{R}}^{N}\times[0,\infty))\).
We are ready to announce the main result, proved in the next section.
Theorem 1.2
2 Proof of the main result
In this section, we prove Theorem 1.2. Namely, it suffices to show that the time partial derivative of u is an \(L^{1}\) function. As the first step, we will show that the time-increment quotients are bounded in \(L^{1}(\Gamma)\), where \(\Gamma={\mathbb{R}}^{N}\times\{0\}\), and thus the limit is still in \(L^{1}({\mathbb{R}}^{N})\).
2.1 Time-increment quotients of the solution are bounded in \(L^{1}({\mathbb{R}}^{N})\)
Proposition 2.1
Proof
2.2 Regularity of time derivative of solution
Proposition 2.2
Proof
Due to the lack of regularity of u in time, we also use the Steklov average of u instead of u itself.
2.3 Proof of Theorem 1.2
Let \(v=u^{\frac{m+1}{2}}\), we claim that \(u^{\frac{m+1}{2}}\in W^{1,1}((\tau,T);L^{1}({\mathbb{R}}^{N}))\).
In fact, since \(\partial_{t}u^{(m+1)/2}\in L^{2}_{\mathrm{loc}} ((0,\infty);L^{2}({\mathbb{R}}^{N}) )\), we can get \(\partial_{t}u^{(m+1)/2}\in L^{1} (K\times(\tau,T) )\) for any compact set \(K\subset\subset{\mathbb{R}}^{N}\).
Declarations
Acknowledgements
The authors would like to thank the referees for helpful suggestions to update our paper. Shan Li is supported in part by NSFC grant (No. 71372189), Project funded by China Postdoctoral Science Foundation (No. 2013M542285) and Social sciences planning project in Sichuan province (No. SC14TJ06).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Fan, MS, Li, S, Zhang, L: Weak solution of the equation for a fractional porous medium with a forcing term. Comput. Math. Appl. 67, 145-150 (2014) View ArticleMathSciNetGoogle Scholar
- Pablo, A, Quirós, F, Rodríguez, A, Vázquez, JL: A fractional porous medium equation. Adv. Math. 226, 1378-1409 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Pablo, A, Quirós, F, Rodríguez, A, Vázquez, JL: A general fractional porous medium equation. Commun. Pure Appl. Math. 65, 1242-1284 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Applebaum, D: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Stud. Adv. Math., vol. 116. Cambridge University Press, Cambridge (2009) View ArticleMATHGoogle Scholar
- Bertoin, J: Lévy Processes. Cambridge Tracts in Math., vol. 121. Cambridge University Press, Cambridge (1996) MATHGoogle Scholar
- Machado, JT, Kiryakova, V, Mainardi, F: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16, 1140-1153 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Ragusa, MA: Necessary and sufficient condition for a VMO function. Appl. Math. Comput. 218(24), 11952-11958 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Bhrawy, AH, Zaky, MA, Baleanu, D: New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Rom. Rep. Phys. 67(2), 1-13 (2015) Google Scholar
- Bhrawy, AH, Zaky, MA, Baleanu, D: A new formula for fractional integrals of Chebyshev polynomials: application for solving multi-term fractional differential equations. Appl. Math. Model. 37(6), 4245-4252 (2013) View ArticleMATHMathSciNetGoogle Scholar
- Doha, EH, Bhrawy, AH, Ezz-Eldien, SS: Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method. Cent. Eur. J. Phys. 11(10), 1494-1503 (2013) View ArticleMATHGoogle Scholar
- Bhrawy, AH, Baleanu, D: A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients. Rep. Math. Phys. 72, 219-233 (2013) View ArticleMATHMathSciNetGoogle Scholar
- Vázquez, JL: The Porous Medium Equation: Mathematical Theory. Oxford Mathematical Monographs. Clarendon, Oxford (2007) MATHGoogle Scholar
- Du, LL, Xiang, XY: A further blow-up analysis for a localized porous medium equation. Appl. Math. Comput. 179, 200-208 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Du, LL, Yao, ZA: Localization of blow-up points for a nonlinear nonlocal porous medium equation. Commun. Pure Appl. Anal. 6, 183-190 (2007) MATHMathSciNetGoogle Scholar
- Du, LL: Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources. J. Comput. Appl. Math. 202, 237-247 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Du, LL: Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources. J. Math. Anal. Appl. 324, 304-320 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Du, LL, Mu, CL, Fan, MS: Global existence and non-existence for a quasilinear degenerate parabolic system with non-local source. Dyn. Syst. 20, 401-412 (2005) View ArticleMATHMathSciNetGoogle Scholar
- Fan, MS, Mu, CL, Du, LL: Uniform blow-up profiles for a nonlocal degenerate parabolic system. Appl. Math. Sci. 1, 13-23 (2007) MATHMathSciNetGoogle Scholar
- Landkof, NS: Foundations of Modern Potential Theory. Grundlehren Math. Wiss., vol. 180. Springer, New York (1972) View ArticleMATHGoogle Scholar
- Stein, EM: Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser., vol. 30. Princeton University Press, Princeton (1970) MATHGoogle Scholar
- Caffarelli, L, Silvestre, L: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245-1260 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Bénilan, P, Crandall, MG: Regularizing effects of homogeneous evolution equations. In: Contributions to Analysis and Geometry, Baltimore, MD, 1980, pp. 23-39. Johns Hopkins University Press, Baltimore (1981) Google Scholar
- Bénilan, P, Gariepy, R: Strong solutions in \(L^{1}\) of degenerate parabolic equations. J. Differ. Equ. 119(2), 473-502 (1995) View ArticleMATHMathSciNetGoogle Scholar