Fractional porous media equations: existence and uniqueness of weak solutions with measure data
The authors study solutions properties for the boundary nonlinear value problem \[ (*):\;\rho u_t(\cdot,\cdot)+(-\Delta)^s[u(\cdot,\cdot)]^m=0\text{ in } \mathbb R^d\times\mathbb R_+\quad \text{and}\quad \rho u(\cdot,\cdot)=\mu\text{ on }\mathbb R^d\times\{0\}, \] where \((s,d,m)\in (0,1)\times (2s,\infty) \times (1,\infty)\), \(\mu\) represents a positive Radon measure on \(\mathbb R^d\) and \(\rho\) is a frame measurable weight function on \(\mathbb R^d\) such that its frame bounds are given, up to multiplicative constants, in terms of the power of \(|x|^{-\gamma}\) for \(\gamma\in [0,2s)\) whenever \(x\) belongs to the complement of the open unit ball in \( \mathbb R^d\) (resp. \(|x|^{-\gamma_0}\) for \(\gamma_0\in [0,\gamma]\) and \(|x|<1\)). Then, the authors state their main result that \((*)\) has a weak solution \(u(\cdot,\cdot)\in L_p(\mathbb R^d,\rho(x)dx)\), the set of \(p\)-integrable functions with respect to the measure \(\rho dx\) for \(p\in [1,\infty]\) and the \(L_\infty\)-norm of \(u(\cdot,t)\) is bounded by \(K_{\gamma,d,s,m}t^{-\alpha}\|u(\cdot,t)\|^{\beta}_{L_1(\mathbb R^d,\rho(x)dx)}\) such that \(\alpha,\beta, K_{\gamma,d,m,s}\) are constants depending on \(\gamma\), \(d\), \(s\), \(m\), where the two first constants are given explicitly. Furthermore, the weak solution satisfies some energy estimates (Theorem 3.2). About the proof of the main result, the authors use an approximation process and substituting \(\mu\) by \(\rho u(\cdot,0)\), where \(u(\cdot,0)\) is a bounded function belonging to \(L_1(\mathbb R^d,\rho(x)dx)\) (Lemma 4.2). The uniqueness of the weak solution for \((*)\) is stated in Theorem 3.4 and due to the toughness for proving this theorem, the authors provide the fundamental ideas, see the fifth section. The authors conclude their work by stating that the operator \(\rho^{-1}(-\Delta)^{s}\) is self-adjoint, positive and generates a Markov semigroup on \(L_2(\mathbb R^d, \rho dx)\) (Theorem 3.7), and the proof is pinpointed in Appendix B.
Stroock-Varopoulos and Caffarelli-Kohn-Nirenberg inequalities
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