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Статьи
Regularity of solutions of the fractional porous medium flow with exponent $1/2$
L. Caffarelliab,
J. L. Vázquezc a Institute for Computational Engineering and Sciences, USA
b School of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712-1082, USA
c Universidad Autónoma de Madrid, Departamento de Matemáticas, 28049, Madrid, Spain
Аннотация:
The object of study is the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is
$u_t=\nabla\cdot(u\nabla(-\Delta)^{-1/2}u)$. For definiteness, the problem is posed in
$\{x\in\mathbb R^N, t\in\mathbb R\}$ with nonnegative initial data
$u(x,0)$ that is integrable and decays at infinity. Previous papers have established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation, as well as the boundedness of nonnegative solutions with
$L^1$ data, for the more general family of equations
$u_t=\nabla\cdot(u\nabla(-\Delta)^{-s}u)$,
$0<s<1$.
Here, the
$C^\alpha$ regularity of such weak solutions is established in the difficult fractional exponent case
$s=1/2$. For the other fractional exponents
$s\in(0,1)$ this Hölder regularity has been proved in an earlier paper. Continuity was under question because the nonlinear differential operator has first-order differentiation. The method combines delicate De Giorgi type estimates with iterated geometric corrections that are needed to avoid the divergence of some essential energy integrals due to fractional long-range effects.
Ключевые слова:
porous medium equation, fractional Laplacian, nonlocal diffusion operator, Hölder regularity.
Поступила в редакцию: 06.01.2015
Язык публикации: английский