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| Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative |
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Citation： |
Mark ALLEN,Luis CAFFARELLI,Alexis VASSEUR.Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative[J].Chinese Annals of Mathematics B,2017,38(1):45~82 |
| Page view： 761
Net amount： 811 |
Authors： |
Mark ALLEN; Luis CAFFARELLI;Alexis VASSEUR |
Foundation： |
This work was supported by NSG grant DMS-1303632, NSF grant
DMS-1500871 and NSF grant DMS-1209420. |
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| Abstract： |
The authors study a porous medium equation with a right-hand side.
The operator has nonlocal diffusion effects given by an inverse
fractional Laplacian operator. The derivative in time is also
fractional and is of Caputo-type, which takes into account
``memory''. The precise model is
$$
\D_t^{\alpha} u - \text{div}(u(-\Delta)^{-\sigma} u) = f, \quad
0<\sigma <\frac12.
$$
This paper poses the problem over $\{t\in \R^+, x\in \R^n\}$ with
nonnegative initial data $u(0,x)\geq 0 $ as well as the right-hand
side $f\geq 0$. The existence for weak solutions when $f,u(0,x)$
have exponential decay at infinity is proved. The main result is
H\"older continuity for such weak solutions. |
Keywords： |
Caputo derivative, Marchaud derivative, Porous medium equation,H"older continuity, Nonlocal diffusion |
Classification： |
35K55, 26A33, 35D10 |
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