imported>Luis |
imported>Nestor |
| Line 1: |
Line 1: |
| $$ | | The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely |
| \newcommand{\dd}{\mathrm{d}}
| |
| \newcommand{\R}{\mathbb{R}}
| |
| $$
| |
|
| |
|
| The Harnack inequality refers to a control of the maximum of a nonnegative solution of an equation by its minimum. Unlike the local case (either [[De Giorgi-Nash-Moser theorem]] or [[Krylov-Safonov theorem]]), for nonlocal equations one needs to assume that the function is nonnegative in the full space.
| | \[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\] |
|
| |
|
| The Harnack inequality is tightly related to [[Holder estimates]] for solutions to elliptic/parabolic equations. For a large class of problems both statements are equivalent. But there are simple cases (stable processes with the spectral measure consisting of atoms) where the Harnack inequality fails but [[Hölder estimates]] still hold true.
| | \[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \] |
|
| |
|
| The result can hold either in the parabolic or elliptic setting. The parabolic Harnack inequality trivially implies the elliptic one. The reverse implication is not automatic, and the proof in the parabolic case may have some extra difficulties compared to the elliptic case.
| | and |
|
| |
|
| == Elliptic case == | | \[ u_t +(-\Delta)^{s}(u^m) = 0 \] |
|
| |
|
| In the elliptic setting, the Harnack inequality refers to the following type of result: if a function $u: \R^n \to \R$ satisfies an elliptic equation $ L_x u (x) = f(x)$ in the unit ball $B_1$ and is nonnegative in the full space $\R^n$, then
| | These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order [[Quasilinear equations]]. |
| \[ \sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + \|f\| \right). \]
| |
|
| |
|
| The norm $\|f\|$ may depend on the type of equation. | | The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which [[instantaneous speed of propagation]] holds. |
|
| |
|
| == Parabolic case == | | This means that the first model presents us with a [[free boundary problem]]. For this model global existence and Hölder continuity of weak solutions have been recently obtained <ref name="CV1"/>, there is almost nothing known about the properties of its free boundary, making it a rich source of open questions. |
|
| |
|
| In the parabolic setting, the Harnack inequality refers to the following type of result: if a function $u: [-1,0] \times \R^n \to \R$ satisfies a parabolic equation $ u_t - L_x u (x) = f(x)$ in the unit cylinder $(-1,0) \times B_1$ and is nonnegative in the full space $[-1,0] \times \R^n$, then
| | For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years <ref name="PQRV" /ref>. |
| \[ \sup_{[-1/2,-1/4] \times B_{1/2}} u \leq \left(\inf_{[-1/4,0] \times B_{1/2}} u + \|f\| \right). \]
| |
| | |
| The norm $\|f\|$ may depend on the type of equation.
| |
| | |
| == Concrete examples ==
| |
| | |
| The Harnack inequality as above is known to hold in the following situations.
| |
| | |
| * '''Generalizad elliptic and parabolic [[Krylov-Safonov]]'''. If $L_x u(x)$ is a symmetric integro-differential operator of the form
| |
| \[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)) K(x,y) \dd y \]
| |
| with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda |y|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda |y|^{-n-s}$.
| |
| | |
| In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $||f||$ refers to $||f||_{L^\infty(B_1)}$ <ref name="CS"/><ref name="lara2011regularity"/>. It is a generalization of [[Krylov-Safonov]] theorem.
| |
| | |
| * '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form
| |
| \[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)- y \cdot \nabla u(x) \chi_{B_1}(y)) K(x,y) \dd y \]
| |
| with uniformly elliptic of variable order: $\lambda |y|^{-n-s_1} \leq K(x,y) \leq \Lambda |y|^{-n-s_2}$ and $0<s_1 < s_2 < 2$ and $s_2 - s_1 < 1$, then
| |
| the elliptic Harnack inequality holds if $f \equiv 0$<ref name="BK"/>. The constants in this result blow up as $s_2 \to 2$, so it does not generalize [[Krylov-Safonov]] theorem. The proof uses probability and was based on a previous result with fixed order <ref name="BL"/>.
| |
| | |
| It is conceivable that a purely analytic proof could be done using the method of the corresponding [[Holder estimate]] <ref name="S"/>, but such proof has never been done.
| |
| | |
| * '''Gradient flows of symmetric Dirichlet forms with variable order'''. If $u_t - L_x u(x)=0$ is the gradient flow of a [[Dirichlet form]]:
| |
| \[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 K(x,y)\, \dd x \dd y. \]
| |
| for kernels $K$ such that $K(x,y)=K(y,x)$ and $\lambda |x-y|^{-n-s_1} \leq K(x,y) \leq \Lambda |x-y|^{-n-s_2}$ for some $0<s_1<s_2<2$ and $|x-y|$ sufficiently small. Then the parabolic Harnack inequality holds if $f \equiv 0$ for some constant $C$ which a priori '''blows up as $s_2 \to 2$''' <ref name="BBCK"/>.
| |
| | |
| It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.
| |
|
| |
|
| == References == | | == References == |
| {{reflist|refs= | | {{reflist|refs= |
| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref> | | <ref name="CV1"> {{Citation | last1=Caffarelli | first1=Luis | last2=Vazquez | first2=Juan | title=Nonlinear Porous Medium Flow with Fractional Potential Pressure | url=http://dx.doi.org/10.1007/s00205-011-0420-4 | publisher=Springer Berlin / Heidelberg | year=2011 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–29}} </ref> |
| <ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
| | |
| <ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Harnack inequalities for non-local operators of variable order | url=http://dx.doi.org/10.1090/S0002-9947-04-03549-4 | doi=10.1090/S0002-9947-04-03549-4 | year=2005 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=357 | issue=2 | pages=837–850}}</ref>
| | <ref name="PQRV">{{Citation | last1=Pablo | first1=Arturo de | last2=Quirós | first2=Fernando | last3=Rodríguez | first3=Ana | last4=Vazquez | first4=Juan Luis | title=A fractional porous medium equation | url=http://www.sciencedirect.com/science/article/pii/S0001870810003130 | doi=DOI: 10.1016/j.aim.2010.07.017 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=1378–1409}} |
| <ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin | first2=David A. | title=Harnack inequalities for jump processes | url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 | year=2002 | journal=Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis | issn=0926-2601 | volume=17 | issue=4 | pages=375–388}}</ref>
| | </ref> |
| <ref name="BBCK">{{Citation | last1=Barlow | first1=Martin T. | last2=Bass | first2=Richard F. | last3=Chen | first3=Zhen-Qing | last4=Kassmann | first4=Moritz | title=Non-local Dirichlet forms and symmetric jump processes | url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 | doi=10.1090/S0002-9947-08-04544-3 | year=2009 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=361 | issue=4 | pages=1963–1999}}</ref> | |
| <ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
| |
| <ref name="lara2011regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | journal=Calculus of Variations and Partial Differential Equations | year=2011 | pages=1--34}}</ref>
| |
| }} | | }} |
| | |
| | [[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]] |
The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely
\[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\]
\[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \]
and
\[ u_t +(-\Delta)^{s}(u^m) = 0 \]
These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order Quasilinear equations.
The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which instantaneous speed of propagation holds.
This means that the first model presents us with a free boundary problem. For this model global existence and Hölder continuity of weak solutions have been recently obtained [1], there is almost nothing known about the properties of its free boundary, making it a rich source of open questions.
For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years Cite error: Invalid <ref> tag; invalid names, e.g. too many
[2]
}}