Time Regularity for Nonlocal Parabolic Equations: Difference between revisions
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One of the phenomena that are exclusive to nonlocal parabolic equations is how the boundary data posed in the complement of a given domain might drastically affect the regularity of the solution. Consider the fractional heat equation of order $\sigma\in(0,2)$ | One of the phenomena that are exclusive to nonlocal parabolic equations is how the boundary data, posed in the complement of a given domain might drastically affect the regularity of the solution. Consider the fractional heat equation of order $\sigma\in(0,2)$ | ||
\begin{alignat*}{3} | \begin{alignat*}{3} | ||
u_t &= \Delta^{\sigma/2} u \quad &&\text{ in } \quad &&B_1\times\mathbb R\\ | u_t &= \Delta^{\sigma/2} u \quad &&\text{ in } \quad &&B_1\times\mathbb R\\ | ||
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If $g$ has a sudden discontinuity in time then it is expected that the nonlocal effect, transmitted into the equation by $\Delta^\sigma$, makes $u_t$ discontinuous in time. A specific example was presented by Chang-Lara and Dávila<ref name="MR3148110"/>. | If $g$ has a sudden discontinuity in time then it is expected that the nonlocal effect, transmitted into the equation by $\Delta^\sigma$, makes $u_t$ discontinuous in time. A specific example was presented by Chang-Lara and Dávila<ref name="MR3148110"/>. | ||
For fully nonlinear, nonlocal parabolic equations it was established by Chang-Lara and Kriventsov<ref name="2015arXiv150507889C"/> that $u_t$ is Holder continuous provided that the boundary is Holder continuous in time. Under the assumption that $g$ is merely bounded, it was also proved that $u$ is Holder continuous in time for every exponent $\beta \in(0,1)$ with an estimate that degenerates as $\beta$ approaches 1. It remains open whether Lipschitz regularity in time also holds under the previous hypothesis. | |||
One application of the result in <ref name="2015arXiv150507889C"/> was to extend the Evans-Krylov estimate for parabolic equations under a mild continuity hypothesis for the boundary data. | |||
== References == | == References == | ||
{{reflist|refs= | {{reflist|refs= | ||
<ref name="MR3148110">{{Citation | last1=Chang-Lara | first1= Héctor | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | url=http://dx.doi.org/10.1007/s00526-012-0576-2 | journal=Calc. Var. Partial Differential Equations | issn=0944-2669 | year=2014 | volume=49 | pages=139--172 | doi=10.1007/s00526-012-0576-2}}</ref> | <ref name="MR3148110">{{Citation | last1=Chang-Lara | first1= Héctor | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | url=http://dx.doi.org/10.1007/s00526-012-0576-2 | journal=Calc. Var. Partial Differential Equations | issn=0944-2669 | year=2014 | volume=49 | pages=139--172 | doi=10.1007/s00526-012-0576-2}}</ref> | ||
<ref name="2015arXiv150507889C">{{Citation | last1=Chang-Lara | first1= Héctor | last2=Kriventsov | first2= Dennis | title=Further Time Regularity for Non-Local, Fully Non-Linear Parabolic Equations | journal=ArXiv e-prints | year=2015}}</ref> | |||
}} | }} | ||
Latest revision as of 10:29, 8 July 2016
This article is a stub. You can help this nonlocal wiki by expanding it.
One of the phenomena that are exclusive to nonlocal parabolic equations is how the boundary data, posed in the complement of a given domain might drastically affect the regularity of the solution. Consider the fractional heat equation of order $\sigma\in(0,2)$ \begin{alignat*}{3} u_t &= \Delta^{\sigma/2} u \quad &&\text{ in } \quad &&B_1\times\mathbb R\\ u &= g \quad &&\text{ on } \quad &&(\mathbb R^n \setminus B_1)\times\mathbb R \end{alignat*} If $g$ has a sudden discontinuity in time then it is expected that the nonlocal effect, transmitted into the equation by $\Delta^\sigma$, makes $u_t$ discontinuous in time. A specific example was presented by Chang-Lara and Dávila[1].
For fully nonlinear, nonlocal parabolic equations it was established by Chang-Lara and Kriventsov[2] that $u_t$ is Holder continuous provided that the boundary is Holder continuous in time. Under the assumption that $g$ is merely bounded, it was also proved that $u$ is Holder continuous in time for every exponent $\beta \in(0,1)$ with an estimate that degenerates as $\beta$ approaches 1. It remains open whether Lipschitz regularity in time also holds under the previous hypothesis.
One application of the result in [2] was to extend the Evans-Krylov estimate for parabolic equations under a mild continuity hypothesis for the boundary data.
References
- ↑ Chang-Lara, Héctor; Dávila, Gonzalo (2014), "Regularity for solutions of non local parabolic equations", Calc. Var. Partial Differential Equations 49: 139--172, doi:10.1007/s00526-012-0576-2, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-012-0576-2
- ↑ 2.0 2.1 Chang-Lara, Héctor; Kriventsov, Dennis (2015), "Further Time Regularity for Non-Local, Fully Non-Linear Parabolic Equations", ArXiv e-prints