Time Regularity for Nonlocal Parabolic Equations - Revision history

imported>Hector at 16:29, 8 July 2016

2016-07-08T16:29:34Z

← Older revision		Revision as of 11:29, 8 July 2016
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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 ~~proven~~ 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.		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.		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.

imported>Hector at 16:26, 8 July 2016

2016-07-08T16:26:27Z

← Older revision		Revision as of 11:26, 8 July 2016
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	{{stub}}		{{stub}}

	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 proven 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>
	}}		}}

imported>Hector at 16:11, 8 July 2016

2016-07-08T16:11:55Z

← Older revision		Revision as of 11:11, 8 July 2016
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			{{stub}}

	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}

imported>Hector at 16:09, 8 July 2016

2016-07-08T16:09:43Z

← Older revision		Revision as of 11:09, 8 July 2016
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	~~Test~~		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<ref name="MR3148110"/>.



			== References ==
			{{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>
			}}

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2016-07-08T15:43:52Z

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