Drift-diffusion equations - Revision history

imported>Luis: /* Divergence-free vector fields */

2015-04-14T15:22:22Z

Divergence-free vector fields

← Older revision		Revision as of 10:22, 14 April 2015
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	=== Divergence-free vector fields ===		=== Divergence-free vector fields ===
	If the vector field $b$ is divergence free, the method of [[De Giorgi-Nash-Moser]] can be adapted to show that the solution $u$ becomes immediately Holder continuous.		If the vector field $b$ is divergence free, the method of [[De Giorgi-Nash-Moser]] can be adapted to show that the solution $u$ becomes immediately Holder continuous. This is a variation of the [[Holder estimates]] for integro-differential equations in variational form.

	In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(BMO)$ <ref name="CV"/> <ref name="KN"/>.		In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(BMO)$ <ref name="CV"/> <ref name="KN"/>.

imported>Luis: /* $C^{1,\alpha}$ estimates */

2012-03-07T19:14:12Z

$C^{1,\alpha}$ estimates

← Older revision		Revision as of 14:14, 7 March 2012
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	Under subcritical assumptions on $b$, one can sometimes show $C^{1,\alpha}$ estimates. These are strong regularity estimates that imply that the solutions are classical.		Under subcritical assumptions on $b$, one can sometimes show $C^{1,\alpha}$ estimates. These are strong regularity estimates that imply that the solutions are classical.

	The result is the following. Assume that $b \in L^\infty(C^{1-2s+\alpha})$ for some $\alpha>0$, then the solution $u$ belongs to $L^\infty(C^{1,\alpha})$. This result ~~holds both if~~ $s \~~geq~~ 1/2$ <ref name="P"/> as if $s \~~leq~~ 1/2$ <ref name="S3"/>.		The result is the following. Assume that $b \in L^\infty(C^{1-2s+\alpha})$ for some $\alpha>0$, then the solution $u$ belongs to $L^\infty(C^{1,\alpha})$. This result has been proved for the case $s \leq 1/2$ <ref name="S3"/>, and it has also been proved for a stationary eigenvalue problem if $s \geq 1/2$ <ref name="P"/>.

	== Scale invariant results ==		== Scale invariant results ==

imported>Luis at 00:06, 5 March 2012

2012-03-05T00:06:40Z

← Older revision		Revision as of 19:06, 4 March 2012
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	A drift-(fractional)diffusion equation refers to an evolution equation of the form		A drift-(fractional)diffusion equation refers to an evolution equation of the form
	\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\]		\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\]
	where $b$ is any vector ~~fields~~. The stationary version can also be of interest		where $b$ is any vector field. The stationary version can also be of interest
	\[ b \cdot \nabla u + (-\Delta)^s u = 0.\]		\[ b \cdot \nabla u + (-\Delta)^s u = 0.\]

imported>Luis: /* Kato classes */

2012-02-23T17:47:12Z

Kato classes

← Older revision		Revision as of 12:47, 23 February 2012
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	=== Kato classes ===		=== Kato classes ===
	The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller making the drift term a small perturbation of the fractional diffusion. The ~~precise~~ assumption ~~varies from article to article and they are~~ tailor made in order to make ~~some~~ [[bootstrapping]] ~~argument~~ possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation <ref name="BJ"/> <ref name="CKS"/>.		The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller, effectively making the drift term a small perturbation of the fractional diffusion. The assumption is tailor made in order to make [[bootstrapping]] arguments possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation <ref name="BJ"/> <ref name="CKS"/>.

	=== $C^{1,\alpha}$ estimates ===		=== $C^{1,\alpha}$ estimates ===

imported>Luis: /* Kato classes */

2011-07-07T13:57:21Z

Kato classes

← Older revision		Revision as of 08:57, 7 July 2011
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	=== Kato classes ===		=== Kato classes ===
	The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller making the drift term a small perturbation of the fractional diffusion. The precise assumption varies from article to article and they are ~~usually~~ tailor made in order to make some [[bootstrapping]] argument possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation <ref name="BJ"/> <ref name="CKS"/>.		The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller making the drift term a small perturbation of the fractional diffusion. The precise assumption varies from article to article and they are tailor made in order to make some [[bootstrapping]] argument possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation <ref name="BJ"/> <ref name="CKS"/>.

	=== $C^{1,\alpha}$ estimates ===		=== $C^{1,\alpha}$ estimates ===

imported>Luis at 13:55, 7 July 2011

2011-07-07T13:55:39Z

← Older revision		Revision as of 08:55, 7 July 2011
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	Under subcritical assumptions on $b$, one can sometimes show $C^{1,\alpha}$ estimates. These are strong regularity estimates that imply that the solutions are classical.		Under subcritical assumptions on $b$, one can sometimes show $C^{1,\alpha}$ estimates. These are strong regularity estimates that imply that the solutions are classical.

	The result is the following. Assume that $b \in L^\infty(C^{1-2s+\alpha})$ for some $\alpha>0$, then the solution $u$ belongs to $L^\infty(C^{1,\alpha})$. This result holds both if $s > 1/2$ <ref name="P"/> as if $s \leq 1/2$ <ref name="S3"/>.		The result is the following. Assume that $b \in L^\infty(C^{1-2s+\alpha})$ for some $\alpha>0$, then the solution $u$ belongs to $L^\infty(C^{1,\alpha})$. This result holds both if $s \geq 1/2$ <ref name="P"/> as if $s \leq 1/2$ <ref name="S3"/>.

	== Scale invariant results ==		== Scale invariant results ==

imported>Luis at 00:31, 4 July 2011

2011-07-04T00:31:40Z

← Older revision		Revision as of 19:31, 3 July 2011
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	=== Kato classes ===		=== Kato classes ===
	The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller making the drift term a small perturbation of the fractional diffusion. The precise assumption varies from article to article and they are usually tailor made in order to make some [[bootstrapping]] argument possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation.		The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller making the drift term a small perturbation of the fractional diffusion. The precise assumption varies from article to article and they are usually tailor made in order to make some [[bootstrapping]] argument possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation <ref name="BJ"/> <ref name="CKS"/>.

	=== $C^{1,\alpha}$ estimates ===		=== $C^{1,\alpha}$ estimates ===
	Under subcritical assumptions on $b$, one can sometimes show $C^{1,\alpha}$ estimates. These are strong regularity estimates that imply that the solutions are classical.		Under subcritical assumptions on $b$, one can sometimes show $C^{1,\alpha}$ estimates. These are strong regularity estimates that imply that the solutions are classical.

	The result is the following. Assume that $b \in L^\infty(C^{1-2s+\alpha})$ for some $\alpha>0$, then the solution $u$ belongs to $L^\infty(C^{1,\alpha})$. This result holds both if $s > 1/2$ <ref name="P"/> as if $s \leq 1/2$ <ref name="S2"/>.		The result is the following. Assume that $b \in L^\infty(C^{1-2s+\alpha})$ for some $\alpha>0$, then the solution $u$ belongs to $L^\infty(C^{1,\alpha})$. This result holds both if $s > 1/2$ <ref name="P"/> as if $s \leq 1/2$ <ref name="S3"/>.

	== Scale invariant results ==		== Scale invariant results ==
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	<ref name="S1">{{Citation \| last1=Silvestre \| first1=Luis \| title=On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion \| url=http://dx.doi.org/10.1016/j.aim.2010.09.007 \| doi=10.1016/j.aim.2010.09.007 \| year=2011 \| journal=Advances in Mathematics \| issn=0001-8708 \| volume=226 \| issue=2 \| pages=2020–2039}}</ref>		<ref name="S1">{{Citation \| last1=Silvestre \| first1=Luis \| title=On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion \| url=http://dx.doi.org/10.1016/j.aim.2010.09.007 \| doi=10.1016/j.aim.2010.09.007 \| year=2011 \| journal=Advances in Mathematics \| issn=0001-8708 \| volume=226 \| issue=2 \| pages=2020–2039}}</ref>
	<ref name="S2">{{Citation \| last1=Silvestre \| first1=Luis \| title=Holder estimates for advection fractional-diffusion equations \| year=To appear \| journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze}}</ref>		<ref name="S2">{{Citation \| last1=Silvestre \| first1=Luis \| title=Holder estimates for advection fractional-diffusion equations \| year=To appear \| journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze}}</ref>
			<ref name="S3">{{Citation \| last1=Silvestre \| first1=Luis \| title=On the differentiability of the solution to an equation with drift and fractional diffusion. \| year=To appear \| journal=Indiana University Mathematical Journal}}</ref>
	<ref name="P">{{Citation \| last1=Priola \| first1=E. \| title=Pathwise uniqueness for singular SDEs driven by stable processes \| year=2010 \| journal=Arxiv preprint arXiv:1005.4237}}</ref>		<ref name="P">{{Citation \| last1=Priola \| first1=E. \| title=Pathwise uniqueness for singular SDEs driven by stable processes \| year=2010 \| journal=Arxiv preprint arXiv:1005.4237}}</ref>
	<ref name="S2">{{Citation \| last1=Silvestre \| first1=Luis \| title=On the differentiability of the solution to an equation with drift and fractional diffusion \| year=2010 \| journal=Arxiv preprint arXiv:1012.2401}}</ref>		<ref name="S2">{{Citation \| last1=Silvestre \| first1=Luis \| title=On the differentiability of the solution to an equation with drift and fractional diffusion \| year=2010 \| journal=Arxiv preprint arXiv:1012.2401}}</ref>

imported>Luis at 00:27, 4 July 2011

2011-07-04T00:27:08Z

← Older revision		Revision as of 19:27, 3 July 2011
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	<ref name="P">{{Citation \| last1=Priola \| first1=E. \| title=Pathwise uniqueness for singular SDEs driven by stable processes \| year=2010 \| journal=Arxiv preprint arXiv:1005.4237}}</ref>		<ref name="P">{{Citation \| last1=Priola \| first1=E. \| title=Pathwise uniqueness for singular SDEs driven by stable processes \| year=2010 \| journal=Arxiv preprint arXiv:1005.4237}}</ref>
	<ref name="S2">{{Citation \| last1=Silvestre \| first1=Luis \| title=On the differentiability of the solution to an equation with drift and fractional diffusion \| year=2010 \| journal=Arxiv preprint arXiv:1012.2401}}</ref>		<ref name="S2">{{Citation \| last1=Silvestre \| first1=Luis \| title=On the differentiability of the solution to an equation with drift and fractional diffusion \| year=2010 \| journal=Arxiv preprint arXiv:1012.2401}}</ref>
			<ref name="BJ">{{Citation \| last1=Bogdan \| first1=Krzysztof \| last2=Jakubowski \| first2=Tomasz \| title=Estimates of heat kernel of fractional Laplacian perturbed by gradient operators \| url=http://dx.doi.org/10.1007/s00220-006-0178-y \| doi=10.1007/s00220-006-0178-y \| year=2007 \| journal=Communications in Mathematical Physics \| issn=0010-3616 \| volume=271 \| issue=1 \| pages=179–198}}</ref>
			<ref name="CKS">{{Citation \| last1=Chen \| first1=Z.Q. \| last2=Kim \| first2=P. \| last3=Song \| first3=R. \| title=Dirichlet heat kernel estimates for fractional Laplacian under gradient perturbation \| year=To appear \| journal=Annals of Probability}}</ref>
	}}		}}

	[[Category:Semilinear equations]]		[[Category:Semilinear equations]]

imported>Luis at 23:59, 3 July 2011

2011-07-03T23:59:34Z

← Older revision		Revision as of 18:59, 3 July 2011
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	=== Kato classes ===		=== Kato classes ===
	The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller. The precise assumption varies from article to article and they are usually tailor made in order to make some [[bootstrapping]] argument possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation.		The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller making the drift term a small perturbation of the fractional diffusion. The precise assumption varies from article to article and they are usually tailor made in order to make some [[bootstrapping]] argument possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation.

	=== $C^{1,\alpha}$ estimates ===		=== $C^{1,\alpha}$ estimates ===

imported>Luis at 23:58, 3 July 2011

2011-07-03T23:58:47Z

← Older revision		Revision as of 18:58, 3 July 2011
Line 22:		Line 22:

	=== Kato classes ===		=== Kato classes ===
	The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller. The precise assumption varies from article to article and they are usually tailor made in order to make some [[bootstrapping]] argument possible.		The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller. The precise assumption varies from article to article and they are usually tailor made in order to make some [[bootstrapping]] argument possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation.

	=== $C^{1,\alpha}$ estimates ===		=== $C^{1,\alpha}$ estimates ===