Open problems - Revision history

imported>Xavi: /* Complete understanding of free boundary points in the fractional obstacle problem */

2016-03-26T14:48:16Z

Complete understanding of free boundary points in the fractional obstacle problem

← Older revision		Revision as of 09:48, 26 March 2016
Line 133:		Line 133:
	# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?		# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?

	UPDATE: It has been recently proved by Barrios, Figalli, and Ros-Oton <ref name="BFR"/> that, when the obstacle $\varphi$ satisfies $\Delta\varphi\leq0$, then regular and singular points do exhaust all free boundary points. Thus, under this ``concavity'' assumption, there are no free boundary points in the ``third category''.		UPDATE: It has been recently proved by Barrios, Figalli, and Ros-Oton <ref name="BFR"/> that, when the obstacle $\varphi$ satisfies $\Delta\varphi\leq0$, then regular and singular points do exhaust all free boundary points. Thus, under this ''concavity'' assumption, there are no free boundary points in the ''third category''.

	= References =		= References =

imported>Xavi: /* References */

2016-03-26T14:47:29Z

References

← Older revision		Revision as of 09:47, 26 March 2016
Line 168:		Line 168:
	<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>		<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>
	<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>		<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
			<ref name="BFR">{{Citation \| last1=Barrios \| first1= B. \| last2=Figalli \| first2= A. \| last3=Ros-Oton \| first3= X. \| title=Global regularity for the free boundary in the obstacle problem for the fractional Laplacian \| journal=preprint arXiv}}</ref>
	}}		}}

imported>Xavi: /* Complete understanding of free boundary points in the fractional obstacle problem */

2016-03-26T14:45:04Z

Complete understanding of free boundary points in the fractional obstacle problem

← Older revision		Revision as of 09:45, 26 March 2016
Line 132:		Line 132:
	# Can there be any point on the free boundary that is neither regular nor singular? It is easy to produce examples in the [[thin obstacle problem]], using the [[extension technique]]. However, it is not clear if such examples can be made in the original formulation of the [[fractional obstacle problem]] because of the decay at infinity requirement.		# Can there be any point on the free boundary that is neither regular nor singular? It is easy to produce examples in the [[thin obstacle problem]], using the [[extension technique]]. However, it is not clear if such examples can be made in the original formulation of the [[fractional obstacle problem]] because of the decay at infinity requirement.
	# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?		# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?

			UPDATE: It has been recently proved by Barrios, Figalli, and Ros-Oton <ref name="BFR"/> that, when the obstacle $\varphi$ satisfies $\Delta\varphi\leq0$, then regular and singular points do exhaust all free boundary points. Thus, under this ``concavity'' assumption, there are no free boundary points in the ``third category''.

	= References =		= References =

imported>Xavi: /* References */

2016-03-26T14:40:22Z

References

← Older revision		Revision as of 09:40, 26 March 2016
Line 161:		Line 161:
	<ref name="li2011one">{{Citation \| last1=Li \| first1= Dong \| last2=Rodrigo \| first2= José L \| title=On a one-dimensional nonlocal flux with fractional dissipation \| journal=SIAM Journal on Mathematical Analysis \| year=2011 \| volume=43 \| pages=507--526}}</ref>		<ref name="li2011one">{{Citation \| last1=Li \| first1= Dong \| last2=Rodrigo \| first2= José L \| title=On a one-dimensional nonlocal flux with fractional dissipation \| journal=SIAM Journal on Mathematical Analysis \| year=2011 \| volume=43 \| pages=507--526}}</ref>
	<ref name="cordobacordoba2005">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Formation of singularities for a transport equation with nonlocal velocity \| url=http://dx.doi.org/10.4007/annals.2005.162.1377 \| journal=Ann. of Math. (2) \| issn=0003-486X \| year=2005 \| volume=162 \| pages=1377--1389 \| doi=10.4007/annals.2005.162.1377}}</ref>		<ref name="cordobacordoba2005">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Formation of singularities for a transport equation with nonlocal velocity \| url=http://dx.doi.org/10.4007/annals.2005.162.1377 \| journal=Ann. of Math. (2) \| issn=0003-486X \| year=2005 \| volume=162 \| pages=1377--1389 \| doi=10.4007/annals.2005.162.1377}}</ref>
	<ref name="ros2014regularity">{{Citation \| last1=Ros-Oton \| first1= Xavier \| last2=Serra \| first2= Joaquim \| title=Regularity theory for general stable operators \| journal=~~arXiv preprint arXiv:1412~~.~~3892~~}}</ref>		<ref name="ros2014regularity">{{Citation \| last1=Ros-Oton \| first1= Xavier \| last2=Serra \| first2= Joaquim \| title=Regularity theory for general stable operators \| journal=J. Differential Equations, to appear.}}</ref>
	<ref name="HDong">{{Citation \| last1=Dong \| first1= Hongjie \| title=Well-posedness for a transport equation with nonlocal velocity \| url=http://dx.doi.org/10.1016/j.jfa.2008.08.005 \| journal=J. Funct. Anal. \| issn=0022-1236 \| year=2008 \| volume=255 \| pages=3070--3097 \| doi=10.1016/j.jfa.2008.08.005}}</ref>		<ref name="HDong">{{Citation \| last1=Dong \| first1= Hongjie \| title=Well-posedness for a transport equation with nonlocal velocity \| url=http://dx.doi.org/10.1016/j.jfa.2008.08.005 \| journal=J. Funct. Anal. \| issn=0022-1236 \| year=2008 \| volume=255 \| pages=3070--3097 \| doi=10.1016/j.jfa.2008.08.005}}</ref>
	<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>		<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>

imported>Luis: /* References */

2016-01-26T17:50:52Z

References

← Older revision		Revision as of 12:50, 26 January 2016
Line 133:		Line 133:
	# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?		# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?

	== References ==		= References =
	{{reflist\|refs=		{{reflist\|refs=
	<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="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>
Line 140:		Line 140:
	<ref name="FV">{{Citation \| last1=Friedlander \| first1=S. \| last2=Vicol \| first2=V. \| title=Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics \| year=2011 \| journal=Annales de l'Institut Henri Poincare (C) Non Linear Analysis}}</ref>		<ref name="FV">{{Citation \| last1=Friedlander \| first1=S. \| last2=Vicol \| first2=V. \| title=Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics \| year=2011 \| journal=Annales de l'Institut Henri Poincare (C) Non Linear Analysis}}</ref>
	<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title= Nonlocal Minimal Surfaces \| url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract \| doi=10.1002/cpa.20331 \| year=2010 \| journal=[[Communications on Pure and Applied Mathematics]] \| issn=0003-486X \| volume=63 \| issue=9 \| pages=1111–1144}}</ref>		<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title= Nonlocal Minimal Surfaces \| url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract \| doi=10.1002/cpa.20331 \| year=2010 \| journal=[[Communications on Pure and Applied Mathematics]] \| issn=0003-486X \| volume=63 \| issue=9 \| pages=1111–1144}}</ref>

	<ref name="CRS16">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Ros-Oton \| first2=Xavier \|last3= Serra \| first3= Joaquim \| title= Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries \| year=2016 \| journal=[[preprint arXiv (2016)]]}}</ref>		<ref name="CRS16">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Ros-Oton \| first2=Xavier \|last3= Serra \| first3= Joaquim \| title= Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries \| year=2016 \| journal=[[preprint arXiv (2016)]]}}</ref>

	<ref name="GS">{{Citation \| last1=Guillen \| first1=N. \| last2=Schwab \| first2=R. \| title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations \| year=2010 \| journal=Arxiv preprint arXiv:1101.0279}}</ref>		<ref name="GS">{{Citation \| last1=Guillen \| first1=N. \| last2=Schwab \| first2=R. \| title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations \| year=2010 \| journal=Arxiv preprint arXiv:1101.0279}}</ref>
	<ref name="CSS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Salsa \| first2=Sandro \| last3=Silvestre \| first3=Luis \| title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian \| url=http://dx.doi.org/10.1007/s00222-007-0086-6 \| doi=10.1007/s00222-007-0086-6 \| year=2008 \| journal=[[Inventiones Mathematicae]] \| issn=0020-9910 \| volume=171 \| issue=2 \| pages=425–461}}</ref>		<ref name="CSS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Salsa \| first2=Sandro \| last3=Silvestre \| first3=Luis \| title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian \| url=http://dx.doi.org/10.1007/s00222-007-0086-6 \| doi=10.1007/s00222-007-0086-6 \| year=2008 \| journal=[[Inventiones Mathematicae]] \| issn=0020-9910 \| volume=171 \| issue=2 \| pages=425–461}}</ref>

imported>Luis: /* Hölder estimates for singular integro-differential equations */

2016-01-26T17:36:42Z

Hölder estimates for singular integro-differential equations

← Older revision		Revision as of 12:36, 26 January 2016
Line 4:		Line 4:
	Consider an integro-differential equation of the form		Consider an integro-differential equation of the form
	\[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\]		\[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\]
	(An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin)		(An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin and not symmetric)

	[[Hölder estimates]] are known to hold under certain 'ellipticity' assumptions for the measures $\mu_x(y)$. In many cases, we consider the absolutely continuous version $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$ and write the assumptions in terms of the kernel $K$. One would expect that the estimates should hold every time the measures $\mu_x$ satisfy.		[[Hölder estimates]] are known to hold under certain 'ellipticity' assumptions for the measures $\mu_x(y)$. In many cases, we consider the absolutely continuous version $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$ and write the assumptions in terms of the kernel $K$. One would expect that the estimates should hold every time the measures $\mu_x$ satisfy.

imported>Luis: /* Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) */

2016-01-25T21:00:18Z

Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$)

← Older revision		Revision as of 16:00, 25 January 2016
Line 60:		Line 60:
	\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]		\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]

	The solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty$, or $b \in L^\infty(BMO)$ and in addition $b$ is divergence free <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b$ is divergence free and $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b$ is divergence free and $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).		The solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty$ <ref name="SilHJ"/>, or $b \in L^\infty(BMO)$ and in addition $b$ is divergence free <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b$ is divergence free and $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b$ is divergence free and $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).

	The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur <ref name="CV"/> can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$ if $\mathrm{div}\, b =0$. The case of arbitrary divergence might be more complicated.		The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur <ref name="CV"/> can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$ if $\mathrm{div}\, b =0$. The case of arbitrary divergence might be more complicated.

imported>Luis: /* References */

2016-01-25T20:59:48Z

References

← Older revision		Revision as of 15:59, 25 January 2016
Line 167:		Line 167:
	<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>		<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>
	<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>		<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>
	<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>		<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	}}		}}

imported>Luis at 20:59, 25 January 2016

2016-01-25T20:59:21Z

← Older revision		Revision as of 15:59, 25 January 2016
Line 167:		Line 167:
	<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>		<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>
	<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>		<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>
			<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>		<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	}}		}}

imported>Luis at 20:58, 25 January 2016

2016-01-25T20:58:28Z

← Older revision		Revision as of 15:58, 25 January 2016
Line 60:		Line 60:
	\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]		\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]

	The solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty(BMO)$ <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).		The solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty$, or $b \in L^\infty(BMO)$ and in addition $b$ is divergence free <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b$ is divergence free and $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b$ is divergence free and $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).

	The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur <ref name="CV"/> can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$.		The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur <ref name="CV"/> can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$ if $\mathrm{div}\, b =0$. The case of arbitrary divergence might be more complicated.

	= Open problems for equations related to fluids =		= Open problems for equations related to fluids =
Line 167:		Line 167:
	<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>		<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>
	<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>		<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>
			<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	}}		}}

← Older revision		Revision as of 09:40, 26 March 2016
Line 161:		Line 161:
	<ref name="li2011one">{{Citation \| last1=Li \| first1= Dong \| last2=Rodrigo \| first2= José L \| title=On a one-dimensional nonlocal flux with fractional dissipation \| journal=SIAM Journal on Mathematical Analysis \| year=2011 \| volume=43 \| pages=507--526}}</ref>		<ref name="li2011one">{{Citation \| last1=Li \| first1= Dong \| last2=Rodrigo \| first2= José L \| title=On a one-dimensional nonlocal flux with fractional dissipation \| journal=SIAM Journal on Mathematical Analysis \| year=2011 \| volume=43 \| pages=507--526}}</ref>
	<ref name="cordobacordoba2005">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Formation of singularities for a transport equation with nonlocal velocity \| url=http://dx.doi.org/10.4007/annals.2005.162.1377 \| journal=Ann. of Math. (2) \| issn=0003-486X \| year=2005 \| volume=162 \| pages=1377--1389 \| doi=10.4007/annals.2005.162.1377}}</ref>		<ref name="cordobacordoba2005">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Formation of singularities for a transport equation with nonlocal velocity \| url=http://dx.doi.org/10.4007/annals.2005.162.1377 \| journal=Ann. of Math. (2) \| issn=0003-486X \| year=2005 \| volume=162 \| pages=1377--1389 \| doi=10.4007/annals.2005.162.1377}}</ref>
	<ref name="ros2014regularity">{{Citation \| last1=Ros-Oton \| first1= Xavier \| last2=Serra \| first2= Joaquim \| title=Regularity theory for general stable operators \| journal=~~arXiv preprint arXiv:1412~~.~~3892~~}}</ref>		<ref name="ros2014regularity">{{Citation \| last1=Ros-Oton \| first1= Xavier \| last2=Serra \| first2= Joaquim \| title=Regularity theory for general stable operators \| journal=J. Differential Equations, to appear.}}</ref>
	<ref name="HDong">{{Citation \| last1=Dong \| first1= Hongjie \| title=Well-posedness for a transport equation with nonlocal velocity \| url=http://dx.doi.org/10.1016/j.jfa.2008.08.005 \| journal=J. Funct. Anal. \| issn=0022-1236 \| year=2008 \| volume=255 \| pages=3070--3097 \| doi=10.1016/j.jfa.2008.08.005}}</ref>		<ref name="HDong">{{Citation \| last1=Dong \| first1= Hongjie \| title=Well-posedness for a transport equation with nonlocal velocity \| url=http://dx.doi.org/10.1016/j.jfa.2008.08.005 \| journal=J. Funct. Anal. \| issn=0022-1236 \| year=2008 \| volume=255 \| pages=3070--3097 \| doi=10.1016/j.jfa.2008.08.005}}</ref>
	<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>		<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>

← Older revision		Revision as of 12:50, 26 January 2016
Line 133:		Line 133:
	# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?		# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?

	== References ==		= References =
	{{reflist\|refs=		{{reflist\|refs=
	<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="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>
Line 140:		Line 140:
	<ref name="FV">{{Citation \| last1=Friedlander \| first1=S. \| last2=Vicol \| first2=V. \| title=Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics \| year=2011 \| journal=Annales de l'Institut Henri Poincare (C) Non Linear Analysis}}</ref>		<ref name="FV">{{Citation \| last1=Friedlander \| first1=S. \| last2=Vicol \| first2=V. \| title=Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics \| year=2011 \| journal=Annales de l'Institut Henri Poincare (C) Non Linear Analysis}}</ref>
	<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title= Nonlocal Minimal Surfaces \| url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract \| doi=10.1002/cpa.20331 \| year=2010 \| journal=[[Communications on Pure and Applied Mathematics]] \| issn=0003-486X \| volume=63 \| issue=9 \| pages=1111–1144}}</ref>		<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title= Nonlocal Minimal Surfaces \| url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract \| doi=10.1002/cpa.20331 \| year=2010 \| journal=[[Communications on Pure and Applied Mathematics]] \| issn=0003-486X \| volume=63 \| issue=9 \| pages=1111–1144}}</ref>

	<ref name="CRS16">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Ros-Oton \| first2=Xavier \|last3= Serra \| first3= Joaquim \| title= Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries \| year=2016 \| journal=[[preprint arXiv (2016)]]}}</ref>		<ref name="CRS16">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Ros-Oton \| first2=Xavier \|last3= Serra \| first3= Joaquim \| title= Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries \| year=2016 \| journal=[[preprint arXiv (2016)]]}}</ref>

	<ref name="GS">{{Citation \| last1=Guillen \| first1=N. \| last2=Schwab \| first2=R. \| title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations \| year=2010 \| journal=Arxiv preprint arXiv:1101.0279}}</ref>		<ref name="GS">{{Citation \| last1=Guillen \| first1=N. \| last2=Schwab \| first2=R. \| title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations \| year=2010 \| journal=Arxiv preprint arXiv:1101.0279}}</ref>
	<ref name="CSS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Salsa \| first2=Sandro \| last3=Silvestre \| first3=Luis \| title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian \| url=http://dx.doi.org/10.1007/s00222-007-0086-6 \| doi=10.1007/s00222-007-0086-6 \| year=2008 \| journal=[[Inventiones Mathematicae]] \| issn=0020-9910 \| volume=171 \| issue=2 \| pages=425–461}}</ref>		<ref name="CSS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Salsa \| first2=Sandro \| last3=Silvestre \| first3=Luis \| title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian \| url=http://dx.doi.org/10.1007/s00222-007-0086-6 \| doi=10.1007/s00222-007-0086-6 \| year=2008 \| journal=[[Inventiones Mathematicae]] \| issn=0020-9910 \| volume=171 \| issue=2 \| pages=425–461}}</ref>

← Older revision		Revision as of 15:59, 25 January 2016
Line 167:		Line 167:
	<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>		<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>
	<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>		<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>
	<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>		<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	}}		}}

← Older revision		Revision as of 15:59, 25 January 2016
Line 167:		Line 167:
	<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>		<ref name="K">{{Citation \| last1=Kiselev \| first1= A. \| title=Regularity and blow up for active scalars \| url=http://dx.doi.org/10.1051/mmnp/20105410 \| journal=Math. Model. Nat. Phenom. \| issn=0973-5348 \| year=2010 \| volume=5 \| pages=225--255 \| doi=10.1051/mmnp/20105410}}</ref>
	<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>		<ref name="CCF2">{{Citation \| last1=Córdoba \| first1= Antonio \| last2=Córdoba \| first2= Diego \| last3=Fontelos \| first3= Marco A. \| title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation \| url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 \| journal=J. Math. Pures Appl. (9) \| issn=0021-7824 \| year=2006 \| volume=86 \| pages=529--540 \| doi=10.1016/j.matpur.2006.08.002}}</ref>
			<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>		<ref name="SilHJ">{{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 \| journal=Adv. Math. \| issn=0001-8708 \| year=2011 \| volume=226 \| pages=2020--2039 \| doi=10.1016/j.aim.2010.09.007}}</ref>
	}}		}}