Semilinear equations - Revision history

imported>Xavi: /* Stationary equations with zeroth order nonlinearity */

2015-05-15T19:27:41Z

Stationary equations with zeroth order nonlinearity

← Older revision		Revision as of 14:27, 15 May 2015
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	If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^\infty$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].		If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^\infty$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].

	Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="OLC"/> <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.		Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, boundedness, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="OLC"/> <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.

	=== Reaction diffusion equations ===		=== Reaction diffusion equations ===

imported>Luis: /* Stationary equations with zeroth order nonlinearity */

2012-03-01T22:14:59Z

Stationary equations with zeroth order nonlinearity

← Older revision		Revision as of 17:14, 1 March 2012
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	Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs.		Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs.
	\[ (-\Delta)^s u = f(u). \]		\[ (-\Delta)^s u = f(u). \]
	If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].		If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^\infty$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].

	Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="OLC"/> <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.		Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="OLC"/> <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.

imported>Luis: /* The most common elliptic equation in the world (provisional title) */

2012-03-01T22:14:40Z

The most common elliptic equation in the world (provisional title)

← Older revision		Revision as of 17:14, 1 March 2012
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	== Some common semilinear equations ==		== Some common semilinear equations ==

	=== ~~The most common elliptic equation in the world (provisional title)~~ ===		=== Stationary equations with zeroth order nonlinearity ===
	Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs.		Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs.
	\[ (-\Delta)^s u = f(u). \]		\[ (-\Delta)^s u = f(u). \]

imported>Luis: /* Hamilton-Jacobi equation with fractional diffusion */

2012-02-07T17:09:26Z

Hamilton-Jacobi equation with fractional diffusion

← Older revision		Revision as of 12:09, 7 February 2012
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	\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]		\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]

	The ~~Cauchy~~ problem is ~~known to be~~ well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.		For Lipschitz initial data, the Cauchy problem always has a viscosity solution which is Lipschitz in space.<ref name="DI"/> The problem is well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.

	The subcritical case $s>1/2$ can be solved with classical [[bootstrapping]] <ref name="DI"/>. The critical case $s=1/2$ was solved using the regularity results for [[drift-diffusion equations]] <ref name="S"/>.		The subcritical case $s>1/2$ can be solved with classical [[bootstrapping]].<ref name="DI"/> The critical case $s=1/2$ was solved using the regularity results for [[drift-diffusion equations]].<ref name="S"/>

	== References ==		== References ==

imported>Luis at 05:00, 7 February 2012

2012-02-07T05:00:25Z

← Older revision		Revision as of 00:00, 7 February 2012
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	If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].		If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].

	Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.		Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="OLC"/> <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.

	=== Reaction diffusion equations ===		=== Reaction diffusion equations ===
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	== References ==		== References ==
	{{reflist\|refs=		{{reflist\|refs=
			<ref name="OLC">{{Citation \| last1=Ou \| first1=Biao \| last2=Li \| first2=Congming \| last3=Chen \| first3=Wenxiong \| title=Classification of solutions for an integral equation \| url=http://dx.doi.org/10.1002/cpa.20116 \| doi=10.1002/cpa.20116 \| year=2006 \| journal=[[Communications on Pure and Applied Mathematics]] \| issn=0010-3640 \| volume=59 \| issue=3 \| pages=330–343}}</ref>
	<ref name="KNS">{{Citation \| last1=Kiselev \| first1=Alexander \| last2=Nazarov \| first2=Fedor \| last3=Shterenberg \| first3=Roman \| title=Blow up and regularity for fractal Burgers equation \| year=2008 \| journal=Dynamics of Partial Differential Equations \| issn=1548-159X \| volume=5 \| issue=3 \| pages=211–240}}</ref>		<ref name="KNS">{{Citation \| last1=Kiselev \| first1=Alexander \| last2=Nazarov \| first2=Fedor \| last3=Shterenberg \| first3=Roman \| title=Blow up and regularity for fractal Burgers equation \| year=2008 \| journal=Dynamics of Partial Differential Equations \| issn=1548-159X \| volume=5 \| issue=3 \| pages=211–240}}</ref>
	<ref name="S">{{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="S">{{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>

imported>Luis at 15:51, 2 July 2011

2011-07-02T15:51:50Z

← Older revision		Revision as of 10:51, 2 July 2011
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	If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].		If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].

	Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained.		Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.

	=== Reaction diffusion equations ===		=== Reaction diffusion equations ===
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	\[ u_t + (-\Delta)^s u = f(u). \]		\[ u_t + (-\Delta)^s u = f(u). \]

	The case $f(u) = u(1-u)$ corresponds to the Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior <ref name="CR"/>, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.		The case $f(u) = u(1-u)$ corresponds to the KPP/Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior <ref name="CR"/>, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.

	=== Burgers equation with fractional diffusion ===		=== Burgers equation with fractional diffusion ===
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	<ref name="S">{{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="S">{{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="CCS">{{Citation \| last1=Chan \| first1=Chi Hin \| last2=Czubak \| first2=Magdalena \| last3=Silvestre \| first3=Luis \| title=Eventual regularization of the slightly supercritical fractional Burgers equation \| url=http://dx.doi.org/10.3934/dcds.2010.27.847 \| doi=10.3934/dcds.2010.27.847 \| year=2010 \| journal=Discrete and Continuous Dynamical Systems. Series A \| issn=1078-0947 \| volume=27 \| issue=2 \| pages=847–861}}</ref>		<ref name="CCS">{{Citation \| last1=Chan \| first1=Chi Hin \| last2=Czubak \| first2=Magdalena \| last3=Silvestre \| first3=Luis \| title=Eventual regularization of the slightly supercritical fractional Burgers equation \| url=http://dx.doi.org/10.3934/dcds.2010.27.847 \| doi=10.3934/dcds.2010.27.847 \| year=2010 \| journal=Discrete and Continuous Dynamical Systems. Series A \| issn=1078-0947 \| volume=27 \| issue=2 \| pages=847–861}}</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 \| doi=10.1051/mmnp/20105410~~ \| year=~~2010~~ \| journal=~~Mathematical Modelling of Natural Phenomena \| issn=0973-5348 \| volume=5 \| issue=4 \| pages=225–255~~}}</ref>		<ref name="K">{{Citation \| last1=Kiselev \| first1=A. \| title=Nonlocal maximum principles for active scalars \| year=to appear \| journal=Advances in Mathematics}}</ref>
	<ref name="DI">{{Citation \| last1=Droniou \| first1=Jérôme \| last2=Imbert \| first2=Cyril \| title=Fractal first-order partial differential equations \| url=http://dx.doi.org/10.1007/s00205-006-0429-2 \| doi=10.1007/s00205-006-0429-2 \| year=2006 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| volume=182 \| issue=2 \| pages=299–331}}</ref>		<ref name="DI">{{Citation \| last1=Droniou \| first1=Jérôme \| last2=Imbert \| first2=Cyril \| title=Fractal first-order partial differential equations \| url=http://dx.doi.org/10.1007/s00205-006-0429-2 \| doi=10.1007/s00205-006-0429-2 \| year=2006 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| volume=182 \| issue=2 \| pages=299–331}}</ref>
	<ref name="CR">{{Citation \| last1=Cabré \| first1=Xavier \| last2=Roquejoffre \| first2=Jean-Michel \| title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire \| url=http://dx.doi.org/10.1016/j.crma.2009.10.012 \| doi=10.1016/j.crma.2009.10.012 \| year=2009 \| journal=Comptes Rendus Mathématique. Académie des Sciences. Paris \| issn=1631-073X \| volume=347 \| issue=23 \| pages=1361–1366}}</ref>		<ref name="CR">{{Citation \| last1=Cabré \| first1=Xavier \| last2=Roquejoffre \| first2=Jean-Michel \| title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire \| url=http://dx.doi.org/10.1016/j.crma.2009.10.012 \| doi=10.1016/j.crma.2009.10.012 \| year=2009 \| journal=Comptes Rendus Mathématique. Académie des Sciences. Paris \| issn=1631-073X \| volume=347 \| issue=23 \| pages=1361–1366}}</ref>
			<ref name="CS">{{Citation \| last1=Cabre \| first1=X. \| last2=Sire \| first2=Yannick \| title=Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates \| year=2010 \| journal=Arxiv preprint arXiv:1012.0867}}</ref>
			<ref name="CC"> {{Citation \| last1=Cabré \| first1=Xavier \| last2=Cinti \| first2=E. \| title=Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian \| year=2010 \| journal=Discrete and Continuous Dynamical Systems (DCDS-A) \| volume=28 \| issue=3 \| pages=1179–1206}} </ref>
			<ref name="LF">{{Citation \| last1=Frank \| first1=R.L. \| last2=Lenzmann \| first2=E. \| title=Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q+ Q-Q^{\alpha+1}= 0$ in $\R$ \| year=2010 \| journal=Arxiv preprint arXiv:1009.4042}}</ref>
			<ref name="FQT">{{Citation \| last1=Felmer \| first1=P. \| last2=Quaas \| first2=A. \| last3=Tan \| first3=J. \| title=Positive Solutions Of Nonlinear Schrodinger Equation With The Fractional Laplacian.}}</ref>
			<ref name="SV"> {{Citation \| last1=Sire \| first1=Yannick \| last2=Valdinoci \| first2=E. \| title=Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result \| publisher=[[Elsevier]] \| year=2009 \| journal=Journal of Functional Analysis \| issn=0022-1236 \| volume=256 \| issue=6 \| pages=1842–1864}} </ref>
			<ref name="PSV">{{Citation \| last1=Palatucci \| first1=G. \| last2=Valdinoci \| first2=E. \| last3=Savin \| first3=O. \| title=Local and global minimizers for a variational energy involving a fractional norm \| year=2011 \| journal=Arxiv preprint arXiv:1104.1725}}</ref>
	}}		}}

imported>Luis at 14:54, 2 July 2011

2011-07-02T14:54:17Z

← Older revision		Revision as of 09:54, 2 July 2011
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	If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].		If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].

	Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, ~~radial symmetry~~, etc...		Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained.

	=== Reaction diffusion equations ===		=== Reaction diffusion equations ===

imported>Luis: /* The most common elliptic equation in the world (provisional title) */

2011-05-31T23:14:19Z

The most common elliptic equation in the world (provisional title)

← Older revision		Revision as of 18:14, 31 May 2011
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	=== The most common elliptic equation in the world (provisional title) ===		=== The most common elliptic equation in the world (provisional title) ===
	Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers ~~has~~ been written on PDEs.		Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs.
	\[ (-\Delta)^s u = f(u). \]		\[ (-\Delta)^s u = f(u). \]
	If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].		If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].

imported>Luis at 23:13, 31 May 2011

2011-05-31T23:13:38Z

← Older revision		Revision as of 18:13, 31 May 2011
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	\[ u_t + (-\Delta)^s u = f(u). \]		\[ u_t + (-\Delta)^s u = f(u). \]

	The case $f(u) = u(1-u)$ corresponds to the Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.		The case $f(u) = u(1-u)$ corresponds to the Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior <ref name="CR"/>, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.

	=== Burgers equation with fractional diffusion ===		=== Burgers equation with fractional diffusion ===
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	where $u = R^\perp \theta$ (and $R$ is the Riesz transform).		where $u = R^\perp \theta$ (and $R$ is the Riesz transform).

	The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem.		The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem. It is believed that solving the supercritical SQG equation could possibly help understand 3D Navier-Stokes equation.

	=== Conservation laws with fractional diffusion ===		=== Conservation laws with fractional diffusion ===
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	<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 \| doi=10.1051/mmnp/20105410 \| year=2010 \| journal=Mathematical Modelling of Natural Phenomena \| issn=0973-5348 \| volume=5 \| issue=4 \| pages=225–255}}</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 \| doi=10.1051/mmnp/20105410 \| year=2010 \| journal=Mathematical Modelling of Natural Phenomena \| issn=0973-5348 \| volume=5 \| issue=4 \| pages=225–255}}</ref>
	<ref name="DI">{{Citation \| last1=Droniou \| first1=Jérôme \| last2=Imbert \| first2=Cyril \| title=Fractal first-order partial differential equations \| url=http://dx.doi.org/10.1007/s00205-006-0429-2 \| doi=10.1007/s00205-006-0429-2 \| year=2006 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| volume=182 \| issue=2 \| pages=299–331}}</ref>		<ref name="DI">{{Citation \| last1=Droniou \| first1=Jérôme \| last2=Imbert \| first2=Cyril \| title=Fractal first-order partial differential equations \| url=http://dx.doi.org/10.1007/s00205-006-0429-2 \| doi=10.1007/s00205-006-0429-2 \| year=2006 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| volume=182 \| issue=2 \| pages=299–331}}</ref>
			<ref name="CR">{{Citation \| last1=Cabré \| first1=Xavier \| last2=Roquejoffre \| first2=Jean-Michel \| title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire \| url=http://dx.doi.org/10.1016/j.crma.2009.10.012 \| doi=10.1016/j.crma.2009.10.012 \| year=2009 \| journal=Comptes Rendus Mathématique. Académie des Sciences. Paris \| issn=1631-073X \| volume=347 \| issue=23 \| pages=1361–1366}}</ref>
	}}		}}

imported>Luis at 23:05, 31 May 2011

2011-05-31T23:05:56Z

← Older revision		Revision as of 18:05, 31 May 2011
Line 44:		Line 44:
	\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]		\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]

	The Cauchy problem is known to be well posed classically if $s \geq 1/2$ ~~<ref name="S"/>~~. For $s<1/2$ there are viscosity solutions that are not $C^1$.		The Cauchy problem is known to be well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.

	The subcritical case $s>1/2$ can be solved with classical [[bootstrapping]]. The critical case $s=1/2$ was solved using the regularity results for [[drift-diffusion equations]].		The subcritical case $s>1/2$ can be solved with classical [[bootstrapping]] <ref name="DI"/>. The critical case $s=1/2$ was solved using the regularity results for [[drift-diffusion equations]] <ref name="S"/>.

	== References ==		== References ==