Quasilinear equations - Revision history

imported>Luis at 02:40, 16 April 2015

2015-04-16T02:40:05Z

← Older revision		Revision as of 21:40, 15 April 2015
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	For instance, the following equations are all quasilinear (and the first two are NOT semilinear)		For instance, the following equations are all quasilinear (and the first two are NOT semilinear)

			=== Mean curvature flow ===
	\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+\|\nabla u\|^2}}\right ) = 0 \]		\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+\|\nabla u\|^2}}\right ) = 0 \]

	~~<center> [[Mean curvature flow]] </center>~~

			=== [[Nonlocal porous medium equation]] ===
	\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * \|x\|^{-n+\alpha} \]		\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * \|x\|^{-n+\alpha} \]

	~~<center> [[Nonlocal porous medium equation]] </center>~~

			=== [[Semilinear equations\|Hamilton-Jacobi with fractional diffusion]] ===
	\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]		\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]

	~~<center> Hamilton-Jacobi with fractional diffusion </center>~~

	Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called [[Fully nonlinear equations]], they include for instance the [[Monge Ampére Equation]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.		Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called [[Fully nonlinear equations]], they include for instance the [[Monge Ampére Equation]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.

imported>Luis at 02:35, 6 February 2012

2012-02-06T02:35:03Z

← Older revision		Revision as of 21:35, 5 February 2012
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	Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called [[Fully nonlinear equations]], they include for instance the [[Monge Ampére Equation]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.		Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called [[Fully nonlinear equations]], they include for instance the [[Monge Ampére Equation]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.


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imported>Nestor at 22:46, 3 June 2011

2011-06-03T22:46:28Z

← Older revision		Revision as of 17:46, 3 June 2011
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	<center> Hamilton-Jacobi with fractional diffusion </center>		<center> Hamilton-Jacobi with fractional diffusion </center>

	Equations which are NOT quasilinear are called [[Fully nonlinear equations]], ~~which~~ include for instance [[Monge Ampére]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.		Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called [[Fully nonlinear equations]], they include for instance the [[Monge Ampére Equation]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.

imported>Nestor at 22:45, 3 June 2011

2011-06-03T22:45:46Z

← Older revision		Revision as of 17:45, 3 June 2011
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	<center> Hamilton-Jacobi with fractional diffusion </center>		<center> Hamilton-Jacobi with fractional diffusion </center>

	Equations which are ~~not~~ quasilinear are called [[Fully nonlinear equations]], which include for instance [[Monge Ampére]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.		Equations which are NOT quasilinear are called [[Fully nonlinear equations]], which include for instance [[Monge Ampére]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.

imported>Nestor at 22:45, 3 June 2011

2011-06-03T22:45:18Z

← Older revision		Revision as of 17:45, 3 June 2011
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	<center> [[Nonlocal porous medium equation]] </center>		<center> [[Nonlocal porous medium equation]] </center>

	\[ u_t + H(x,t,u,\nabla u) ~~+ (-\Delta)^s u~~ = 0.\]		\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]

	<center> Hamilton-Jacobi with fractional diffusion </center>		<center> Hamilton-Jacobi with fractional diffusion </center>

	Equations which are not quasilinear are called [[Fully nonlinear equations]], which include for instance [[Monge Ampére]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.		Equations which are not quasilinear are called [[Fully nonlinear equations]], which include for instance [[Monge Ampére]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.

imported>Nestor at 22:45, 3 June 2011

2011-06-03T22:45:01Z

← Older revision		Revision as of 17:45, 3 June 2011
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			Quasilinear equations are those which are linear in all terms except for the highest order derivatives (whether they are of fractional order or not).

	For instance, the following equations are all quasilinear (and the first two are NOT semilinear)		For instance, the following equations are all quasilinear (and the first two are NOT semilinear)

imported>Nestor: Created page with "For instance, the following equations are all quasilinear (and the first two are NOT semilinear) \[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] ..."

2011-06-03T22:43:22Z

Created page with "For instance, the following equations are all quasilinear (and the first two are NOT semilinear) \[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] ..."

New page

For instance, the following equations are all quasilinear (and the first two are NOT semilinear)

\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]

<center> [[Mean curvature flow]] </center>

\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]

<center> [[Nonlocal porous medium equation]] </center>

\[ u_t + H(x,t,u,\nabla u) + (-\Delta)^s u = 0.\]

<center> Hamilton-Jacobi with fractional diffusion </center>

Equations which are not quasilinear are called [[Fully nonlinear equations]], which include for instance [[Monge Ampére]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.