Category:Quasilinear equations and Quasilinear equations: Difference between pages

Revision as of 17:46, 3 June 2011

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)

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

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

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

Hamilton-Jacobi with fractional diffusion

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.

The following 2 pages are in this category, out of 2 total.

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-A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not). For instance, the following equations are quasilinear
+Quasilinear equations are those which are linear in all terms except for the  highest order derivatives  (whether they are of fractional order or not).
-\[ \mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
+For instance,  the following equations are all quasilinear (and the first two are NOT semilinear)
-\[ u_t = \mbox{div} \left ( u^p \nabla u\right ) \]
-\[ (-\Delta)^{s}+H(x,u,\nabla u)=0 (s>1/2) \]
+\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
+<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>
+\[ u_t  + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]
+<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.