Is there a quantitative measure for the nonlinearity of PDEs?

[hitmre]

Hi all,

I understand some PDE is linear like
$\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}=0$
while some PDE is nonlinear like
$\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x}=0$

Some PDE is weak nonlinear and some is strong nonlinear.

I am wondering whether there is any quantitative measure of the nonlinearity? Many thank!



Joseph

 

[fzero]

Let's write our differential equation in the form

$$ \hat{D}[f] + \hat{L}f =0,$$

where $\hat{L}$ is a linear operator and $\hat{D}$ contains all of the nonlinear terms. We can assume that $f$ belongs to a Banach space $X$ and $\hat{D}:X\rightarrow Y$, with $Y$ another a Banach space. Then it seems that a common definition of weak nonlinearity is that $\hat{D}$ is Lipschitz continuous. This means that, given $f_1,f_2\in X$ and a metric $d_X(,)$ on $X$, that

$$ d_Y ( \hat{D}[f_1], \hat{D}[f_1] )\leq K d_X (f_1,f_2) $$

for some constant $K\geq 0$. The wiki describes how this works for functions, rather than operators, but the ideas are the same.

This suggests that for $f_1\neq f_2$, the quantity

$$ C[f_1,f_2] = \frac{ d_Y ( \hat{D}[f_1], \hat{D}[f_1] )}{d_X (f_1,f_2)} $$

would be studied as a measure of nonlinearity. There could be domains of $X$ where this is bounded and constant, while in other domains it is not. Unfortunately I am not familiar with the literature on these concepts, but perhaps the extra terminology would help your search.

 

[hitmre]

Thank you fzero!

It seems a good idea. I am wondering whether this definition
$$ C[f_1,f_2] = \frac{ d_Y ( \hat{D}[f_1], \hat{D}[f_2] )}{d_X (f_1,f_2)} $$
is widely use.

Many thanks!
joseph

Let's write our differential equation in the form

$$ \hat{D}[f] + \hat{L}f =0,$$

where $\hat{L}$ is a linear operator and $\hat{D}$ contains all of the nonlinear terms. We can assume that $f$ belongs to a Banach space $X$ and $\hat{D}:X\rightarrow Y$, with $Y$ another a Banach space. Then it seems that a common definition of weak nonlinearity is that $\hat{D}$ is Lipschitz continuous. This means that, given $f_1,f_2\in X$ and a metric $d_X(,)$ on $X$, that

$$ d_Y ( \hat{D}[f_1], \hat{D}[f_1] )\leq K d_X (f_1,f_2) $$

for some constant $K\geq 0$. The wiki describes how this works for functions, rather than operators, but the ideas are the same.

This suggests that for $f_1\neq f_2$, the quantity

$$ C[f_1,f_2] = \frac{ d_Y ( \hat{D}[f_1], \hat{D}[f_1] )}{d_X (f_1,f_2)} $$

would be studied as a measure of nonlinearity. There could be domains of $X$ where this is bounded and constant, while in other domains it is not. Unfortunately I am not familiar with the literature on these concepts, but perhaps the extra terminology would help your search.