# Is there a quantitative measure for the nonlinearity of PDEs?

## Main Question or Discussion Point

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!

Best regards,

Joseph

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fzero
Homework Helper
Gold Member
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.

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.