Is there a quantitative measure for the nonlinearity of PDEs?

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Hi all,

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

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
 

Answers and Replies

  • #2
fzero
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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.
 
  • #3
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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.
 

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