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Is there a quantitative measure for the nonlinearity of PDEs?

  1. Sep 18, 2013 #1
    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
     
  2. jcsd
  3. Sep 18, 2013 #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.
     
  4. Sep 19, 2013 #3
    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

     
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