What exactly does subcritical, critical and supercritical mean in the context of PDEs?
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I have seen these terms thrown around a lot in the PDE literature but have struggled to find a definition of what they actually mean. If anything, I get the idea that these notions of sub/super- criticality are specific to the equation at hand and perhaps don't have a definition that applies more broadly.
For instance, one thing I have read is that the harmonic map equation $Delta u = |nabla u|^2$ is critical, whereas if one changes the exponent to some $p<2$ then the equation becomes sub-critical. I have read elsewhere that this criticality is 'with respect to scaling', and I am also not sure what this refers to.
Could someone please explain these notions, either in general context or in the context of the equation above? Other examples may also be useful, preferably elliptic and even better if they are geometrically motivated! (I have found some descriptions of criticality for wave equations arising in GR etc. but these have been a little difficult to understand). Thanks!
pde definition
asked Dec 4 '18 at 23:12
jl2jl2
500312
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$begingroup$
I have seen these terms thrown around a lot in the PDE literature but have struggled to find a definition of what they actually mean. If anything, I get the idea that these notions of sub/super- criticality are specific to the equation at hand and perhaps don't have a definition that applies more broadly.
For instance, one thing I have read is that the harmonic map equation $Delta u = |nabla u|^2$ is critical, whereas if one changes the exponent to some $p<2$ then the equation becomes sub-critical. I have read elsewhere that this criticality is 'with respect to scaling', and I am also not sure what this refers to.
Could someone please explain these notions, either in general context or in the context of the equation above? Other examples may also be useful, preferably elliptic and even better if they are geometrically motivated! (I have found some descriptions of criticality for wave equations arising in GR etc. but these have been a little difficult to understand). Thanks!
pde definition
asked Dec 4 '18 at 23:12
jl2jl2
500312
$endgroup$
add a comment |
$begingroup$
I have seen these terms thrown around a lot in the PDE literature but have struggled to find a definition of what they actually mean. If anything, I get the idea that these notions of sub/super- criticality are specific to the equation at hand and perhaps don't have a definition that applies more broadly.
For instance, one thing I have read is that the harmonic map equation $Delta u = |nabla u|^2$ is critical, whereas if one changes the exponent to some $p<2$ then the equation becomes sub-critical. I have read elsewhere that this criticality is 'with respect to scaling', and I am also not sure what this refers to.
Could someone please explain these notions, either in general context or in the context of the equation above? Other examples may also be useful, preferably elliptic and even better if they are geometrically motivated! (I have found some descriptions of criticality for wave equations arising in GR etc. but these have been a little difficult to understand). Thanks!
pde definition
asked Dec 4 '18 at 23:12
jl2jl2
500312
$endgroup$
I have seen these terms thrown around a lot in the PDE literature but have struggled to find a definition of what they actually mean. If anything, I get the idea that these notions of sub/super- criticality are specific to the equation at hand and perhaps don't have a definition that applies more broadly.
For instance, one thing I have read is that the harmonic map equation $Delta u = |nabla u|^2$ is critical, whereas if one changes the exponent to some $p<2$ then the equation becomes sub-critical. I have read elsewhere that this criticality is 'with respect to scaling', and I am also not sure what this refers to.
Could someone please explain these notions, either in general context or in the context of the equation above? Other examples may also be useful, preferably elliptic and even better if they are geometrically motivated! (I have found some descriptions of criticality for wave equations arising in GR etc. but these have been a little difficult to understand). Thanks!
pde definition
pde definition
asked Dec 4 '18 at 23:12
jl2jl2
500312
asked Dec 4 '18 at 23:12
jl2jl2
500312
asked Dec 4 '18 at 23:12
jl2jl2
500312
asked Dec 4 '18 at 23:12
jl2jl2
500312
asked Dec 4 '18 at 23:12
jl2jl2
500312
500312
add a comment |
add a comment |
1 Answer
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The equation being critical with respect to scaling likely comes from the fact that if $u$ solves the equation, then so does $tilde{u}(x) = u(lambda x)$ where $lambda$ is some positive scaling factor.
I am also curious about this, as I have seen this come up in discussions of several PDEs with no strict definition. I have also seen references to certain quantities related to the PDE being super/sub/critical. E.g $L^p$ and $H^p$ norms can be super/sub/critical for different values of $p$. Values of $p$ for which $|u|_p = |tilde{u}|_p$ (there may be a boundedness condition instead of equality here) are called critical exponents from what I can gather.
As you can see, I am no expert and I would really like to see a good explanation/source for this because I have had no luck.
answered Dec 5 '18 at 0:50
whpowell96whpowell96
56615
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1 Answer
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1 Answer
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$begingroup$
The equation being critical with respect to scaling likely comes from the fact that if $u$ solves the equation, then so does $tilde{u}(x) = u(lambda x)$ where $lambda$ is some positive scaling factor.
I am also curious about this, as I have seen this come up in discussions of several PDEs with no strict definition. I have also seen references to certain quantities related to the PDE being super/sub/critical. E.g $L^p$ and $H^p$ norms can be super/sub/critical for different values of $p$. Values of $p$ for which $|u|_p = |tilde{u}|_p$ (there may be a boundedness condition instead of equality here) are called critical exponents from what I can gather.
As you can see, I am no expert and I would really like to see a good explanation/source for this because I have had no luck.
answered Dec 5 '18 at 0:50
whpowell96whpowell96
56615
$endgroup$
add a comment |
$begingroup$
The equation being critical with respect to scaling likely comes from the fact that if $u$ solves the equation, then so does $tilde{u}(x) = u(lambda x)$ where $lambda$ is some positive scaling factor.
I am also curious about this, as I have seen this come up in discussions of several PDEs with no strict definition. I have also seen references to certain quantities related to the PDE being super/sub/critical. E.g $L^p$ and $H^p$ norms can be super/sub/critical for different values of $p$. Values of $p$ for which $|u|_p = |tilde{u}|_p$ (there may be a boundedness condition instead of equality here) are called critical exponents from what I can gather.
As you can see, I am no expert and I would really like to see a good explanation/source for this because I have had no luck.
answered Dec 5 '18 at 0:50
whpowell96whpowell96
56615
$endgroup$
add a comment |
$begingroup$
The equation being critical with respect to scaling likely comes from the fact that if $u$ solves the equation, then so does $tilde{u}(x) = u(lambda x)$ where $lambda$ is some positive scaling factor.
I am also curious about this, as I have seen this come up in discussions of several PDEs with no strict definition. I have also seen references to certain quantities related to the PDE being super/sub/critical. E.g $L^p$ and $H^p$ norms can be super/sub/critical for different values of $p$. Values of $p$ for which $|u|_p = |tilde{u}|_p$ (there may be a boundedness condition instead of equality here) are called critical exponents from what I can gather.
As you can see, I am no expert and I would really like to see a good explanation/source for this because I have had no luck.
answered Dec 5 '18 at 0:50
whpowell96whpowell96
56615
$endgroup$
The equation being critical with respect to scaling likely comes from the fact that if $u$ solves the equation, then so does $tilde{u}(x) = u(lambda x)$ where $lambda$ is some positive scaling factor.
I am also curious about this, as I have seen this come up in discussions of several PDEs with no strict definition. I have also seen references to certain quantities related to the PDE being super/sub/critical. E.g $L^p$ and $H^p$ norms can be super/sub/critical for different values of $p$. Values of $p$ for which $|u|_p = |tilde{u}|_p$ (there may be a boundedness condition instead of equality here) are called critical exponents from what I can gather.
As you can see, I am no expert and I would really like to see a good explanation/source for this because I have had no luck.
answered Dec 5 '18 at 0:50
whpowell96whpowell96
56615
answered Dec 5 '18 at 0:50
whpowell96whpowell96
56615
answered Dec 5 '18 at 0:50
whpowell96whpowell96
56615
answered Dec 5 '18 at 0:50
whpowell96whpowell96
56615
56615
add a comment |
add a comment |
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