Smoothness of solution to transport equation IVP
$begingroup$
Consider the following PDE (the unknown is $uin C^1(mathbb{R}^{n+1},mathbb{R})$):
$$partial_tu(x,t)+sum_{k=1}^{n} a_kpartial_k u(x,t)= cu(x,t)$$
$$u(x,0)=g(x)$$
where $g in C(mathbb{R}^n, mathbb{R})$, $ain mathbb{R}^n$ and $cin mathbb{R}$.
My textbook shows that if $u$ is a solution, then we must have $u(x,t)=g(x-ta)e^{ct}=:v(x,t)$; and, it says, $v$ is a solution.
It seems to me it is assuming $gin C^1$, because (here $k=1,...,n$) $partial_k v(x,t)= lim_{sto 0}{e^{ct}(g(x-ta+se_k)-g(x-ta))/s }$ exists iff $partial_k g(x-ta)$ exists, and in that case $partial_k v(x,t)=e^{ct}partial_k g(x-ta)$ so that $vin C^1$ iff $g in C^1$.
After assuming $g in C^1$ I am able to verify that $v$ solves the equation.
Am I correct in saying that $g$ must be in $C^1$ so that the equation can be solved?
PS Is 'it' the correct pronoun when referring to what the text is saying? Or should I use 'he/she' because it is understood that the author is saying that? (Probably a stupid question)
pde regularity-theory-of-pdes transport-equation
edited Jan 16 at 16:38
Harry49
6,21331132
asked Jan 16 at 15:37
useruser
13610
$endgroup$
add a comment |
$begingroup$
Consider the following PDE (the unknown is $uin C^1(mathbb{R}^{n+1},mathbb{R})$):
$$partial_tu(x,t)+sum_{k=1}^{n} a_kpartial_k u(x,t)= cu(x,t)$$
$$u(x,0)=g(x)$$
where $g in C(mathbb{R}^n, mathbb{R})$, $ain mathbb{R}^n$ and $cin mathbb{R}$.
My textbook shows that if $u$ is a solution, then we must have $u(x,t)=g(x-ta)e^{ct}=:v(x,t)$; and, it says, $v$ is a solution.
It seems to me it is assuming $gin C^1$, because (here $k=1,...,n$) $partial_k v(x,t)= lim_{sto 0}{e^{ct}(g(x-ta+se_k)-g(x-ta))/s }$ exists iff $partial_k g(x-ta)$ exists, and in that case $partial_k v(x,t)=e^{ct}partial_k g(x-ta)$ so that $vin C^1$ iff $g in C^1$.
After assuming $g in C^1$ I am able to verify that $v$ solves the equation.
Am I correct in saying that $g$ must be in $C^1$ so that the equation can be solved?
PS Is 'it' the correct pronoun when referring to what the text is saying? Or should I use 'he/she' because it is understood that the author is saying that? (Probably a stupid question)
pde regularity-theory-of-pdes transport-equation
edited Jan 16 at 16:38
Harry49
6,21331132
asked Jan 16 at 15:37
useruser
13610
$endgroup$
add a comment |
$begingroup$
Consider the following PDE (the unknown is $uin C^1(mathbb{R}^{n+1},mathbb{R})$):
$$partial_tu(x,t)+sum_{k=1}^{n} a_kpartial_k u(x,t)= cu(x,t)$$
$$u(x,0)=g(x)$$
where $g in C(mathbb{R}^n, mathbb{R})$, $ain mathbb{R}^n$ and $cin mathbb{R}$.
My textbook shows that if $u$ is a solution, then we must have $u(x,t)=g(x-ta)e^{ct}=:v(x,t)$; and, it says, $v$ is a solution.
It seems to me it is assuming $gin C^1$, because (here $k=1,...,n$) $partial_k v(x,t)= lim_{sto 0}{e^{ct}(g(x-ta+se_k)-g(x-ta))/s }$ exists iff $partial_k g(x-ta)$ exists, and in that case $partial_k v(x,t)=e^{ct}partial_k g(x-ta)$ so that $vin C^1$ iff $g in C^1$.
After assuming $g in C^1$ I am able to verify that $v$ solves the equation.
Am I correct in saying that $g$ must be in $C^1$ so that the equation can be solved?
PS Is 'it' the correct pronoun when referring to what the text is saying? Or should I use 'he/she' because it is understood that the author is saying that? (Probably a stupid question)
pde regularity-theory-of-pdes transport-equation
edited Jan 16 at 16:38
Harry49
6,21331132
asked Jan 16 at 15:37
useruser
13610
$endgroup$
Consider the following PDE (the unknown is $uin C^1(mathbb{R}^{n+1},mathbb{R})$):
$$partial_tu(x,t)+sum_{k=1}^{n} a_kpartial_k u(x,t)= cu(x,t)$$
$$u(x,0)=g(x)$$
where $g in C(mathbb{R}^n, mathbb{R})$, $ain mathbb{R}^n$ and $cin mathbb{R}$.
My textbook shows that if $u$ is a solution, then we must have $u(x,t)=g(x-ta)e^{ct}=:v(x,t)$; and, it says, $v$ is a solution.
It seems to me it is assuming $gin C^1$, because (here $k=1,...,n$) $partial_k v(x,t)= lim_{sto 0}{e^{ct}(g(x-ta+se_k)-g(x-ta))/s }$ exists iff $partial_k g(x-ta)$ exists, and in that case $partial_k v(x,t)=e^{ct}partial_k g(x-ta)$ so that $vin C^1$ iff $g in C^1$.
After assuming $g in C^1$ I am able to verify that $v$ solves the equation.
Am I correct in saying that $g$ must be in $C^1$ so that the equation can be solved?
PS Is 'it' the correct pronoun when referring to what the text is saying? Or should I use 'he/she' because it is understood that the author is saying that? (Probably a stupid question)
pde regularity-theory-of-pdes transport-equation
pde regularity-theory-of-pdes transport-equation
edited Jan 16 at 16:38
Harry49
6,21331132
asked Jan 16 at 15:37
useruser
13610
edited Jan 16 at 16:38
Harry49
6,21331132
asked Jan 16 at 15:37
useruser
13610
edited Jan 16 at 16:38
Harry49
6,21331132
edited Jan 16 at 16:38
Harry49
6,21331132
edited Jan 16 at 16:38
Harry49
6,21331132
6,21331132
asked Jan 16 at 15:37
useruser
13610
asked Jan 16 at 15:37
useruser
13610
asked Jan 16 at 15:37
useruser
13610
13610
add a comment |
add a comment |
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