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Non piecewise $C^1$ solutions of conservation laws
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I studied the following theorem:(Rankine-Hugoniot condition)
Let $u:mathbbR times [0,+infty) rightarrow mathbbR $ be a piecewise $C^1$ function. Then $u$ is a weak solution if and only if the two of the following conditions are satisfied:
i) $u$ is a classical solution of in the domain where $u$ is $C^1$
ii) $u$ satisfies the jump condition
$$(u_+ -u_-) eta_t +sumlimits_j=1^df_j(u_+) -f_j(u_-) eta_x=0$$
My doubts...
i)Can we find an example of the conservation law where solutions are not piecewise $C^1$(i.e the solution does not have a piece wise $C^1$ representative) so that we cannot apply Rankine-Hugoniot condition across the jump?
ii) Is there any weaker versions of this theorem so that piecewise $C^1$ can be relaxed?
differential-equations pde regularity-theory-of-pdes hyperbolic-equations
asked Jul 26 at 6:53
Rosy
643
add a comment |Â
up vote
2
down vote
favorite
I studied the following theorem:(Rankine-Hugoniot condition)
Let $u:mathbbR times [0,+infty) rightarrow mathbbR $ be a piecewise $C^1$ function. Then $u$ is a weak solution if and only if the two of the following conditions are satisfied:
i) $u$ is a classical solution of in the domain where $u$ is $C^1$
ii) $u$ satisfies the jump condition
$$(u_+ -u_-) eta_t +sumlimits_j=1^df_j(u_+) -f_j(u_-) eta_x=0$$
My doubts...
i)Can we find an example of the conservation law where solutions are not piecewise $C^1$(i.e the solution does not have a piece wise $C^1$ representative) so that we cannot apply Rankine-Hugoniot condition across the jump?
ii) Is there any weaker versions of this theorem so that piecewise $C^1$ can be relaxed?
differential-equations pde regularity-theory-of-pdes hyperbolic-equations
asked Jul 26 at 6:53
Rosy
643
For $gin L^infty(mathbbR)$, $g(x-at)$ is a weak solution of $u_t+au_x=0$, $u(x,0)=g(x)$ (you have to know that, as you asked weak solution of transport(advection) equation). $u_t+au_x=0$ can be written as $u_t+(f(u))_x=0$ with $f(u)=au$. Now it suffices to take your favorite "wild" $gin L^infty(mathbbR)$.
– user539887
Jul 29 at 9:41
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I studied the following theorem:(Rankine-Hugoniot condition)
Let $u:mathbbR times [0,+infty) rightarrow mathbbR $ be a piecewise $C^1$ function. Then $u$ is a weak solution if and only if the two of the following conditions are satisfied:
i) $u$ is a classical solution of in the domain where $u$ is $C^1$
ii) $u$ satisfies the jump condition
$$(u_+ -u_-) eta_t +sumlimits_j=1^df_j(u_+) -f_j(u_-) eta_x=0$$
My doubts...
i)Can we find an example of the conservation law where solutions are not piecewise $C^1$(i.e the solution does not have a piece wise $C^1$ representative) so that we cannot apply Rankine-Hugoniot condition across the jump?
ii) Is there any weaker versions of this theorem so that piecewise $C^1$ can be relaxed?
differential-equations pde regularity-theory-of-pdes hyperbolic-equations
asked Jul 26 at 6:53
Rosy
643
I studied the following theorem:(Rankine-Hugoniot condition)
Let $u:mathbbR times [0,+infty) rightarrow mathbbR $ be a piecewise $C^1$ function. Then $u$ is a weak solution if and only if the two of the following conditions are satisfied:
i) $u$ is a classical solution of in the domain where $u$ is $C^1$
ii) $u$ satisfies the jump condition
$$(u_+ -u_-) eta_t +sumlimits_j=1^df_j(u_+) -f_j(u_-) eta_x=0$$
My doubts...
i)Can we find an example of the conservation law where solutions are not piecewise $C^1$(i.e the solution does not have a piece wise $C^1$ representative) so that we cannot apply Rankine-Hugoniot condition across the jump?
ii) Is there any weaker versions of this theorem so that piecewise $C^1$ can be relaxed?
differential-equations pde regularity-theory-of-pdes hyperbolic-equations
asked Jul 26 at 6:53
Rosy
643
asked Jul 26 at 6:53
Rosy
643
asked Jul 26 at 6:53
Rosy
643
asked Jul 26 at 6:53
Rosy
643
643
For $gin L^infty(mathbbR)$, $g(x-at)$ is a weak solution of $u_t+au_x=0$, $u(x,0)=g(x)$ (you have to know that, as you asked weak solution of transport(advection) equation). $u_t+au_x=0$ can be written as $u_t+(f(u))_x=0$ with $f(u)=au$. Now it suffices to take your favorite "wild" $gin L^infty(mathbbR)$.
– user539887
Jul 29 at 9:41
add a comment |Â
For $gin L^infty(mathbbR)$, $g(x-at)$ is a weak solution of $u_t+au_x=0$, $u(x,0)=g(x)$ (you have to know that, as you asked weak solution of transport(advection) equation). $u_t+au_x=0$ can be written as $u_t+(f(u))_x=0$ with $f(u)=au$. Now it suffices to take your favorite "wild" $gin L^infty(mathbbR)$.
– user539887
Jul 29 at 9:41
For $gin L^infty(mathbbR)$, $g(x-at)$ is a weak solution of $u_t+au_x=0$, $u(x,0)=g(x)$ (you have to know that, as you asked weak solution of transport(advection) equation). $u_t+au_x=0$ can be written as $u_t+(f(u))_x=0$ with $f(u)=au$. Now it suffices to take your favorite "wild" $gin L^infty(mathbbR)$.
– user539887
Jul 29 at 9:41
For $gin L^infty(mathbbR)$, $g(x-at)$ is a weak solution of $u_t+au_x=0$, $u(x,0)=g(x)$ (you have to know that, as you asked weak solution of transport(advection) equation). $u_t+au_x=0$ can be written as $u_t+(f(u))_x=0$ with $f(u)=au$. Now it suffices to take your favorite "wild" $gin L^infty(mathbbR)$.
– user539887
Jul 29 at 9:41
add a comment |Â
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For $gin L^infty(mathbbR)$, $g(x-at)$ is a weak solution of $u_t+au_x=0$, $u(x,0)=g(x)$ (you have to know that, as you asked weak solution of transport(advection) equation). $u_t+au_x=0$ can be written as $u_t+(f(u))_x=0$ with $f(u)=au$. Now it suffices to take your favorite "wild" $gin L^infty(mathbbR)$.
– user539887
Jul 29 at 9:41