What are $delta$-shock solutions?
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I studied that a measure valued solution of a conservation law $u_t+f(u)_x=0$ is a measurable map $eta : y rightarrow eta_y in Prob(mathbb{R^n})$ which satisfies $partial_t(eta_y, lambda) +partial_x (eta_y, f(lambda)) =0$ in the sense of distribution on $mathbb{R^2_+}$.
In particular, when the conservation law admits $L^{infty}$ solution then $eta_y=delta_{u(y)}$.
Now I am trying to read "Delta-shock Wave Type Solution of
Hyperbolic Systems of Conservation Laws" by Danilov and Shelkovich [1]. In this article
- what do they mean by $delta$-shocks?
- in which sense these $delta$-shocks are different from the shocks of the conservation laws?
According to the definition which I stated in the begining any shock solution $u in L^{infty}$ can be written as a dirac measure $eta_y=delta_{u(y)}$. So:
- are all shocks $delta$-shocks?
Please suggest me the reference
[1] V.G. Danilov, V.M. Shelkovich (2005): "Delta-shock wave type solution of hyperbolic systems of conservation laws", Quart. Appl. Math. 63, 401-427. doi:10.1090/S0033-569X-05-00961-8
measure-theory pde weak-derivatives hyperbolic-equations
edited Dec 20 '18 at 9:34
Harry49
7,49431341
asked Dec 19 '18 at 18:38
RosyRosy
1426
$endgroup$
add a comment |
$begingroup$
I studied that a measure valued solution of a conservation law $u_t+f(u)_x=0$ is a measurable map $eta : y rightarrow eta_y in Prob(mathbb{R^n})$ which satisfies $partial_t(eta_y, lambda) +partial_x (eta_y, f(lambda)) =0$ in the sense of distribution on $mathbb{R^2_+}$.
In particular, when the conservation law admits $L^{infty}$ solution then $eta_y=delta_{u(y)}$.
Now I am trying to read "Delta-shock Wave Type Solution of
Hyperbolic Systems of Conservation Laws" by Danilov and Shelkovich [1]. In this article
- what do they mean by $delta$-shocks?
- in which sense these $delta$-shocks are different from the shocks of the conservation laws?
According to the definition which I stated in the begining any shock solution $u in L^{infty}$ can be written as a dirac measure $eta_y=delta_{u(y)}$. So:
- are all shocks $delta$-shocks?
Please suggest me the reference
[1] V.G. Danilov, V.M. Shelkovich (2005): "Delta-shock wave type solution of hyperbolic systems of conservation laws", Quart. Appl. Math. 63, 401-427. doi:10.1090/S0033-569X-05-00961-8
measure-theory pde weak-derivatives hyperbolic-equations
edited Dec 20 '18 at 9:34
Harry49
7,49431341
asked Dec 19 '18 at 18:38
RosyRosy
1426
$endgroup$
$begingroup$
Could you please interpret $delta-$ shock as a measure valued solution.. i.e. can we write it like $eta_y$ in the definition of measure valued solution which i mentioned. Thanks in advance
$endgroup$
– Rosy
Dec 21 '18 at 10:34
add a comment |
$begingroup$
I studied that a measure valued solution of a conservation law $u_t+f(u)_x=0$ is a measurable map $eta : y rightarrow eta_y in Prob(mathbb{R^n})$ which satisfies $partial_t(eta_y, lambda) +partial_x (eta_y, f(lambda)) =0$ in the sense of distribution on $mathbb{R^2_+}$.
In particular, when the conservation law admits $L^{infty}$ solution then $eta_y=delta_{u(y)}$.
Now I am trying to read "Delta-shock Wave Type Solution of
Hyperbolic Systems of Conservation Laws" by Danilov and Shelkovich [1]. In this article
- what do they mean by $delta$-shocks?
- in which sense these $delta$-shocks are different from the shocks of the conservation laws?
According to the definition which I stated in the begining any shock solution $u in L^{infty}$ can be written as a dirac measure $eta_y=delta_{u(y)}$. So:
- are all shocks $delta$-shocks?
Please suggest me the reference
[1] V.G. Danilov, V.M. Shelkovich (2005): "Delta-shock wave type solution of hyperbolic systems of conservation laws", Quart. Appl. Math. 63, 401-427. doi:10.1090/S0033-569X-05-00961-8
measure-theory pde weak-derivatives hyperbolic-equations
edited Dec 20 '18 at 9:34
Harry49
7,49431341
asked Dec 19 '18 at 18:38
RosyRosy
1426
$endgroup$
I studied that a measure valued solution of a conservation law $u_t+f(u)_x=0$ is a measurable map $eta : y rightarrow eta_y in Prob(mathbb{R^n})$ which satisfies $partial_t(eta_y, lambda) +partial_x (eta_y, f(lambda)) =0$ in the sense of distribution on $mathbb{R^2_+}$.
In particular, when the conservation law admits $L^{infty}$ solution then $eta_y=delta_{u(y)}$.
Now I am trying to read "Delta-shock Wave Type Solution of
Hyperbolic Systems of Conservation Laws" by Danilov and Shelkovich [1]. In this article
- what do they mean by $delta$-shocks?
- in which sense these $delta$-shocks are different from the shocks of the conservation laws?
According to the definition which I stated in the begining any shock solution $u in L^{infty}$ can be written as a dirac measure $eta_y=delta_{u(y)}$. So:
- are all shocks $delta$-shocks?
Please suggest me the reference
[1] V.G. Danilov, V.M. Shelkovich (2005): "Delta-shock wave type solution of hyperbolic systems of conservation laws", Quart. Appl. Math. 63, 401-427. doi:10.1090/S0033-569X-05-00961-8
measure-theory pde weak-derivatives hyperbolic-equations
measure-theory pde weak-derivatives hyperbolic-equations
edited Dec 20 '18 at 9:34
Harry49
7,49431341
asked Dec 19 '18 at 18:38
RosyRosy
1426
edited Dec 20 '18 at 9:34
Harry49
7,49431341
asked Dec 19 '18 at 18:38
RosyRosy
1426
edited Dec 20 '18 at 9:34
Harry49
7,49431341
edited Dec 20 '18 at 9:34
Harry49
7,49431341
edited Dec 20 '18 at 9:34
Harry49
7,49431341
7,49431341
asked Dec 19 '18 at 18:38
RosyRosy
1426
asked Dec 19 '18 at 18:38
RosyRosy
1426
asked Dec 19 '18 at 18:38
RosyRosy
1426
1426
$begingroup$
Could you please interpret $delta-$ shock as a measure valued solution.. i.e. can we write it like $eta_y$ in the definition of measure valued solution which i mentioned. Thanks in advance
$endgroup$
– Rosy
Dec 21 '18 at 10:34
add a comment |
$begingroup$
Could you please interpret $delta-$ shock as a measure valued solution.. i.e. can we write it like $eta_y$ in the definition of measure valued solution which i mentioned. Thanks in advance
$endgroup$
– Rosy
Dec 21 '18 at 10:34
$begingroup$
Could you please interpret $delta-$ shock as a measure valued solution.. i.e. can we write it like $eta_y$ in the definition of measure valued solution which i mentioned. Thanks in advance
$endgroup$
– Rosy
Dec 21 '18 at 10:34
$begingroup$
Could you please interpret $delta-$ shock as a measure valued solution.. i.e. can we write it like $eta_y$ in the definition of measure valued solution which i mentioned. Thanks in advance
$endgroup$
– Rosy
Dec 21 '18 at 10:34
add a comment |
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$begingroup$
Could you please interpret $delta-$ shock as a measure valued solution.. i.e. can we write it like $eta_y$ in the definition of measure valued solution which i mentioned. Thanks in advance
$endgroup$
– Rosy
Dec 21 '18 at 10:34