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gregarki khayal
gregarki khayal
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I am interested in the time when a quasi-linear $p$-system produces shocks.

Let $\mathbb T$ be 1-d torus: $[0, 1]$ with periodic boundary conditions. Fix $p$, $r \in C^\infty(\mathbb T)$. For each $n \ge 1$, let $f_n \in C_b^\infty(\mathbb R)$, $f_n \not= const$, and consider quasi-linear $p$-system on $\mathbb T$: $$ \partial_tp(t,x) = f_n(r(t,x))\partial_xr(t,x), \quad \partial_tr(t,x) = \partial_xp(t,x), $$ $$ (p,r)(0,x) = (p,r)(x). $$ It is clear that smooth solution $(p_n,r_n)(t,x)$ exists till some finite time $T_n > 0$, then the shock appears. Intuitively, if as $n \to \infty$, $f_n$ behaves more and more like a constant function, then $T_n \to \infty$.

  1. I guess that if $\sup_K |\frac{d}{dr}\sqrt{f_n}| \to 0$ on every compact $K \subseteq \mathbb R$, then $T_n \to \infty$. I believe this because of the observation for the scalar equation. Indeed, for the scalar PDE $$ \partial_t u(t,x) = f_n(u)\partial_xu(t,x), \quad u(0,\cdot) = u \in C^\infty(\mathbb T), $$ we have that ($T_n$ still stands for the time when shock appears) $$ T_n \ge \big(\sup_{x\in\mathbb T}|f'_n(u(x))u'(x)|\big)^{-1}. $$ This can be proved by the fact that all the characteristic lines are straight, and shock does not appear until any two lines cross. For the vector system, the characteristic lines are $$ \frac{d}{dt}x_t = \pm\sqrt{f_n(r(t, x_t))}, \quad x_0 \in \mathbb T, $$ but I have no idea when they will cross.

  2. For fixed $n$ and $t < T_n$, can we control $|r_n(t,\cdot)|_\infty$, $|\partial_xr_n(t,\cdot)|_\infty$, ..., by $f_n$ and the derivatives of the initial data $(p,r)$?

    can we give some sufficient condition (on $f_n$), such that $T_n \to 0$?

I guess that if $\sup_K |\frac{d}{dr}\sqrt{f_n}| \to 0$ on every compact $K \subseteq \mathbb R$, then $T_n \to \infty$. Indeed, if we consider the scalar PDE $$ \partial_t u(t,x) = f_n(u)\partial_xu(t,x), \quad u(0,\cdot) = u \in C^\infty(\mathbb T), $$ we have that ($T_n$ still stands for the time when shock appears) $$ T_n \ge \big(\sup_{x\in\mathbb T}|f'_n(u(x))u'(x)|\big)^{-1}. $$ This can be proved by the fact that all the characteristic lines are straight, and shock does not appear until any two lines cross. For the vector system $(p, r)$, the characteristic lines are $$ \frac{d}{dt}x_t = \pm\sqrt{f_n(r(t, x_t))}, \quad x_0 \in \mathbb T, $$ but I have no idea when they will cross.

  1. For fixed $n$ and $t < T_n$, can we control $|r_n(t,\cdot)|_\infty$, $|\partial_xr_n(t,\cdot)|_\infty$, ..., by $f_n$ and the derivatives of the initial data $(p,r)$?

I am interested in the time when a quasi-linear $p$-system produces shocks.

Let $\mathbb T$ be 1-d torus: $[0, 1]$ with periodic boundary conditions. Fix $p$, $r \in C^\infty(\mathbb T)$. For each $n \ge 1$, let $f_n \in C_b^\infty(\mathbb R)$, $f_n \not= const$, and consider quasi-linear $p$-system on $\mathbb T$: $$ \partial_tp(t,x) = f_n(r(t,x))\partial_xr(t,x), \quad \partial_tr(t,x) = \partial_xp(t,x), $$ $$ (p,r)(0,x) = (p,r)(x). $$ It is clear that smooth solution $(p_n,r_n)(t,x)$ exists till some finite time $T_n > 0$, then the shock appears. Intuitively, if as $n \to \infty$, $f_n$ behaves more and more like a constant function, then $T_n \to \infty$.

  1. I guess that if $\sup_K |\frac{d}{dr}\sqrt{f_n}| \to 0$ on every compact $K \subseteq \mathbb R$, then $T_n \to \infty$. I believe this because of the observation for the scalar equation. Indeed, for the scalar PDE $$ \partial_t u(t,x) = f_n(u)\partial_xu(t,x), \quad u(0,\cdot) = u \in C^\infty(\mathbb T), $$ we have that ($T_n$ still stands for the time when shock appears) $$ T_n \ge \big(\sup_{x\in\mathbb T}|f'_n(u(x))u'(x)|\big)^{-1}. $$ This can be proved by the fact that all the characteristic lines are straight, and shock does not appear until any two lines cross. For the vector system, the characteristic lines are $$ \frac{d}{dt}x_t = \pm\sqrt{f_n(r(t, x_t))}, \quad x_0 \in \mathbb T, $$ but I have no idea when they will cross.

  2. For fixed $n$ and $t < T_n$, can we control $|r_n(t,\cdot)|_\infty$, $|\partial_xr_n(t,\cdot)|_\infty$, ..., by $f_n$ and the derivatives of the initial data $(p,r)$?

I am interested in the time when a quasi-linear $p$-system produces shocks.

Let $\mathbb T$ be 1-d torus: $[0, 1]$ with periodic boundary conditions. Fix $p$, $r \in C^\infty(\mathbb T)$. For each $n \ge 1$, let $f_n \in C_b^\infty(\mathbb R)$, $f_n \not= const$, and consider quasi-linear $p$-system on $\mathbb T$: $$ \partial_tp(t,x) = f_n(r(t,x))\partial_xr(t,x), \quad \partial_tr(t,x) = \partial_xp(t,x), $$ $$ (p,r)(0,x) = (p,r)(x). $$ It is clear that smooth solution $(p_n,r_n)(t,x)$ exists till some finite time $T_n > 0$, then the shock appears. Intuitively, if as $n \to \infty$, $f_n$ behaves more and more like a constant function, then $T_n \to \infty$.

  1. can we give some sufficient condition (on $f_n$), such that $T_n \to 0$?

I guess that if $\sup_K |\frac{d}{dr}\sqrt{f_n}| \to 0$ on every compact $K \subseteq \mathbb R$, then $T_n \to \infty$. Indeed, if we consider the scalar PDE $$ \partial_t u(t,x) = f_n(u)\partial_xu(t,x), \quad u(0,\cdot) = u \in C^\infty(\mathbb T), $$ we have that ($T_n$ still stands for the time when shock appears) $$ T_n \ge \big(\sup_{x\in\mathbb T}|f'_n(u(x))u'(x)|\big)^{-1}. $$ This can be proved by the fact that all the characteristic lines are straight, and shock does not appear until any two lines cross. For the vector system $(p, r)$, the characteristic lines are $$ \frac{d}{dt}x_t = \pm\sqrt{f_n(r(t, x_t))}, \quad x_0 \in \mathbb T, $$ but I have no idea when they will cross.

  1. For fixed $n$ and $t < T_n$, can we control $|r_n(t,\cdot)|_\infty$, $|\partial_xr_n(t,\cdot)|_\infty$, ..., by $f_n$ and the derivatives of the initial data $(p,r)$?
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gregarki khayal
gregarki khayal
  • 199
  • 9

The time when a quasi-linear hyperbolic system produces shocks

I am interested in the time when a quasi-linear $p$-system produces shocks.

Let $\mathbb T$ be 1-d torus: $[0, 1]$ with periodic boundary conditions. Fix $p$, $r \in C^\infty(\mathbb T)$. For each $n \ge 1$, let $f_n \in C_b^\infty(\mathbb R)$, $f_n \not= const$, and consider quasi-linear $p$-system on $\mathbb T$: $$ \partial_tp(t,x) = f_n(r(t,x))\partial_xr(t,x), \quad \partial_tr(t,x) = \partial_xp(t,x), $$ $$ (p,r)(0,x) = (p,r)(x). $$ It is clear that smooth solution $(p_n,r_n)(t,x)$ exists till some finite time $T_n > 0$, then the shock appears. Intuitively, if as $n \to \infty$, $f_n$ behaves more and more like a constant function, then $T_n \to \infty$.

  1. I guess that if $\sup_K |\frac{d}{dr}\sqrt{f_n}| \to 0$ on every compact $K \subseteq \mathbb R$, then $T_n \to \infty$. I believe this because of the observation for the scalar equation. Indeed, for the scalar PDE $$ \partial_t u(t,x) = f_n(u)\partial_xu(t,x), \quad u(0,\cdot) = u \in C^\infty(\mathbb T), $$ we have that ($T_n$ still stands for the time when shock appears) $$ T_n \ge \big(\sup_{x\in\mathbb T}|f'_n(u(x))u'(x)|\big)^{-1}. $$ This can be proved by the fact that all the characteristic lines are straight, and shock does not appear until any two lines cross. For the vector system, the characteristic lines are $$ \frac{d}{dt}x_t = \pm\sqrt{f_n(r(t, x_t))}, \quad x_0 \in \mathbb T, $$ but I have no idea when they will cross.

  2. For fixed $n$ and $t < T_n$, can we control $|r_n(t,\cdot)|_\infty$, $|\partial_xr_n(t,\cdot)|_\infty$, ..., by $f_n$ and the derivatives of the initial data $(p,r)$?