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Consider the Cauchy Problem of finding $u(x,t)$ such that $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0,x\in\mathbb{R},t>0$$ $$u(x,0) = u_0(x), x\in\mathbb{R}$$ Which choices of the following functions for $u_{0}$ yield a $C^{1}$ solution $u(x,t)$ for all $x\in\mathbb{R},t>0$

  1. $u_{0}(x)=\frac{1}{1+x^{2}}$

  2. $u_{0}(x)=x$

  3. $u_{0}(x)=1+x^{2}$

  4. $u_{0}(x)=1+2x$.

If I use the existence and uniqueness theorem for Cauchy problem i get the corresponding determinant is non zero so all are true according to me. But in answer key only option 2nd and 4th is given. Please help me to solve the problem. Thanks a lot.

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neelkanth
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    First of all, the theorem you tried to use gives local in time existence/uniqueness/regularity. It is not global in time, which is the subject of this question ("for all $t>0$"). To see what's going on, you can make a traffic analogy. Here the velocity of cars is to the right and is proportional to the density of cars (if only real traffic worked like that, eh?). This means that increasing functions have no problems: the cars behind you are going slower than you, and the cars ahead of you are going faster than you, so you never have a shock. – Ian May 24 '16 at 12:04
  • If your function is decreasing somewhere, then you have cars behind you going faster than you, which tends to make the density behind you higher than the density ahead of you, until the density is eventually forced to be discontinuous. This is heuristic, of course, but you can formalize it using the method of characteristics: a shock occurs when two characteristic lines cross each other. – Ian May 24 '16 at 12:04
  • Sir i am student of pure mathematics...i am not getting the concepts physics... – neelkanth May 24 '16 at 16:32
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    And that's fine...but as I said, the heuristic that I just described is formalized in the method of characteristics. – Ian May 24 '16 at 16:34

1 Answers1

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In the present case of the inviscid Burgers' equation, the method of characteristics gives the unique solution, for short times and small smooth initial data. However, the method of characteristics fails when reaching the breaking time $t_b$ given by (see e.g. these posts (1) (2)) $$ t_b = -\frac{1}{\min u_0'(x)} \, . $$ At this time, the characteristics cross, and a shock wave (discontinuity of the solution) is generated. Here, we compute the derivatives of $u_0$ and draw the characteristics in $x$-$t$ plane for each case:

  1. $\min u_0'(x) = -\frac{3\sqrt{3}}{8}$. A shock occurs at $t_b \approx 1.54$ s.

1

  1. $\min u_0'(x) = 1$. No shock occurs for $t>0$.

2

  1. $\min u_0'(x) = -\infty$. Characteristics intersect at $t_b = 0$.

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  1. $\min u_0'(x) = 2$. No shock occurs for $t>0$.

4

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