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Consider the cauchy problem of finding u=u(x,t) such that $u_{t}+uu_{x}=0$ for $x \in \mathbb R$ , $t \gt 0$ with

$ u(x,0)= u_0(x)$ for $x \in \mathbb R$

then, which choices of the following functions for $u_0$ yields a $C^1$ (here, $C^1$ is the space of continuously differentiable functions) solution $u(x,t) $ for $x \in \mathbb R$ , $t \gt 0$

(a) $\frac{1}{1+x^2}$

(b) $ x $

(c) $1+ x^2$

(d) $1+2x$

I have no idea on how to proceed in this problem.

Abhishek Shrivastava
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1 Answers1

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The characteristics of the Burgers equation are lines with equations $x=x_0+ t u_0(x_0)$. If these lines intersect, the solution forms a shock, which is a region where it loses smoothness. Such an intersection occurs whenever a smaller value of $x_0$ has a larger value of $u(x_0)$. That is to say, the initial values need to be a smooth nondecreasing function of $x$ in order for the solution to be smooth.

  • :Thanks a lot for your answer but could you please elaborate your answer ... I am looking for descriptive answer using solution of the given quasi linear pde and initial conditions . I have been taught a very basic course in first order partial differential equation and have never come across the terminology you have used.so,i am not able to catch up with your explanation.Also,correct choices are (b) and (d). – Abhishek Shrivastava Jun 06 '16 at 18:19