Problem 4: Nonlinear scalar conservation equations. [20 marks] An important class of first order PDEs are the so-called nonlinear scalar conservation equations in one spatial variable. The corresponding Cauchy problem is du +3./(w) = 0, on (0,00) (0,0) = (*) on {t=0) * R. Clearly, the above PDE is quasi-linear as it can be seen by rewriting it into the equivalent form df 2u+afula,u=0. a(w): du In analogy to the transport equation appearing in Problem 3, the function a(u) is called the local propagation velocity. The most famous among all conservation equations is the Burgers equation (Johann Marting Burgers 1895 - 1981) where (u) - Answer the following questions, 1 .) We consider the following Canely problem for the Burgers equation au + ud,u = 0, on (0,60) R. 2 (0,4) = (x) on {t=0) R. where -{: 1 if 1 Find all characteristics associated to the Cauchy problem above! b) Plot the characteristics found above on the same plane (where on the s-axis you put the time variable while the z-axis remains as usual associated to the I coordi- nate) for different values of the integration constant. What happens to the solution (t, x) at the point (1, 1)? Explain your answer! c) Find the solution u(t, ) of the given Cauchy problem. For which values of t does the solution exist? d) Plot the solution found in the previous part for t = 0,1/4.1/2, 3/4. At which time the solution develops a singularity? This problems signalizes that the Cauchy problem of a nonlinear equation does not have in general a global solution for all times. We started with a contimous initial profile to that developed a singularity in a finite time! Problem 4: Nonlinear scalar conservation equations. [20 marks] An important class of first order PDEs are the so-called nonlinear scalar conservation equations in one spatial variable. The corresponding Cauchy problem is du +3./(w) = 0, on (0,00) (0,0) = (*) on {t=0) * R. Clearly, the above PDE is quasi-linear as it can be seen by rewriting it into the equivalent form df 2u+afula,u=0. a(w): du In analogy to the transport equation appearing in Problem 3, the function a(u) is called the local propagation velocity. The most famous among all conservation equations is the Burgers equation (Johann Marting Burgers 1895 - 1981) where (u) - Answer the following questions, 1 .) We consider the following Canely problem for the Burgers equation au + ud,u = 0, on (0,60) R. 2 (0,4) = (x) on {t=0) R. where -{: 1 if 1 Find all characteristics associated to the Cauchy problem above! b) Plot the characteristics found above on the same plane (where on the s-axis you put the time variable while the z-axis remains as usual associated to the I coordi- nate) for different values of the integration constant. What happens to the solution (t, x) at the point (1, 1)? Explain your answer! c) Find the solution u(t, ) of the given Cauchy problem. For which values of t does the solution exist? d) Plot the solution found in the previous part for t = 0,1/4.1/2, 3/4. At which time the solution develops a singularity? This problems signalizes that the Cauchy problem of a nonlinear equation does not have in general a global solution for all times. We started with a contimous initial profile to that developed a singularity in a finite time
