Partial Differential Equations III/IV

[6pt] Exercise Sheet 3

Lecturer: Alpár R. Mészáros

1. Existence of classical solutions. Consider the conservation law

   $u_t+u^3{u}_x=0$ $\text{for }(x,t)\in ℝ\times (0,T),$

   $u(x,0)=u_0(x)$ $\text{for }x\in ℝ,$

   where

   Find the largest value of $T$ for which this conservation law has a classical solution $u:ℝ\times [0,T)\to ℝ$.

2. Breakdown of classical solutions. Consider the conservation law

   $u_t+u{u}_x=0$ $\text{for }(x,t)\in ℝ\times (0,\mathrm{\infty }),$

   $u(x,0)={\displaystyle \frac{1}{1+x^2}}$ $\text{for }x\in ℝ.$

   • Show that $u_t$ and $u_x$ blow up at $(x_{\mathrm{c}},t_{\mathrm{c}})$ where $t_{\mathrm{c}}=8/(3\sqrt{3})$ and $x_{\mathrm{c}}$ is to be determined.

   • Sketch the characteristics up until time $t_{\mathrm{c}}$.

   • Find an implicit equation for $u$ that does not contain partial derivatives.

3. Geometric interpretation of $t_{\mathrm{c}}$: crossing characteristics. Consider the conservation law

   $u_t+c(u)u_x=0$ $\text{for }(x,t)\in ℝ\times (0,\mathrm{\infty }),$

   $u(x,0)=u_0(x)$ $\text{for }x\in ℝ,$

   where $c$, $u_0\in C^1(ℝ)$, $u_0$ and $u_0^{\prime }$ are bounded. Assume that

   is nonempty and suppose that there exists $s_{\mathrm{c}}\in I$ such that

   $$t_c=\underset{s\in I}{\min }\frac{-1}{c{(u_0(s))}_s}=\underset{s\in I}{\min }\frac{-1}{c^{\prime }(u_0(s))u_0^{\prime }(s)}=\frac{-1}{c^{\prime }(u_0(s_{\mathrm{c}}))u_0^{\prime }(s_{\mathrm{c}})}.$$

   • Let $t>t_{\mathrm{c}}$ and $x=s_{\mathrm{c}}+c(u_0(s_{\mathrm{c}}))t$. Show that there exists $s\in ℝ$, $s\ne s_{\mathrm{c}}$, such that $x=s+c(u_0(s))t$. Therefore the point $(x,t)$ lies on (at least) two characteristics. Hint: see the proof of Theorem 3.5. Let $h(s,t)=s+c(u_0(s))t$. Show that . Recall that $h(s,t)\to \pm \mathrm{\infty }$ as $s\to \pm \mathrm{\infty }$.

   • Suppose that there are two characteristics given by $x_1(t)=s_1+tc(u_0(s_1))$ and $x_2(t)=s_2+tc(u_0(s_2))$ with $s_1,s_2\in I$, $s_1\ne s_2$, that are crossing each other at point $(x,t)$. Show that $t\ge t_{\mathrm{c}}$, that is the time coordinate of the crossing point must be greater than the critical time. Hint: use the fact that the function $s\mapsto c(u_0(s))$ is Lipschitz continuous.

4. The Fundamental Lemma of the Calculus of Variations. Let $g:\mathrm{\Omega }\to ℝ$ be a continuous function on an open set $\mathrm{\Omega }\subseteq {ℝ}^n$. Prove that if

   $${\int }_{\mathrm{\Omega }}g(𝒙)\psi (𝒙)𝑑𝒙=0\mathit{\hspace{1em}}\text{for all }\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega }),$$

   then $g=0$. Hint: Assume for contradiction that $g\ne 0$.

5. Weak formulation of scalar conservation laws. State the weak formulation of the conservation law

   $u_t+(\ln |u|+1)u_x=0$ $\text{for }(x,t)\in ℝ\times (0,\mathrm{\infty }),$

   $u(x,0)=u_0(x)$ $\text{for }x\in ℝ,$

   where

   Hint:

   $${\int }_0^u(\ln |s|+1)𝑑s={\int }_0^u\left(\frac{ds}{ds}\ln |s|+1\right)𝑑s.$$

6. Weak formulation of scalar conservation laws in several spatial variables. Scalar conservation laws in several spatial variables take the form

   $u_t+\mathrm{div}𝒇(u)=0$ $\text{for }(𝒙,t)\in {ℝ}^n\times (0,\mathrm{\infty }),$ (1)

   $u(𝒙,0)=u_0(𝒙)$ $\text{for }𝒙\in {ℝ}^n,$

   where $𝒇:ℝ\to {ℝ}^n$ and $u_0:{ℝ}^n\to ℝ$ are prescribed, and $u:{ℝ}^n\times [0,\mathrm{\infty })\to ℝ$ is the unknown. We assume that $𝒇\in C^1(ℝ;{ℝ}^n)$, $u_0\in L^1({ℝ}^n)$. The weak formulation of (1) is the following: Find $u\in L^{\mathrm{\infty }}({ℝ}^n\times (0,\mathrm{\infty }))$ such that

   $${\int }_0^{\mathrm{\infty }}{\int }_{{ℝ}^n}[u(𝒙,t){\phi }_t(𝒙,t)+𝒇(u(𝒙,t))\cdot \nabla \phi (𝒙,t)]𝑑𝒙𝑑t+{\int }_{{ℝ}^n}{u}_0(𝒙)\phi (𝒙,0)𝑑𝒙=0$$ (2)

   for all $\phi \in C_c^{\mathrm{\infty }}({ℝ}^n\times [0,\mathrm{\infty }))$.

   • Let $u\in C^1({ℝ}^n\times [0,\mathrm{\infty }))$ be a bounded, classical solution, i.e., let $u$ satisfy (1). Show that $u$ is also a weak solution, i.e., that it satisfies (2).

   • Let $u\in L^{\mathrm{\infty }}({ℝ}^n\times [0,\mathrm{\infty }))$ be a weak solution. In addition, assume that $u$ is continuously differentiable, $u\in C^1({ℝ}^n\times [0,\mathrm{\infty }))$. Show that $u$ is a classical solution.

7. The Rankine-Hugoniot condition. Consider the initial value problem

   $u_t+u{u}_x=0$ $\text{for }(x,t)\in ℝ\times (0,\mathrm{\infty }),$

   $u(x,0)=u_0(x)$ $\text{for }x\in ℝ,$

   where

   • Use the Rankine-Hugoniot condition to find the shock that originates from $x=0$ and the shock that originates from $x=1$. Show that these shocks meet at $(x,t)=(3/2,1)$.

   • Write down the weak solution for .

   • Find a weak solution for all $t>0$ by introducing a new shock at time $t=1$. Sketch the characteristics and the shocks.

   • Is the weak solution found in (iii) satisfying Lax’s entropy condition? Is it an entropy solution? Justify your answers!

8. The Lax entropy condition. Find all values of the positive constants $a$, $b$, $c$, $d$ for which the function

   is an admissible weak solution of Burger’s equations for $x\in ℝ$, $t\in [0,\mathrm{\infty })$, where admissible means that it satisfies the Lax entropy condition. For these values of the parameters, is the weak solution an entropy solution? Justify your answer!

   Hint: The PDE and the Rankine-Hugoniot condition give three equations for $a$, $b$, $c$, $d$. This question is reproduced from Shearer & Levy (2015, page 212).

9. The Lax entropy condition again. Consider the PDE

   $$u_t+\frac{1}{3}{(u^3)}_x=0.$$

   • State the Rankine-Hugoniot condition for a shock wave of the form

     (3)

     where $u_{-}$ and $u_{+}$ are constants.

   • Find all shock waves of the form (3) with $u_{-}=1$ that satisfy the Lax entropy condition.

   • Is the solution found by the arguments in (ii) an entropy solution? Justify your answer!

   This question is reproduced from Shearer & Levy (2015, page 212).

10. Rarefaction waves and shocks. Consider the initial value problem

    $u_t+u^4{u}_x=0$ $\text{for }(x,t)\in ℝ\times (0,\mathrm{\infty }),$

    $u(x,0)=u_0(x)$ $\text{for }x\in ℝ,$

    where

    • Sketch the characteristics. Explain why the problem has no classical solution for $t>0$.

    • Find a flux function $f(u)$ such that the PDE can be written in the form of a conservation law. Hence write down the Rankine-Hugoniot jump condition for a shock of the form $x=\sigma (t)$.

    • Derive the following weak solution for :

    • Explain what happens at time $t=15/4$. Use the Rankine-Hugoniot condition to find a weak solution for all $t>0$. Sketch the characteristics and the shocks. Hint: The shape of the shock changes after time $t=15/4$; it is no longer a straight line.

11. Watching paint dry. Consider the initial value problem

    $u_t+u^2{u}_x=0$ $\text{for }(x,t)\in ℝ\times (0,\mathrm{\infty }),$

    $u(x,0)=u_0(x)$ $\text{for }x\in ℝ,$

    where

    This PDE models paint flowing down a wall, where $u(x,t)$ is the thickness of the paint at height $x$ at time $t$. The initial condition corresponds to applying a strip of paint at time $t=0$. See J. Ockendon, S. Howison, A. Lacey & A. Movchan (2003) Applied Partial Differential Equations, Oxford, page 6.

    • Derive the following weak solution for :

      where $\sigma (t)=1+t/3$.

    • Explain what happens at time $t=3/2$. Use the Rankine-Hugoniot condition to find a weak solution for all $t>0$. Sketch the characteristics and the shocks. Hint: The shape of the shock changes after time $t=3/2$; it is no longer a straight line.

12. Traffic flow. Consider the traffic flow model

    $${\rho }_t+v_m(1-2\rho ){\rho }_x=0$$

    where $\rho (x,t)\in [0,1]$ is the traffic density and $v_m$ is the speed limit. We saw this model in Example 2.1.

    • Show that this PDE can be written in the form of a conservation law with flux $f(\rho )=\rho v(\rho )$, where $v(\rho )=v_m(1-\rho )$ is the velocity of the traffic.

    • Suppose that the initial density is constant except for a block of higher density, slower moving traffic:

      Find the weak solution $\rho (x,t)$ that satisfies the entropy condition.

    • Sketch the solution $\rho (x,t)$ as a function of $x$ at times $t=0$, $6/v_m$, $24/v_m$. Describe the effect as $t\to \mathrm{\infty }$ of the initial block of slower moving traffic.

    This question is based on a traffic model proposed in the paper M.J. Lighthill & G.B. Whitham (1955) On Kinematic Waves. II. A Theory of Traffic Flow on Long Crowded Roads, Proc. R. Soc. Lond. A, 229, 317–345.

13. Strong solutions. Show that if $u$ is continuous and piecewise smooth and satisfies a scalar conservation law piecewise, then $u$ is a weak solution of the conservation law. To be concrete, let $g:[0,\mathrm{\infty })\to ℝ$ be continuously differentiable and let $u:ℝ\times [0,\mathrm{\infty })\to ℝ$ be bounded and continuous, and be continuously differentiable on the sets , $V_r=\{(x,t):t>0,x>g(t)\}$. Assume that $u$ satisfies $u_t+f{(u)}_x=0$ in $V_l$ and $V_r$, in the classical sense, and satisfies $u(x,0)=u_0(x)$, where $u_0\in C(ℝ)$ and $f\in C^1(ℝ)$ are prescribed. Show that $u$ is a weak solution of the scalar conservation law

    $u_t+f{(u)}_x=0$ $\text{for }(x,t)\in ℝ\times (0,\mathrm{\infty }),$

    $u(x,0)=u_0(x)$ $\text{for }x\in ℝ.$

    Solutions of this form are called strong solutions. For those familiar with measure theory, $u$ satisfies the classical form of the PDE almost everywhere.