Entropy Solutions, Shocks, and Rarefactions

Let \(U\) be a bounded domain in \(\mathbb {R}^d\) and \(\Gamma \) be a closed subset of \(\partial U\). Consider the energy

on the admissible class

Suppose that \(u\) is a minimizer for the calculus of variations problem

find and justify the PDE boundary value problem solved by \(u\). Show that there is at most one solution of this PDE in the admissible class \(\mathcal {A}\).

Let \(U\) be a bounded domain and \(u\) solve the heat equation \[ \begin {cases} \partial _t u=\Delta u+cu & \text {in } U\times (0,\infty ),\\ u(x,0)=g(x),\\ u(x,t)=0 & \text {on } \partial U\times (0,\infty ). \end {cases} \]

Classify, with justification, the optimal range of values for \(c\in \mathbb {R}\) so that for any smooth initial data \(g\) the solution \(u\) above converges to 0 as \(t\to \infty \).

Find the entropy solution of the following scalar conservation law, arising in traffic flow modeling,

with initial data

After the rarefaction hits the shock you should write the explicit ODE IVP for the shock location \(\gamma (t)\) and a piecewise definition of the solution using this shock location \(\gamma (t)\), but you don’t actually need to solve the ODE IVP to get the explicit formula for \(\gamma (t)\).

Let \(f:\mathbb {R}^d\to \mathbb {R}\) smooth. Show that there is at most one smooth solution of \[ \begin {cases} -\Delta u=f & \text {in } \mathbb {R}^d,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0. \end {cases} \]

Give a (non-trivializing) additional condition on \(f\) so that you can show existence of such a solution.

Let \(U\) a bounded domain in \(\mathbb {R}^n\). Consider a smooth solution \(u\) of the equation \[ \begin {cases} u_t=\Delta u+c(x,t)u & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ),\\ u(x,0)=f(x). \end {cases} \]

Show that, if \(c(x, t) \le -\gamma \) for all \(x\) and \(t\), then

Consider the scalar conservation law,

Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \]

Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.

Show that \(u(x)=\frac {1}{4\pi |x|}\) is the fundamental solution of the Laplace equation in \(\mathbb {R}^3\), \(-\Delta u=\delta _{0}\) in the sense of distributions, i.e. \[ \int _{\mathbb {R}^3} u(x)\Delta \varphi (x)\,dx=-\varphi (0) \]

for all \(\varphi \) smooth and compactly supported. You should provide a fully detailed proof, not just a formal computation.

Let \(b(x)\) a smooth vector field on a bounded domain \(U\) in \(\mathbb {R}^n\). Consider a smooth scalar solution \(u\) of the equation \[ \begin {cases} u_t=\Delta u+\nabla \cdot (b(x)u) & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ),\\ u(x,0)=f(x). \end {cases} \]

1. Show that if \(\nabla \cdot b = 0\) then for all \(t > 0\)

   \[ \sup _{x\in U}u(x,t)\le \sup _U |f|. \]
2. Show that if \(\nabla \cdot b \ne 0\) then for all \(t > 0\) \[ \sup _{x\in U}u(x,t)\le e^{At}\sup _U |f| \quad \text {with } A=\sup _U \nabla \cdot b. \]

Consider the scalar conservation law,

Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1-x, & 0<x<1. \end {cases} \]

Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.

Let \(p(t)>\alpha >0\) be a smooth function defined for all \(t\). Prove that for any solution of \[ x''+p(t)x'+x=0, \tag {4} \] \(x(t)\to 0\) as \(t\to \infty \).

Consider Newton’s equation \[ \ddot {x}+\frac {x}{2+x^2}=\epsilon \cos t \] Prove that there is an \(\epsilon _0>0\) such that if \(\abs {\epsilon }<\epsilon _0\) then there is a \(2\pi \)-periodic solution.

Find and sketch the solution of the differential equation \[ u_t+uu_x=0 \] on the domain \(-\infty <x<\infty \), \(t>0\) subject to the initial condition \(u(0,x)=u_0(x)\), where

1. \[ u_0(x)= \begin {cases} 0; & \text {if }x<0,\\ x, & \text {if }0<x<1,\\ 1, & \text {if }x>1; \end {cases} \]
2. \[ u_0(x)= \begin {cases} 1; & \text {if }x<0,\\ 1-x, & \text {if }0<x<1,\\ 0, & \text {if }x>1. \end {cases} \]

For which (if any) of these initial profiles does the solution of the problem \[ u_t+uu_x=\epsilon u_{xx}, \] \(\epsilon >0\), converges to a traveling wave profile? What is the speed of the travelling wave solution?

Let \(\delta \in \mathcal {D}'(\mathbb {R})\) denote the ‘delta function’. Find \(u\in \mathcal {D}'(\mathbb {R})\) that solves the following linear partial differential equation: \[ \partial ^2 u=\delta . \] Is the solution unique?

Hint: You can solve this equation by guessing the distribution and then verifying that is solves the equation.

Consider the problem of a thin layer of paint of thickness \(h(x,t)\) and speed \(u(x,y,t)\) flowing down a wall, see Fig. 1. The paint is assumed to be uniform in the \(z\)-direction. The balance between gravity and viscosity (fluid friction) means that the velocity satisfies the equation \[ \frac {\partial ^2u}{\partial y^2}=-c, \] where \(c\) is a positive constant. This is supplemented by the boundary conditions \[ u(x,0,t)=0,\qquad \left .\frac {\partial u(x,y,t)}{\partial y}\right |_{y=h}=0. \] The density of paint per unit length in the \(x\)-direction is \(\rho _0h(x,t)\) where \(\rho _0\) is a constant, and the corresponding flux is \[ q(x,t)=\rho _0\int _0^h u(x,y,t)d y. \]

1. Using conservation of paint, and solving for \(u(x,y,t)\) in terms of \(h(x,t)\) and \(y\), derive the following PDE for the thickness \(h\): \[ \frac {\partial h}{\partial t}+ch^2\frac {\partial h}{\partial x}=0. \]
2. Set \(c=1\). Show that the characteristics are straight lines and that the Rankine-Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {[h^3/3]^+_-}{[h]^+_-}. \]
3. A stripe of paint is applied at \(t=0\) so that \[ h(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \] Show that, for small enough \(t\), \[ h= \begin {cases} 0, & x<0,\\ (x/t)^{1/2}, & 0<x<t,\\ 1, & t<x<S(t),\\ 0, & S(t)<x, \end {cases} \] where the shock is \(x=S(t)=1+t/3\).
4. Explain why this solution changes at \(t=3/2\), and show that thereafter \[ \frac {dS}{dt}=\frac {S}{3t}. \]

1. Let \(A\) be an \(n\times n\) real matrix. Find necessary and sufficient conditions on \(A\) so that for all \(x_0\in \mathbb {R}^n\), the solution \(\varphi (t;x_0)\) of (1) remains bounded for \(t\ge 0\). Prove your result. \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \tag {1} \]
2. Suppose that \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) is such that \(f(0)=0\) and the Jacobian matrix \(A=df(0)\) satisfies the condition in (a). Do all solutions \(\psi (t;x_0)\) of the nonlinear equation (2) still remain bounded for \(t\ge 0\), at least if \(x_0\) is close to \(0\)? Explain. \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0. \end {cases} \tag {2} \]

Consider the problem of a thin layer of paint of thickness \(h(x,t)\) and speed \(u(x,y,t)\) flowing down a wall. The paint is assumed to be uniform in the \(z\)-direction. The balance between gravity and viscosity (fluid friction) means that the velocity satisfies the equation \[ \frac {\partial ^2u}{\partial y^2}=-c, \] where \(c\) is a positive constant. This is supplemented by the boundary conditions \[ u(x,0,t)=0,\qquad \left .\frac {\partial u(x,y,t)}{\partial y}\right |_{y=h}=0. \] The density of paint per unit length in the \(x\)-direction is \(\rho _0h(x,t)\) where \(\rho _0\) is a constant, and the corresponding flux is \[ q(x,t)=\rho _0\int _0^h u(x,y,t)d y. \]

1. Sketch a figure illustrating the physical problem.
2. Using conservation of paint, and solving for \(u(x,y,t)\) in terms of \(h(x,t)\) and \(y\), derive the following PDE for the thickness \(h\): \[ \frac {\partial h}{\partial t}+ch^2\frac {\partial h}{\partial x}=0. \]
3. Set \(c=1\). Show that the characteristics are straight lines and that the Rankine-Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {[h^3/3]^+_-}{[h]^+_-}. \]
4. A stripe of paint is applied at \(t=0\) so that \[ h(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \] Show that, for small enough \(t\), \[ h= \begin {cases} 0, & x<0,\\ (x/t)^{1/2}, & 0<x<t,\\ 1, & t<x<S(t),\\ 0, & S(t)<x, \end {cases} \] where the shock is \(x=S(t)=1+t/3\). Explain why this solution changes at \(t=3/2\), and show that thereafter \[ \frac {dS}{dt}=\frac {S}{3t}. \]

Does the partial differential equation \[ \frac {\partial u}{\partial t}-\frac {\partial u}{\partial x}=-u \] subject to the restriction that \(u=\exp (-x^2)\) on the curve \(t=x^2\) have a solution? If it does have a solution, indicate for what ranges of \(x\) and \(t\) this solution is uniquely defined and if it does not have a solution, explain why.

The Ornstein-Uhlenbeck (OU) differential equation is \[ \frac {\partial p}{\partial t} =\frac {\partial }{\partial x}(kxp)+D\frac {\partial ^2p}{\partial x^2}. \]

1. What type of partial differential equation is this? What calculation is needed to justify this statement?
2. If \(t\) has units of time and \(x\) has units of length, what are the units of the parameters \(k\) and \(D\)?
3. Express this equation in nondimensional variables? How many nondimensional parameters remain after nondimensionalization?
4. Find the solution of the OU equation on the domain \(-\infty <x<\infty \), \(t\ge 0\) with initial data \(p(x,0)=\delta (x-x_0)\) for arbitrary \(x_0\). Give an interpretation of this solution in terms of a probability.

1. Suppose an initial profile \(u_0(x)\) is monotone with \(\lim _{x\to -\infty }u_0(x, )=u_L\), \(\lim _{x\to \infty }u_0(x)=u_R\). Under what conditions on \(u_L\) and \(u_R\) does the solution of Burger’s equation \[ \frac {\partial u}{\partial t} +u\frac {\partial u}{\partial x}=0, \] evolve into a shock, and what is its asymptotic speed?
2. Consider the viscous Burger’s equation \[ \frac {\partial u}{\partial t} +u\frac {\partial u}{\partial x} =\epsilon \frac {\partial ^2u}{\partial x^2}. \] with \(\epsilon >0\). Under what conditions on \(u_L\) and \(u_R\), where \(\lim _{x\to -\infty }u(x,t)=u_L\), \(\lim _{x\to \infty }u(x,t)=u_R\) does this equation have a travelling wave solution, and what is its speed? Verify your result.

1. Find all values of \(i^i\).
2. Find all solutions of the equation \(\cos z=2\).
3. Find all solutions of the equation \(\tan z=i\).

Consider the van der Pol equation \[ \ddot {x}+x+\epsilon (x^2-1)\dot {x}=\Gamma \cos (\omega t), \qquad 0<\epsilon \ll 1 \] with \(\Gamma =O(1)\) and \(\omega \ne 1/3,1,3\). Using the method of multiple scales show that the solution is attracted to \[ x(t)=\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2\ge 2(1-\omega ^2)^2\) and \[ x(t)=2\left [1-\frac {\Gamma ^2}{2(1-\omega ^2)^2}\right ]^{1/2}\cos t +\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2<2(1-\omega ^2)^2\). Explain why this result breaks down when \(\omega =1/3,1,3\).

1. Does a solution of the equation \(u_x+xu_y=u\) with \(u(x,x^2)=x\) exist for \(-\infty <x<\infty \), \(y>0\)? Explain why or why not.
2. Does a solution of the equation \(xu_x-yu_y=u\) with \(u(x,x^2)=x\) exist for \(-\infty <x<\infty \), \(y>0\)? Explain why or why not.

Consider the equation \[ \frac {\partial u}{\partial t} +\frac {\partial }{\partial x}\left (\frac {u^2}{2}\right ) =\epsilon \frac {\partial ^2u}{\partial x^2}, \] for \(-\infty <x<\infty \) and \(t>0\).

1. Look for a traveling wave solution \(u(x,t)=U(z)\), \(z=(x-Vt)/\epsilon \), with velocity \(V>0\) and \(U(z)\to U_\pm \) as \(z\to \pm \infty \). Solve the resulting ODE for \(U(z)\) and deduce that \[ V=\frac {[U^2/2]_{-\infty }^{\infty }}{[U]_{-\infty }^{\infty }} \]
2. Discuss how the traveling wave solution relates to shock solutions of the quasilinear equation obtained by setting \(\epsilon =0\).
3. Using phase-plane analysis, show that the wave profile \(U\) is monotonically decreasing, that is, \(dU/dz<0\).

Consider the one-dimensional parabolic PDE \[ \frac {\partial u}{\partial t} +\frac {\partial }{\partial x}\left (\frac {u^n}{n}\right ) =\epsilon \frac {\partial ^2u}{\partial x^2}. \]

1. Show that a traveling wave solution \(u(x,t)=U(z)\) with \(z=(x-Vt)/\epsilon \) satisfies \[ \frac {dU}{dz}=\frac {U^n}{n}-VU+\text {constant}, \] and deduce that \[ V=\frac {[U^n/n]_{-\infty }^{\infty }}{[U]_{-\infty }^{\infty }}. \]
2. Discuss how the traveling wave solution relates to shock solutions of the quasilinear equation obtained by setting \(\epsilon =0\). Also show that, when \(n=2\), \(U\) can only tend to \(U(\pm \infty )\) as \(z\to \pm \infty \) if \(dU/dz<0\).

Let \(p(t)>0\) be a smooth function that is periodic of least period \(T>0\). Show that every solution approaches zero as \(t\to \infty \). \[ x''+p(t)x'+x=0 \]

Show that for small \(\norm {\epsilon }\), there is a \(2\pi \)-periodic solution near zero to the inhomogeneous equation \[ \ddot {y}+5y=\epsilon \sin ^3 t. \]

Suppose that \(\rho (x,t)\) is the number density of cars evolving according to the traffic model \[ \frac {\partial \rho }{\partial t}+\frac {\partial (\rho u)}{\partial x}=0. \] with \(u\) the car speed. Take \(u=1-\rho \). A queue is building up at a traffic light \(x=1\) so that, when the light turns green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & x<0\text { and }x>1,\\ x, & 0<x<1. \end {cases} \]

1. Use the method of characteristics to solve the Cauchy problem. In particular, construct the characteristic projections and identify any shocks or rarefaction waves.
2. Hence, show that a collision first occurs at \(x=1/2\) when \(t=1/2\) and, thereafter, there is a shock \(S(t)\). Use the Rankine-Hugoniot condition to derive the following expression for the speed of the shock: \[ \frac {dS}{dt}=\frac {S+t-1}{2t}. \]

Show that for small \(\norm {\epsilon }\), there is a \(2\pi \)-periodic solution to the forced Duffing’s equation \[ \ddot {y}+3y+\epsilon y^3=\cos t. \]

Consider the problem \(u''(x)=x\delta '(x)\), where \(\delta \) is the “delta function”. Explain what it means for this problem to have a solution \(u\) in the sense of distributions. Find a distributional solution \(u\) and prove that it does in fact solve the problem in the sense of distributions.

Let \(A\) be a real \(m\times n\) matrix, and let \(b\in \mathbb {R}^m\) be a column vector. Consider the problem \[ A^*Ax=A^*b. \tag {*} \]

1. Explain the relationship between \((*)\) and the problem \[ \min _{x\in \mathbb {R}^n}\lVert Ax-b\rVert ^2. \]
2. Under what conditions on \(A\), \(A^*\), and \(b\) is it guaranteed that \((*)\) has a solution \(x\)? Under what conditions is it guaranteed that \((*)\) has a unique solution?
3. Given \(\varepsilon >0\), consider the problem \[ (A^*A+\varepsilon I)x_\varepsilon =A^*b. \tag {**} \] Formulate an associated minimization problem, describe existence and uniqueness, and explain how \(\lim _{\varepsilon \to 0+}x_\varepsilon \) is related to solutions of \((*)\).

Construct the Green’s function for the problem \[ u''(x)+\alpha ^2u(x)=f(x),\quad \text {on }(0,1),\qquad u(0)=u(1)=0. \] Write the solution \(u\) in terms of the Green’s function, and determine values of \(\alpha \) for which the Green’s function does not exist.

Consider a modified predator-prey model in the first quadrant. Show that it has at least one nonconstant periodic solution. \[ \begin {aligned} \dot {x} &= x(8-x)-\frac {9xy}{x+1},\\ \dot {y} &= y\left (1-\frac {y}{x}\right ) \end {aligned} \]

Show that there is a solution that goes to \(\infty \) as \(t\to \infty \). \[ y''-\cos ^2(t)y'+\sin ^2(t)y=0. \]

Show that the planar system \[ \begin {aligned} \dot {x} &= x(x^2+y^2-3x-1)-y,\\ \dot {y} &= y(x^2+y^2-3x-1)+x \end {aligned} \] has at least one nonconstant periodic solution.

1. Find all conditions on \(\alpha ,\beta ,\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \tag {1} \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Consider the integral equation \[ \phi (x)-\lambda \int _0^\pi \sin (x+t)\phi (t)\,dt=\cos (x)+\sin (x), \qquad 0\le x\le \pi . \]

1. Find the unique solution when \(\lambda \ne \pm 2/\pi \). Why is a unique solution guaranteed to exist?
2. Show that there is no solution when \(\lambda =2/\pi \).
3. Show that when \(\lambda =-2/\pi \) there is a one-parameter family of solutions of the form \[ \phi (x)=f_p(x)+\alpha f_h(x). \] Determine \(f_h(x)\).

Consider the first-order equation \[ u=x_1u_{x_1}+x_2u_{x_2}+\frac 12\left (u_{x_1}^2+u_{x_2}^2\right ), \] for the function \(u(x_1,x_2)\).

1. Write down the characteristic equations \[ \begin {aligned} \dot {p} &= -D_xF-D_zF\,p,\\ \dot {z} &= D_pF\cdot p,\\ \dot {x} &= D_pF, \end {aligned} \] associated with this problem.
2. Verify that the Cauchy data \(u(x_1,0)=\frac 12(1-x_1^2)\), \(x_1\in \mathbb {R}\), is noncharacteristic, and find an admissible set of initial conditions.
3. Find a solution \(u(x_1,x_2)\) to the original equation which satisfies the given Cauchy data.

Compute the unique entropy solution of \[ \begin {aligned} u_t+\left (\frac {u^2}{2}\right )_x &= 0 && \text {in }\mathbb {R}\times (0,\infty ),\\ u &= g && \text {on }\mathbb {R}\times \{0\}, \end {aligned} \] where \[ g(x)= \begin {cases} 0, & \text {if }x<0,\\ x, & \text {if }0\le x\le 1,\\ 0, & \text {if }x>1. \end {cases} \] Draw a sketch documenting your answer, illustrating what happens at all times \(t>0\).

Determine whether the planar system \[ \begin {aligned} \dot {x} &= x-x^3-xy^2,\\ \dot {y} &= 2y-y^5-yx^4 \end {aligned} \] has any nonconstant periodic solutions.

Suppose that \(u\in C^2\) solves \[ u_{tt}-\operatorname {div}(aDu)=0,\qquad \text {in }\mathbb {R}^n\times (0,\infty ), \] where \(a\in C^1(\mathbb {R}^n)\) satisfies \(\frac 12\le a(x)\le 1\) for all \(x\).

1. Assunimg \(\operatorname {supp}_x u(x,t)\subset B(0,R)\subset \mathbb {R}^n\) for all \(0\le t\le T\), show that the energy \[ E(t)=\frac 12\int _{\mathbb {R}^n}u_t^2+a\lvert Du\rvert ^2\dd x,\qquad 0\le t\le T, \] is conserved.
2. Using an appropriately defined local energy, establish that solutions exhibit finite propagation speed: if for some \(r>0\), \(u(x,0)\) and \(u_t(x,0)\) are zero for \(x\in B(x_0,r)\subset \mathbb {R}^n\), then \(u\) is zero in the cone \(K=\{(x,t):0\le t\le r,\lvert x-x_0\rvert <r-t\}\).

Consider Burger’s equation \[ \begin {aligned} u_t+\left (\frac {u^2}{2}\right )_x &= 0, && \text {in }\mathbb {R}\times (0,\infty ),\\ u &= g, && \text {on }\mathbb {R}\times \{0\}. \end {aligned} \]

1. Construct an integral solution satisfying the Rankine-Hugoniot condition which exhibits “nonphysical shocks.”
2. Write down the explicit entropy condition for this problem and explain why it rules out the solution in part (a).
3. Construct another integral solution which satisfies the entropy condition, and explain why this condition does not allow shocks.

Show that the planar system \[ \begin {aligned} \dot {x} &= y,\\ \dot {y} &= (1-3x^2-y^2)y-x \end {aligned} \] has at least one nonzero periodic solution.

Traffic flow may be modeled by the equation \[ \frac {\partial }{\partial x}\bigl (G(\rho )\bigr )+\frac {\partial }{\partial t}\rho =0 \] where \(\rho (x,t)\) is the density of traffic flow along a highway. If \(\rho _{\max }\) is the maximum density of cars in a bumper to bumper situation, then \[ G(\rho )=c\rho \left (1-\frac {\rho }{\rho _{\max }}\right ) \] where \(c>0\) is the free speed constant. Suppose that the initial concentration is given by \[ \rho (x,o)= \begin {cases} \frac 12\rho _{\max }, & \text {if }x<0,\\ 0, & \text {if }x\ge 0. \end {cases} \] Describe the weak solution. Does a shock form? Interpret your result in terms of traffic flow.

Find a solution \(u(x,y)\) of Laplace’s equation on the domain \(-\infty <x<\infty \), \(0<y<\infty \) for which \(u(x,0)=x^{1/2}\) for \(0<x<\infty \). What is \(u(x,0)\) for \(-\infty <x<0\)?

Show that the system \[ \begin {aligned} \dot {x}&=x+y-x^3\\ \dot {y}&=-x+y-y^3 \end {aligned} \] has at least one nonzero periodic solution.

Traffic flow may be modeled by the equation \[ \frac {\partial }{\partial x}\bigl (G(\rho )\bigr )+\frac {\partial }{\partial t}\rho =0 \] where \(\rho (x,t)\) is the density of traffic flow along a highway. If \(\rho _{\max }\) is the maximum density of cars in a bumper to bumper situation, then \[ G(\rho )=c\rho \left (1-\frac {\rho }{\rho _{\max }}\right ) \] where \(c>0\) is the free speed constant. Suppose that the initial concentration is given by \[ \rho (x,o)= \begin {cases} \frac 12\rho _{\max }, & \text {if }x<0,\\ 0, & \text {if }x\ge 0. \end {cases} \] Find the shock curve and describe the weak solution. Interpret your result in terms of traffic flow.

Consider the scalar differential equation \[ \ddot {x}+x=-\epsilon f(x,\dot {x}) \] with \(\lvert \epsilon \rvert \ll 1\). Let \(y=\dot {x}\).

1. Show that if \(E(x,y)=(x^2+y^2)/2\) then \[ \dot {E}=-\epsilon f(x,y)y. \] Hence show that an approximate periodic solution of the form \(x=A\cos t+O(\epsilon )\) exists if \[ \int _0^{2\pi } f(A\cos t,-A\sin t)\sin td t=0. \]
2. Let \(E_n=E(x(2\pi n),y(2\pi n))\) with \(x(t)=A_n\cos t+O(\epsilon )\) for \(2\pi n\le t<2\pi (n+1)\). Show that to lowest order \(E_n\) satisfies a difference equation of the form \[ E_{n+1}=E_n+\epsilon F(E_n) \] and write down \(F(E_n)\) explicitly as an integral. Hence deduce that a periodic orbit with approximate amplitude \(A^*=\sqrt {2E^*}\) exists if \(F(E^*)=0\) and this orbit is stable if \[ \epsilon \frac {dF}{dE}(E^*)<0. \]
3. Using the above result, find the approximate amplitude of the periodic orbit of the van der Pol equation \[ \ddot {x}+x+\epsilon (x^2-1)\dot {x}=0 \] and verify that it is stable.

Paint flowing down a wall has thickness \(u(x,t)\) satisfying (for \(t>0\)) \[ \frac {\partial u}{\partial t}+u^2\frac {\partial u}{\partial x}=0. \]

1. Show that the characteristics are straight lines and that the Rankine–Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {\left [u^3/3\right ]_-^+}{[u]_-^+}. \]
2. A stripe of paint is applied at \(t=0\) so that \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1\\ 1, & 0<x<1 \end {cases} \] For sufficiently small \(t\), determine \(u\) in the domains \(x<0\), \(0<x<t\), \(t<x<S(t)\) and \(S(t)<x\), where the shock is \(x=S(t)=1+t/3\)
3. Explain why the solution changes at \(t=3/2\) and show that thereafter \(\dot {S}=S/3t\).

1. Using Green’s functions, find the solution to the equation \[ u''=f(x),\qquad u(0)=0,\qquad u(1)=1. \]
2. Find the necessary and sufficient condition on \(f(x)\) for a solution to \[ u''+m^2\pi ^2u=f(x),\qquad u(0)=u(1)=0 \] to exist.

Find a solution \(u(x,y)\) of Laplace’s equation in 2 dimensions for which \(u(x,0)=|x|^{1/2}\).

For each \(n\in \N \), consider the Cauchy problem \[ \begin {aligned} -\Delta u_n &= 0, && \text {in } U,\\ u_n &= \frac {1}{n^2}\sin nx, && \text {on } \{(x,y):y=0\},\\ \frac {\partial u_n}{\partial y} &= \frac {1}{n}\sin nx, && \text {on } \{(x,y):y=0\}, \end {aligned} \] where \(U=\{(x,y):0<y<1\}\). Find a sequence \(\{u_n\}\) of solutions to these problems, prove that \(\{u_n\}\) does not converge to zero, and explain why this implies that the Cauchy problem above is not “well posed”.

Denote \(U=\{x\in \mathbb {R}^2:x_1>0\}\), and \(\Gamma =\{x\in \mathbb {R}^2:x_1=1\}\). Use the method of characteristics to solve the first-order problem \[ \begin {aligned} x_1u_{x_1}+uu_{x_2}-x_2 &= 0, && \text {in } U,\\ u(1,x_2) &= 2x_2, && \text {on } \Gamma . \end {aligned} \] (Hint: to solve the characteristic equations, note that the linear combinations \(x_2+z\) and \(x_2-z\) decouple.) Determine the region around \(\Gamma \) upon which the solution is well-defined.

1. Find all conditions on \(\alpha \), \(\beta \), \(\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Let \(A\in \mathbb {R}^{n\times m}\), \(B\in \mathbb {R}^{r\times m}\), and \(b\in \mathbb {R}^n\) be given. Assume \(R(B^*)=N(A)\). For fixed \(\mu \ge 0\), consider the problem \[ \min _{x\in \mathbb {R}^m} \|Ax-b\|_2^2+\mu \|Bx\|_2^2. \]

1. Find the necessary conditions (analogous to the normal equations for the standard least-squares problem) and prove that the necessary conditions have a solution when \(\mu =0\).
2. Prove that the necessary conditions have a unique solution if \(\mu >0\).

Consider the fourth-order differential equation \[ \begin {aligned} u^{(4)} &= f, && \text {on } (0,1),\\ u(0) &= u(1), && u'(0)=u'(1),\\ u''(1) &= -2, && u^{(3)}(1)=1. \end {aligned} \]

1. Find the least restrictive constraint(s) on \(f\in L^2(0,1)\) under which a solution to the problem exists.
2. With fixed \(f\) chosen such that a solution exists, is the solution unique? If the solution is unique, prove it. Otherwise provide a counterexample.

Conservation Laws. Consider the first-order equation \[ (G(u))_x+u_y=0, \] where \(G\in C^\infty \), with initial condition \(u(x,0)=h(x)\).

1. Using the method of characteristics, derive the general solution of the initial value problem. Assuming \(h\in C^\infty \), find a condition on \(G'(h(x))\) which implies the solution \(u(x,y)\) will develop a singularity in finite time \(y>0\).
2. Set \(G(u)=\frac {1}{2}u^2\). Given \(u_0\in \mathbb {R}\), find explicit weak solution(s) for \[ h(x)= \begin {cases} u_0, & x<0,\\ 0, & x>0. \end {cases} \] Show that your solutions are weak solutions. Which properties do your solutions exhibit: shocks, rarefaction, nonuniqueness?

Construct the Green’s function for the equation \[ u''(x)+\alpha ^2u(x)=f(x),\qquad u(0)=u(1),\qquad u'(0)=u'(1) \] and express the solution in terms of the Green’s function. For what values of \(\alpha \) does the Green’s function fail to exist?

Picard’s Existence Theorem.

1. Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Show that there is an \(\epsilon >0\) so that the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0 \end {cases} \] has a unique solution \(y\in C^1([t_0-\epsilon ,t_0+\epsilon ],\mathbb {R}^n)\).
2. Suppose that the hypothesis on \(f\) was replaced by \(f(x,t)\in C^0(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\). Would all or part of the conclusions in (a.) continue to hold?

Periodic Coefficients. Prove that if \(\lvert \epsilon \rvert \ne 0\) is small enough, then all solutions are bounded \[ \ddot {u}+[1+\epsilon \sin (3t)]u=0. \] [Convert to a system of equations. Give a condition for the boundedness of all solutions. Verify that the condition is satisfied in this case.]

Three Species Predator-Prey Model. Consider the following Lotka-Volterra model for populations of three species \(x,y,z\ge 0\). \[ \begin {cases} \dot {x}=x(4-y-z),\\ \dot {y}=y(1-z),\\ \dot {z}=z(x+y-5);\\ x(0)=x_0,\quad y(0)=y_0,\quad z(0)=z_0. \end {cases} \] The only stationary point for this system with all positive coordinates is at \(P=(2,3,1)\). Show that there is an open \(U\subset \mathbb {R}^2\) such that \((2,3)\in U\), a function \(f(x,y)\in C^1(U)\) and \(\epsilon >0\) so that for initial points which are \(\epsilon \)-close to \(P\), that is for \(\lvert (x_0,y_0,z_0)-P\rvert <\epsilon \), if \(z_0=f(x_0,y_0)\) then the solution converges to \(P\) as \(t\to \infty \), but if \(z_0\ne f(x_0,y_0)\) then the solution eventually goes away from \(P\), that is for some \(t_0>0\), \(\lvert (x(t_0),y(t_0),z(t_0))-P\rvert \ge \epsilon \).

1. Prove that \(x\delta (x)=0\) in the sense of distribution.
2. Formulate the differential equation \(xu'-u=0\) as a problem in the sense of distribution and find its general solution, also in the sense of distribution.

Peano Existence Theorem. Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Consider the system of ODE’s \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \qquad (\spadesuit ) \]

1. Find an example that shows that the solution of \((\spadesuit )\) may not be unique. Give another example that shows that the solution may not exist for all time.
2. Prove the Peano Existence theorem: Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R})\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Then there is an \(\epsilon >0\) and a function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) which solves \((\spadesuit )\).

Conservation Laws. Consider the first order equation \[ \begin {cases} \dfrac {\partial u}{\partial t}+(\tan u)\dfrac {\partial u}{\partial x}=0, & \text {for }x\in \mathbb {R}\text { and }t\ge 0,\\ u(x,0)=\varphi (x), & \text {for }x\in \mathbb {R}. \end {cases} \qquad (\diamond ) \]

1. Find a solution \(u(x,t)\) to \((\diamond )\) where \(\varphi (x)=\operatorname {Arctan}(x)\).
2. For \[ \varphi (x)= \begin {cases} 0, & \text {if }x<0,\\ \frac {\pi }{4}, & \text {if }x>0, \end {cases} \] find two weak solutions to \((\diamond )\). Prove that one of them is a weak solution.
3. State the entropy (Rankine-Hugoniot) condition. Which weak solution of part (b.) satisfies this condition?

1. Determine the conditions on \(f(x)\), \(\alpha \) and \(\beta \) for which there solutions of \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=\alpha ,\qquad u'(1)-u(1)=\beta . \]
2. Find the “best” solution of \[ \frac {d^2u}{dx^2}=1,\qquad u(0)=1,\qquad u'(1)-u(1)=1. \]

1. Find all conditions on \(\alpha \), \(\beta \), \(\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Find an integral representation for the solution of the telegrapher’s equation \[ u_{tt}+au_t+bu=u_{xx} \] with \(u(0,t)=f(t)\) and \(u_x(0,t)=g(t)\) specified for all time.

Let \(E\) be an open subset of \(\mathbb {R}^n\) containing \(x_0\) and assume that \(\mathbf {f}\) is a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\).

1. Use the method of successive approximations to prove that the initial value problem \(x'=\mathbf {f}(x)\), \(x(0)=x_0\), has a unique solution on some interval \([-a,a]\), and give an estimate for \(a\).
2. Indicate how the result in (a) can be used to prove a similar result for the nonautonomous case.

Consider the equation \[ a(x,y)u_x+b(x,y)u_y=c(x,y,u) \] where \(a\), \(b\) and \(c\) are suitably smooth.

1. Define what is meant by an integral surface for the above equation.
2. Write down the characteristic equations and show how they can be used to construct an integral surface that contains the curve \(x=\phi (s)\), \(y=\psi (s)\), \(u=\eta (s)\), where \(\phi \), \(\psi \) and \(\eta \) are \(C^1\).
3. What additional conditions, if any, must be placed on the data to guarantee a locally unique solution? Explain carefully.
4. Demonstrate the above method by solving \[ \begin {aligned} u_x+2u_y&=-u,\\ u(0,y)&=f(y) \end {aligned} \] where \(f\) is of class \(C^1\).

Use the above and a Gronwall inequality argument (note: \(\lvert f(t,x)\rvert \le \lvert f(t,x)-f(t,0)\rvert +\lvert f(t,0)\rvert \)) to show that solutions of (2.2) are defined on all of \(\mathbb {R}\).

Define what is meant by a fundamental matrix solution \(\Phi (t)\) of (2.3).

Consider the problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0,\\ u(x,0) &= F(x),\\ u_t(x,0) &= 0,\\ u(0,t)=u(\pi ,t) &= 0 \end {aligned} \tag {7.1} \]

1. Use separation of variables to formally solve (7.1) when \(F\in C[0,\pi ]\).
2. Suppose that \(F\in C^4[0,\pi ]\) and that \(F''(0)=F''(\pi )=0\). Prove that the series you obtain in (a) defines a classical solution of (7.1) under the foregoing conditions on \(F\).

Consider the integral equation \[ u(x)=f(x)+\frac {1}{2}\int _0^1 xtu(t)\,dt,\qquad \text {for }0\le x\le 1 \] where \(f(x)\) is a prescribed function.

1. Show that the equation has at most one solution.
2. Derive a formula for the solution.
3. Calculate the solution for the special case \(f(x)=5x/6\).

Define: \(z\) is a uniformly Liapunov stable solution of (2).

Define what is meant by a weak solution of (3).

Under what additional condition are the weak solutions unique?

Assume that \(u\in H^1(\mathbb {T}^n)\) is a weak solution to \(Lu=f(x)\) on \(\mathbb {T}^n\), where \(f\in C^\infty (\mathbb {T}^n)\). Show that \(u\in C^\infty (\mathbb {T}^n)\).

Use properties of the Leray-Schauder degree to deduce that (2.13) has a \(T\)-periodic solution for all \(\epsilon \).

Find the general solution, in the sense of distribution theory, of the differential equation \[ x^2u'(x)=\delta (x^2-1). \]

Show that solutions of initial value problems for (2) are defined on \((-\infty ,\infty )\).

Let \(\operatorname {trace} A(t)\equiv 0\), \(t\in (-\infty ,\infty )\) and let \(\overline {X}(t)\) be a matrix solution of (2) show that \(\det \overline {X}(t)=\text {constant}\), \(t\in (-\infty ,\infty )\).

Let \(\overline {X}(t)\) be a matrix solution of (2) with \(\det \overline {X}(t_0)\ne 0\), some \(t_0\). Characterize all other matrix solutions of (2) and provide a characterization of all solutions of (2).

Assume that \(\overline {X}(t)\) is a matrix solution of (2) with \(\overline {X}(0)\) nonsingular and \(\overline {X}(t)\) uniformly bounded on \([0,\infty )\). Also assume \[ \liminf _{t\to \infty }\operatorname {Re}\int _{0}^{t}\operatorname {trace} A(s)\,ds >-\infty . \] Prove that \(\overline {X}^{-1}(t)\) is uniformly bounded on \([0,\infty )\) and no solution \(\varphi (t)\) of (2), not identically zero, can satisfy \(\varphi (t)\longrightarrow 0\) as \(t\longrightarrow \infty \).

Let \(E\) be an open subset of \(\mathbb {R}^n\). Let \(f\) be a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\), and let \(x_0\) belong to \(E\).

1. Use the method of successive approximations to prove that the initial value problem \(x'=f(x)\), \(x(0)=x_0\) has a unique solution on some interval \([-a,a]\), and give an estimate for \(a\).
2. Suggest how the result in a) can be used to prove a similar result for the nonautonomous case.

The quasilinear Cauchy problem \[ xu_x+yuu_y+xy=0,\qquad u=5\quad \text {when }xy=1, \] has the solution \[ u(x,y)=-1+\sqrt {38-2xy}. \]

1. Derive this solution by the method of characteristics. (Hint: Along a characteristic \(x(s)\), \(y(s)\), \(u(s)\) find \(d(xy)/ds\) and then \(du/dz\) where \(z=xy\).)
2. Verify that \(u(x,y)\) is a solution in the plane domain defined by \(xy<19\).