See the syllabus on my web page.

FINAL EXAM: ITV 1111, Tuesday December 18th, 1:30-3:30.

Classes and Reading:

- Week 1 (08/27-08/31) : Introduction, 1.1, 1.2, 1.3

- Week 2 (09/03-09/07) : 2.1, beginning of 2.2
- Week 3 (09/10-09/10): 2.2
- Week 4 (09/17-09/21): End of 2.2
- Week 5 (09/24-09/28): No class. Students must read 2.3.1 and 2.3.2 and then write a summary of what they think is important (3-5 pages).
- Week 6 (10/01-10/05): 2.3.3 (mostly a.), 2.3.4.

- Week 7 (10/08-10/12): 2.4.1
- Week 8 (10/15-10/19): end of 2.4.1, 2.4.2, 2.4.3.

Midterm: Thursday November 8th, 12:30-13:45.

Homework assignments:

- For Thursday October 4th, do in chapter 2, exercises 3, 5, 6, 14, 17. Exercises 20, 22, 24 are no longer part of the assignment.
- For
Tuesday November 20th, do in chapter 2, section 2.5, exercises 20, 22,
24 and in chapter 3, section 2.5, exercises 3, 5, 11, 20 (the last 2
exercises are facultative as the corresponding material was not covered
yet).

Projects assignments:

Students must choose a project and complete it by the assigned time. Examples of projects that may be chosen are given below. It is however possible to choose another project, please see with me if you wish to do so.

The 1st assignment is for Oct. 18th. A tentative list of possible projects follows:

- Understand and explain the derivation of the linear elasticity equations.

- Find explicit solutions to incompressible Stokes equations and
Maxwell equations in dimensions 2 and 3. Use this to understand some
properties of the solutions.

- Study the notion of weak solutions to incompressible
Navier-Stokes equations by Leray. See for instance P-L Lions,
Mathematical topic in fluid mechanic: Incompressible models.

- Study the "porous medium equation" du/dt = laplacian (u^2). In
particular try to understand the modeling assumptions behind this
equation (you can check chapter 2 in Vazquez). Check that
u=(t^{-n/(n+2)}-k |x|^2/t)_+ where ()_+ is the
positive part and k=1/(4(d+2)) is a weak solution. Use this explicit
weak solution to deduce some general properties of the equation. For
further, one can check the book by J-L Vazquez, The porous medium
equation: Mathematical Theory.

- Study the existence of solution to the elliptic equation -laplacian u=f(u) for a smooth non linear function of u, using the variationnal approach. You can see section 8.1.2 in Evans (and try to understand example 3). In particular try to determine which kind on conditions on f are needed (does f(u)=u^2 work for instance?).
- Write a finite difference code for the Poisson equation -laplacian u=f(x) in the box [0, 1]^2 with a given Dirichlet condition on the boundary.
- Write a finite difference code for the heat equation in dimension
2 in the box [0, 1]^2 with a given Dirichlet condition on the boundary.

The 2nd assignment is for Dec. 6th. A tentative list of projects is:

- Understand and explain the notion of viscosity solutions for Hamilton-Jacobi equation with convex Hamiltonian. See for instance Bardi M., Capuzzo Dolcetta I. Optimal control and viscosity solutions of Hamilton-Jacobi Bellman equations.
- Write a finite difference code for the Burgers equation using Lax-Friedrichs scheme (or another one if you prefer).
- Implement numerically the Hopf-Lax formula for the eikonal
equation |nabla u|=1 in dimension 2 in the square with boundary
condition f. Can you find (numerically) the location of the shocks?

- Let M be a Riemanian manifold with some metric g. Explain the notion of gradient and use that to pose the eikonal |nabla u|=1/c(x) on M. Generalize the notion of characteristics. Find the relation between g and c s.t. the characteristics are also geodesics.
- Explain the kinetic formulation for entropy solutions of the Burgers equation. See for example Perthame, kinetic formulation of conservation laws.
- Write the 1d wave equation as the system partial_t v=\partial_x
w; partial_t w=partial_x v. Use a finite difference upwind scheme to
solve the system and compare the solution to the explicit formula for
various initial data. You can try to do the same in dimension 2.

- Study the 1d wave equation over the interval [0, 1]. Try to use that to explain the difference in sound between a piano and a harpsichord (1 string per key).
- Study the multi-d wave equation over a torous using Fourier series.

Ex. 1: Sphere embedded in the torus?

Assume that we have a function phi, C^1, from the sphere S^2 on the torus [0, 1]^2. Assume as well that there exists L(v) s.t. for any C^2 curve w from [0,1] on S^2, the length of w on the sphere is given by

int_0^1 L(d\ds(phi(w(s))) ds.

1. Explain why L is convex.

2. Show that the geodesics on the sphere necessarily include all curves whose image by phi is a line.

3. Conclude.

Ex. 2: Characteristics

Compute the explicit solutions to the following problems

1. d/dt u + e^x d/dx u=0, u(t=0)=u^0.

2. d/dt u + d/dx(e^x u)=0, u(t=0)=u^0.

3. |du/dt|^2+|nabla u|^2-u^2=0, u(t=0)=u^0.

Ex. 3: Legendre transform

1. Compute the Legendre transform of the function f(x)=|x|^p for any real number p>1.

2. Deduce that x.y<= |x|^p/p + |y|^q/q with 1/p+1/q=1.

3. Show that for any M

sum_i x_i y_i <= M^p sum_i |x_i|^p/p + M^{-q} sum_i |y_i|^q/q

4. Optimize in M and conclude that

sum_i x_i y_i <= (sum_i |x_i|^p/p)^{1/p} (sum_i |y_i|^q/q)^{1/q}.

Ex. 4: Eikonal equation vs Burgers equation

1. Let phi be a C^2 solution to

d/dt phi + (d/dx phi)^2=0, x in R.

Define u=dphi/dx and prove that u solve the Burgers equation

d/dt u + d/dx(u^2)=0.

2. Compute the Lagrangian corresponding to H(p)=p^2 and show that the Hopf-Lax formula reads

phi(t,x)=x^2/(2t) + inf_y (-y x/t +y^2/2t+ phi(0,y)).

3. We assume that phi(t=0) is bounded. Prove that the infimum is attained.

4. Prove that phi is differentiable at (t,x) if the minimum problem admits a unique solution in a neighborhood of that point. Denoting y(t,x) the corresponding minimum, prove that

dphi/dx=(x-y(t,x))/t.

5. At the points where phi is differentiable we pose u=dphi/dx (so u is only defined almost everywhere). Show that u is a weak solution to Burgers equation.

Training exercises:

Those exercises are not mandatory or part of any assignment. It is recommended that the students try to do them in order to make sure that they understand the corresponding material. On the students' wishes, they can of course be returned to the professor to check that the solution is correct.

Week 2:

Ex 2., 3, 4 in 1.5; Ex 1 in 2.5

Ex. 1: Modeling. Demographers sometimes use the following model to study the evolution of a population

d/dt u + d/dx u = -d(x) u, u(t,x=0) is the integral in x of b(x) u(t,x).

Can you give an interpretation for all the terms in this context?

Ex. 2: Explicit solutions to transport Eq. One solves

d/dt u + x d/dx u=0.

1. Check that any function U( x exp t) is a solution when U is smooth.

2. Check that it is a weak solution when U is not smooth.

Ex. 3: Explicit solutions, second part (harder). One wishes to solve

d/dt u + b(x) d/dx u=0.

In order to do that, we introduce the flow of the ordinary differential equation:

d/dt X(t,s,x)= b(X(t,s,x)), X(t=s,s,x)=x.

We assume that we know there exists a unique solution to this flow problem and that the solution satisfies X(t,s,X(s,r,x))=X(t,r,x) (try to explain why!).

1. Assuming that u is a classical solution to the transport equation, show that u(t,X(t,0,x))=u(t=0,x).

2. Prove that the unique weak solution is always given by u(t=0, X(0,t,x)).

Enigma (really hard): We study

d/dt u+ d/dx(b(x) u)=0

with b(x)=1 if x<0, and b(x)=2 if x>0.

1. Write the definition of a weak solution and check that it has a meaning.

2. Find the expression of the weak solution.

3. We change b to b(x)=0 if x>0. Why can there be a problem?

Week 3:

In chapter 2, section 2.5, ex 2.

Ex. 1: Limit of continuous solutions to discrete ones. We solve -laplacian u=f in dimension n.

1. Take f=e^{-n} times the indicatrix function of the ball of radius 1. Take any point x different from 0 and show the limit of u(x) as e tends to 0.

2. Do the same computation with several small spheres and recover Newton's law of gravitation.

Ex. 2: Explicit solutions for non homogeneous equations. We wish to solve -div (D gradient u)=f with D a constant symmetric matrix with strictly positive egeinvalues.

1. Recall that there exists a symmetric and invertible matrix S s.t. S*S=D.

2. We change variables and define x=Sy and v(y)=u(Sy). Write down the equation satisfied by v.

3. Find the explicit formula for v and then u.

Hint: If this exercise is too difficult, try it first in dimension 2 with D= 4 0

0 1

Ex. 3: Asymptotic behavior at long distance. (a bit hard) Denote u the solution to -laplacian u=f. Assume f is compactly supported.

1. Show that at long distance u behaves like C_n M/|x|^{n-2} where M is the integral of f.

2. (harder) Find the next term in the asymptotic behavior (hint: it uses the integral of x f(x)).

Enigma: Continuity conditions.

Assume we can solve in dimension 2 -div( D(x) gradient u)=f where D is a discontinuous function. More precisely D=identity matrix for x_1<0 and

D=4 0 for x_1>0.

0 1

Show that if u is a weak solution (what does that mean here?) then D(x) gradient u is continuous across the line x_1=0.

Week 4

Ex. 1: Maximum principle. Denote u a C^2 solution to -laplacian u=f in O, u=g on the boundary of O. Assume g>0 and f>0 everywhere. We wish to deduce that u>0 everywhere as well.

1. Assume that u has a minimum at x_0 inside O. Recall why gradient u(x_0)=0 and why laplacian u(x_0) is non negative.

2. Conclude that u cannot have such a minimum and deduce the desired result.

Ex. 2: Mean value theorem for non homogeneous diffusion. Denote u a C^2 solution to -div (D gradient u)=0 in O, u=g on the boundary of O. Here D is a given symmetric matrix with positive eigenvalues.

1. Can you guess an equivalent mean value theorem in this case? Hint: The sphere is replaced by an ellipsoid given by D.

2. Prove the corresponding theorem!

Ex. 3: Green function for the Neumann problem. We look for the Green function which will let us solve -laplacian u=f in O with nu.gradient u=g on the boundary of O where nu is the outward normal.

1. Give the definition of the corresponding Green function.

2. Compute a formula for it in terms of the Green function G in the whole space as we did in class. Hint: The intermediary problem is now

-laplacien_y G^x=0 in O, nu.gradient_y G^x=nu.gradient_y G(x-y).

Enigma: Why is gravitation not destroying anything? Assume that the universe is infinite with a periodic distribution of masses m(x). The gravitational potential u should solve laplacian u= m. Using the Green function for the whole space, one sees that the corresponding integral giving u is infinite. How to correctly understand this problem?

Week 5

In chapter 2, section 2.5, ex. 15, 17, 18.

Ex. 1: Bounds on solutions to the heat equation in a domain. Let O be a bounded domain and u be a solution to u_t-laplacian u=f in O and u=0 on the boundary of O.

1. We define M(t) the supremum of |f(t,x)| in x for a fixed t. Find an explicit solution U(t) to the equation u_t-laplacian u=M(t) in O.

2. Define v=u-M. Apply the maximum principle to v and deduce a bound on u.

Ex. 2: Explicit solutions for non homogeneous heat equations. Find the explicit solution to the equation u_t-div(D gradient u)=0 with u(t=0)=f, for a constant, symmetric, positive matrix D.

Enigma: When can we solve the heat equation backward? Can you give conditions on f s.t. one can solve u_t+laplacian u=0 with u(t=0)=f?

Week 6

In Chapter 2, exercises 16, 19.

Ex. 1: Implicit scheme for the heat equation. We study the following numerical scheme

(u_{n+1,i}-u_{n,i})/dt=(u_{n+1,i+1}+u_{n+1,i-1}-2u_{n+1,i})/dx^2.

Assume that u_{n,i} is positive for all i. Give conditions on dt, dx s.t. u_{n+1,i} be positive for all i.

Ex. 2: Energy method for the convergence to equilibrium. Assume we have a smooth solution to u_t-laplacian u=0 in O, u=C on the boundary of O with C a constant.

1. Write down the energy for u-C.

2. Deduce that as time tends to infinity, u converges to C.

3. Try to do the same if u=f on the boundary of O with f a non constant function. Hint: Introduce g, solution to -laplacian g=0 in O, g=f on the boundary of O and study u-g.

Week 7

In Chapter 2, exercise 21.

Ex. 1: Explicit scheme for the heat equation.

1. Using the upwind scheme for the linear transport equation, write an explicit scheme for the wave equation.

2. Study the stability of this scheme.

Enigma: Wave equation with damping. Can you write a wave equation for which the oscillations slowly vanish? Study then the corresponding model, find the explicit solutions...