SOLUTIONS TO HW 3

TERENCE TAO

1. Exercise 1.50 (Stability)

We have

ũ = u_{lin} + DN(ũ) + e

and

u = u_{lin} + DN(u)

so it is natural to study the difference v := ũ − u. Subtracting two equations and using the linearity of D, we have

v = D[N(u + v) − N(u)] + e.

Now we take S norms of both sides and use the triangle inequality:

∥v∥_{S} ≤ ∥D[N(u + v) − N(u)]∥_{S} + ∥e∥_{S}.

Using the hypotheses on D and e we conclude

∥v∥_{S} ≤ C_{0}∥N(u + v) − N(u)∥_{N} + ∥e∥_{S}.

Now from hypothesis, both u and u + v = ũ have an S-norm of ε or less, and hence by the Lipschitz properties of N we have ∥N(u + v) − N(u)∥_{N} ≤ \frac{1}{2C_{0}} ∥v∥_{S}. We thus have

∥v∥_{S} ≤ \frac{1}{2}∥v∥_{S} + ∥e∥_{S}.

Since the S norm of v is clearly finite, we conclude that ∥v∥_{S} ≤ 2∥e∥_{S} as claimed. If we specialise to the case when ũ is the solution to the equation ũ = ũ_{lin} + DN(ũ) for some ũ_{lin} ∈ B_{ε/2}, we thus have e = ũ_{lin} − u_{lin} and hence

∥ũ − u∥_{S} ≤ 2∥ũ_{lin} − u_{lin}∥_{S}

which is the Lipschitz claim.

2. Exercise 1.52 (Second order Duhamel)

Blah

UCLA Department of Mathematics, Los Angeles, CA 90095-1596.

E-mail address: tao@math.ucla.edu