Well - Posedness - MIT OpenCourseWare

18.336 spring 2009

lecture 2

02/05/09

Well-Posedness

Def.: A PDE is called well-posed (in the sense of Hadamard), if

(1) a solution exists (2) the solution is unique (3) the solution depends continuously on the data

(initial conditions, boundary conditions, right hand side)

Careful: Existence and uniqueness involves boundary conditions

Ex.: uxx + u = 0

a)

u(0)

=

0,

u(

2

)

=

1

unique

solution

u(x)

=

sin(x)

b) u(0) = 0, u() = 1 no solution

c) u(0) = 0, u() = 0 infinitely many solutions: u(x) = A sin(x)

Continuous dependence depends on considered metric/norm.

We typically consider || ? ||L , || ? ||L2 , || ? ||L1 .

Ex.:

ut = uxx u(0, t) = u(1, t) = 0

u(x, 0) = u0(x)

heat equation

boundary conditions initial conditions

well-posed

ut = -uxx u(0, t) = u(1, t)

u(x, 0) = u0(x)

backwards heat equation boundary conditions

initial conditions

no continuous dependence

on initial data [later]

Notions of Solutions

Classical solution

kth order PDE u Ck

Ex.: 2u = 0 u C

ut + ux = 0 u(x, 0) C1

u(x, t) C1

Weak solution kth order PDE, but u / Ck.

1

Ex.: Discontinuous coefficients

(b(x)ux)x

=

0

u(0)

=

0

u(1) = 1

b(x)

=

1 2

x

<

1 2

x

1 2

u(x) =

4 3

x

2 3

x

+

1 3

x

<

1 2

x

1 2

Ex.: Conservation laws

ut

+

(

1 2

u2)x

=

0

Burgers' equation

Image by MIT OpenCourseWare.

Fourier Methods for Linear IVP

IVP = initial value problem

ut = ux advection equation

ut = uxx heat equation

ut = uxxx Airy's equation

ut = uxxxx

w=+

a) on whole real axis: u(x, t) =

eiwxu^(w, t)dw

Fourier transform

w=- +

b) periodic case x [-, [: u(x, t) = u^k(t)eikx Fourier series (FS)

Here case b).

k=-

u

nu

PDE: (x, t) - (x, t) = 0

t

insert FS:

+

xn du^k

dt

(t)

-

(ik)nu^k(t) eikx

=

0

k=-

Since (eikx)kZ linearly independent:

du^k dt

=

(ik)nu^k(t)

ODE

for

each

Fourier

coefficient

2

Solution:

u^k(t) = e(ik)nt u^k(0)

Fourier

coefficient

of

initial

conditions:

u^k

(0)=

1 2

-

u0(x)e-ikxdx

+

u(x, t) = u^k(0)eikxe(ik)nt

k=-

n = 1: n = 2: n = 3: n = 4:

u(x, t) = u^k(0)eik(x+t)

k

u(x, t) = u^k(0)eikxe-k2t

k

u(x, t) = u^k(0)eik(x-k2t)

k

u(x, t) = u^k(0)eikxek4t

k

all waves travel to left with velocity 1

frequency k decays with e-k2t

frequency k travels to right

with velocity k2 dispersion

all frequencies are amplified

unstable

Message:

For linear PDE IVP, study behavior of waves eikx.

The ansatz u(x, t) = e-iwteikx yields a dispersion relation of w to k.

The wave eikx is transformed by the growth factor e-iw(k)t.

Ex.:

wave equation: heat equation: conv.-diffusion: Schro?dinger: Airy equation:

utt = c2uxx ut = duxx ut = cux +duxx iut = uxx ut = uxxx

w = ?ck w = -idk2 w = -ck-idk2 w = -k2 w = k3

conservative dissipative dissipative dispersive dispersive

|e?ickt| = 1 |e-dk2t| 0 |eickte-dk2t| 0 |eik2t| = 1 |e-ik3t| = 1

3

MIT OpenCourseWare

18.336 Numerical Methods for Partial Differential Equations

Spring 2009

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