Editing Well-posed problem (section)

==Energy method==

The energy method is useful for establishing both uniqueness and continuity with respect to initial conditions (i.e. it does not establish existence).
The method is based upon deriving an upper bound of an energy-like functional for a given problem.

'''Example''':
Consider the diffusion equation on the unit interval with homogeneous [[Dirichlet boundary conditions]] and suitable initial data <math>f(x)</math> (e.g. for which <math>f(0)=f(1)=0</math>).

<math display="block">	
\begin{align}
u_t&=D u_{xx}, && 0<x<1,\, t>0,\, D>0,\\
u(x,0)&=f(x),\\
u(0,t)&=0,\\
u(1,t)&=0,\\
\end{align}
</math>

Multiply the equation <math>u_t=D u_{xx}</math> by <math>u</math> and integrate in space over the unit interval to obtain

<math display="block">\begin{align}
&&\int_0^1 uu_t dx&=D\int_0^1 uu_{xx}dx
\\\Longrightarrow 
&&\int_0^1 \frac{1}{2}\partial_t  u^2 dx&=Duu_x\Big|_0^1-D\int_0^1(u_x)^2dx 
\\\Longrightarrow  
&&\frac{1}{2}  \partial_t\|u\|_2^2&=0-D\int_0^1(u_x)^2dx\leq 0
\end{align}</math>

This tells us that <math>\|u\|_2</math> ([[p-norm]]) cannot grow in time.
By multiplying by two and integrating in time, from <math>0</math> up to <math>t</math>, one finds

<math display="block">\|u(\cdot,t)\|_2^2 \leq \|f(\cdot)\|_2^2</math>

This result is the '''energy estimate''' for this problem.

To show uniqueness of solutions, assume there are two distinct solutions to the problem, call them <math>u</math> and <math>v</math>, each satisfying the same initial data. 
Upon defining <math>w=u-v</math> then, via the linearity of the equations, one finds that <math>w</math> satisfies

<math display="block">	
\begin{align}
w_t&=D w_{xx}, &&0<x<1,\, t>0,\, D>0,\\
w(x,0)&=0,\\
w(0,t)&=0,\\
w(1,t)&=0,\\
\end{align}
</math>

Applying the energy estimate tells us <math>\|w(\cdot,t)\|_2^2 \leq 0</math> which implies <math>u=v</math> ([[almost everywhere]]).

Similarly, to show continuity with respect to initial conditions, assume that <math>u</math> and <math>v</math> are solutions corresponding to different initial data <math>u(x,0)=f(x)</math> and <math>v(x,0)=g(x)</math>.
Considering <math>w=u-v</math> once more, one finds that <math>w</math> satisfies the same equations as above but with <math>w(x,0)=f(x)-g(x)</math>. 
This leads to the energy estimate <math>\|w(\cdot,t)\|_2^2 \leq D\|f(\cdot)-g(\cdot)\|_2^2</math> which establishes continuity (i.e. as <math>f</math> and <math>g</math> become closer, as measured by the <math>L^2</math> norm of their difference, then <math>\|w(\cdot,t)\|_2 \to 0</math>).

The [[maximum principle]] is an alternative approach to establish uniqueness and continuity of solutions with respect to initial conditions for this example.
The existence of solutions to this problem can be established using [[Heat equation#Solving_the_heat_equation_using_Fourier_series|Fourier series]].