I am trying to solve the inviscid Burgers' equation with help of Newton's method in Banach spaces. So I want to solve $\partial_t u(x,t) + u(x,t) \partial_x u(x,t) = 0$ with u(x,0)=f(x), let's choose f(x)=1.
As my Banach space I would like to use $U:=C^1[0,1]^2= \left\{ v: [0,1]^2 \rightarrow \mathbb{R} | v(\cdot, t)\in C^1[0,1], v(x, \cdot )\in C^1[0,1] \right\}$ with norm
$\| u |:= \max\limits_{|\alpha|\leq 2, ,\alpha_i\leq 1 }\| \partial^\alpha u \|_{C[0,1]^2}= \max\limits_{|\alpha|\leq m, \alpha_i\leq 1 }\max \left\{ | \partial^\alpha u(x,t) | : (x,t)\in [0,1]\times [0,1] \right\}.$
So let's define $F:U\rightarrow C[0,1]^2$ as
$F(u)(x,t)=\partial_t u(x,t)+u(x,t)\partial_x u(x,t)$.
I calculate the derivative $F'$:
$F'(u)(v)=\lim\limits_{h\rightarrow 0}\frac{1}{h} \left[ F(u+hv)-F(u) \right]\\ =\lim\limits_{h\rightarrow 0} \frac{1}{h}\left[ \partial_t u + h \partial_t v + (u+hv)\cdot (\partial_x u + h \partial_x v) - \partial_t u - u \partial_x u \right]\\ =\lim\limits_{h\rightarrow 0}\frac{1}{h}\left[ h \partial_t v + h v\partial_x u + hu\partial_x v + h^2 v \partial_x v \right]\\ =\partial_t v + v \partial_x u + u \partial_x v$.
So Newton's method gives me
$ F'(u_{n})(\delta_n)= -F(u_n)\\ \Rightarrow\partial_t u_{n+1} - \partial_t u_n+ (u_{n+1}-u_n)\cdot \partial_x u_n + u_n \cdot \left( \partial_x u_{n+1} - \partial_x u_n \right)= -\partial_t u_n -u_n \partial_x u_n\\ \Leftrightarrow \partial_t u_{n+1} +u_{n+1}\partial_x u_n + u_n \partial_x u_{n+1} = u_n \partial_x u_n$
So at each step I would solve the linearized boundary value problem
$ \begin{cases} \partial_t u_{n+1} +u_{n+1}\partial_x u_n + u_n \partial_x u_{n+1} = u_n \partial_x u_n \\ u_{n+1}(x,0)= 1\quad \forall x (0,1). \end{cases} $
Now my question is if this converges. First I would say, it it obvious that $F$ is Lipschitz-continous, because for $u,w\in U$ we get
$||F'(u)(v)-F'(w)(v)|| =||\partial_t v + v \partial_x u + u \partial_x v - \partial_t v - v \partial_x w - w \partial_x v||\\ =|| v \partial_x u + u \partial_x v - v \partial_x w - w \partial_x v||\\ =|| v(\partial_x u -\partial_x w) + \partial_x v(u-w)||\\ \leq || v(\partial_x u -\partial_x w) ||+|| \partial_x v(u-w)|| \leq ||v|| \cdot \left( || \partial_x u -\partial_x w ||+|| u-w|| \right) \\ \leq 2\cdot ||v|| \cdot ||u-w||$.
So when I could prove, that $[F'(u^*)]^{-1}$ exists, then I would have convergence of the Newton's method, right? How could I prove that? Or is my idea wrong at some other point?
Best regards