I want to prove the existence of a solution to the following system $$\eqalign{ & {u_{tt}}-{u_{ttxx}}+{u_{xxxx}}-{({u_x}({v_x} + \frac{1}{2}u_x^2))_x}=0 \cr & {v_{tt}}- {({v_x} + \frac{1}{2}u_x^2)_x} =0 \cr & u(t,0) = u(t,1) = 0={u_x}(t,0) = {u_x}(t,1) = 0 \cr & {v_x}(t,0) = {v_x}(t,1) = 0 \cr & u(0,x) = {u_0},{u_t}(0,x) = {u_1} \cr & v(0,x) = {v_0},{v_t}(0,x) = {v_1} \cr} $$
The energy associated with this system is given by $$E(t) = {\left\| {{u_t}} \right\|^2}+{\left\| {{u_{tx}}} \right\|^2} + {\left\| {{u_{xx}}} \right\|^2} + {\left\| {{v_t}} \right\|^2} + {\left\| {{v_x} + \frac{1}{2}u_x^2} \right\|^2}$$ and satifies $$E'(t) = 0 \, ,$$ so the system is conservative.
As usual, we proceed by approximation, then we pass to the limit.
For this purpose, we consider the spaces
$$\eqalign{
& H_0^2(0,1) = \{ u \in {H^2}(0,1),u(0) = u(1) = {u_x}(0) = {u_x}(1)\} \cr
& V = \{ v \in {H^1}(0,1),{v_x}(0) = {v_x}(1),\int\limits_0^1 {v = 0} \} \cr} $$
and define $u^m$ and $v^m$ by
$$\eqalign{
& {u^m} = \sum\limits_{i = 1}^m {a_i^m(t){e_i}(x)} \cr
& {v^m} = \sum\limits_{i = 1}^m {b_i^m(t){\sigma _i}(x)} \cr} $$
where $e_i$ and $\sigma_i$ are a basis for the two spaces above.
So, we need to solve the approximate variational problem :
$$\eqalign{
& \left( {u_{tt}^m,{e_i}} \right)+\left( {u_{ttx}^m,{e_{xi}}} \right) + \left( {u_{xx}^m,{e_{ixx}}} \right) + \left( {u_x^m(v_x^m + \frac{1}{2}{{\left( {u_x^m} \right)}^2},{e_{ix}}} \right) = 0 \cr
& \left( {v_{tt}^m,{\sigma _i}} \right) + \left( {v_x^m + \frac{1}{2}{{\left( {u_x^m} \right)}^2},{\sigma _{ix}}} \right) = 0 \cr} $$
I found that
$${\left\| {{u^m_x}} \right\|^2} + {\left\| {{u^m_{tx}}} \right\|^2} + {\left\| {{u^m_t}} \right\|^2} + {\left\| {{u^m_{tt}}} \right\|^2} < C$$
where $C$ is positive constant.
By using the usual procedure, I have obtained these estimations, with some conditions on the initial datum \begin{equation} \left(u^{m},u_{t}^{m}\text{ },u_{tt}^{m}\text{ }\right) \text{ are bounded in }L^{2}(0,T;\left( H_{0}^{2}(0,L)\right) ^{2}\times \left(H_{0}^{1}(0,L)\right)), \end{equation} \begin{equation} \left( v^{m},v_{t}^{m}\text{ },v_{tt}^{m}\text{ }\right) \text{ are bounded in }L^{2}(0,T;\ V ^{2}\times L^{2}(0,L)), \end{equation} for the sequence $u^m$ there is no problem, but for $v^m$ I don't know whether my choice of space is correct or not, or even whether the space $V$ is dense in $H^1$. I need some help please.