Here $\Gamma(t)$ is a family of smooth compact connected and oriented surfaces in $\mathbb{R}^n$. $u$ solves the PDE $\dot{u} + u\nabla_\Gamma \cdot v - \Delta_\Gamma u = 0$ where $\dot{u} = u_t + v \cdot \nabla u$ is the material derivative and $v$ is a velocity field on $\Gamma$. The $\nabla_{\Gamma(t)}$ is the surface gradient (for fixed $t$) obtained by projecting the ordinary gradient onto the tangent space.
Multiplying this PDE by a test function, integrating by parts, setting the test function to $u$ and doing some manipulations, we can get $\frac{1}{2}\frac{d}{dt}\int_{\Gamma(t)} u^2 + \int_{\Gamma(t)} |\nabla_\Gamma u|^2 + \frac{1}{2}\int_{\Gamma(t)} u^2 \nabla_{\Gamma(t)} \cdot v = 0$ How can we get the following estimate from the above: $\sup_{t \in [0,T]} \lVert u(t) \rVert^2_{L^2(\Gamma(t))} + \int_0^T\lVert \nabla_{\Gamma(t)} u(t) \rVert^2_{L^2(\Gamma(t))} < c\lVert u_0 \rVert^2_{L^2(\Gamma_0)}\;?$
Here, $u(t=0) = u_0$ is known and we can assume that $\nabla_\Gamma \cdot v$ is in $L^\infty(\cup (\{t\}\times \Gamma(t)))$ (and the constant $c$ depends on $v$, $T$ and $\Gamma$).
Clearly integrating in time between 0 and $T$ is what we need to do but I don't know how to deal with the third term in the first equation.