Cfrgtkky

Consider the function
$$f(x,vec{v}):=g(I_t+nablacdot(Ivec{v}))$$
where $I=I(x,t)$ is the image intensity function, $g:mathbb{R}to[0,infty)$ is continuous and non negative, $vec{v}in H^1(Omega)times H^1(Omega)$. It is assumed that $I_tin L^2(Omega),I_x,I_yin L^infty(Omega)$. I have to show that $f$ is Lipschitz wrt $vec{v}$ provided $g$ is uniformly Lipschitz in every closed ball of radius $r$, $bar{B(r)}subseteqmathbb{R}$ with Lipschitz constant $L_r$. This is what I have tried so far.
begin{align*}
|f(x,vec{v})-f(x,vec{v}_0)|&=|g(I_t+nablacdot(Ivec{v}))-g(I_t+nablacdot(I:vec{v}_0))|\
&le L_r|I_t+nablacdot(Ivec{v})-I_t+nablacdot(Ivec{v}_0)|\
&le L_r|nablacdot{I(vec{v}-vec{v}_0)}|\
&le L_r|nabla Icdot(vec{v}-vec{v}_0))|+L_r|Inablacdot(vec{v}-vec{v}_0))|
end{align*}
Using the given hypothesis, we can show that that the first part satisfies the bounds
$$|nabla Icdot(vec{v}-vec{v}_0))|lesqrt 2CL_r|vec{v}-vec{v}_0|$$
where $C=max{|I_x|^2,|I_y|^2}$. I am stuck with the second part. How to get the required bound for the second part ? Do we need any additional requirement for $I$ ? Any help will be appreciated. Thank You.