Let $L, T, \lambda> 0$ be fixed, and let $f \in C^1([0,T];H^1(0,L))$, $g \in C^1([0,T];H^1(0,L)) \cap C^2([0,T];L^2(0,L))$ and $v^0 \in H^1(0,L)$. Consider the problem $$ \begin{cases} \partial_t v + \lambda \partial_x v = f(x,t) & \text{in }(0,L)\times(0,T)\\ v(0,t) = g(0,t) & \text{on }(0,T)\\ v(x,0) = v^0(x) & \text{on }(0,L), \end{cases} $$ where the unknown is $v \colon (0,L)\times (0,T) \rightarrow \mathbb{R}$. My question is:
Is it possible, or not, to obtain an inequality of the form $$ \|v(\cdot, t)\|_{H^1(0,L)} \leq C \big( \|v^0\|_{H^1(0,L)} + \max_{\tau \in [0,T]}\|g(\cdot, \tau)\|_{H^1(0,L)} \big), $$ for $C>0$ independant of $v$, where $v \in C^0([0,T]; H^1(0,L)) \cap C^1([0,T]; L^2(0,L))$ is the solution to above problem?
Having in mind to latter use Gronwall's lemma, for $t \in [0,T]$ fixed, I compute \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \int_0^L v(x,t)^2 dt &= 2 \int_0^L v \partial_tv dt \\ &= 2 \int_0^L v(- \lambda \partial_x v + f)dt \\ &= - \lambda \int_0^L \partial_x(v^2)dt + 2 \int_0^L vf dt\\ &= - \lambda (v(L,t)^2 - v(0,t)^2) + 2 \int_0^L vf dt\\ &\leq \lambda g(0,t)^2 + \|v(\cdot, t)\|_{L^2(0,L)}^2 + \|f(\cdot, t)\|_{L^2(0,L)}^2 \\ &\leq \lambda C_0 \|g(\cdot, t)\|_{H^1(0,L)}^2 + \|v(\cdot, t)\|_{L^2(0,L)}^2 + \|f(\cdot, t)\|_{L^2(0,L)}^2, \end{align*} where $C_0>0$ and Poincaré inequality is used for the last estimation. However, I do not know if it is possible to use a similar procedure for the $L^2$-norm of $\partial_x v$. I tried the following computations: \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \int_0^L (\partial_x v(x,t))^2 dt &= 2 \int_0^L \partial_x v \partial_{xt }v dt \\ &= 2 \int_0^L \partial_x v(- \lambda \partial_{xx} v + \partial_x f)dt \\ &= - \lambda \int_0^L \partial_x(\partial_x v^2)dt + 2 \int_0^L \partial_x v \partial_x f dt\\ &= - \lambda (\partial_x v(L,t)^2 - \partial_x v(0,t)^2) + 2 \int_0^L \partial_x v \partial_x f dt\\ &\leq \partial_x v(0,t)^2 + \|\partial_x v(\cdot, t)\|_{L^2(0,L)}^2 + \|f(\cdot, t)\|_{H^1(0,L)}^2, \end{align*} and I do not know how to estimate the term $\partial_x v(0,t)^2$ without making appear the time derivative of $g$.