estimation with Sobolev Space $L^2(0,T,H^1)$

Question

if we have the relation $$ \dfrac{1}{2} ||w||^2_{L^2(\Omega)} + ||\nabla w||^2_{L^2((0,T);L^2(\Omega))} \leq ||f||_{L^2((0,T);L^2(\Omega))} ||w||_{L^2((0,T);L^2(\Omega))} $$ where $f$ is data, how we prouve that $$ ||w||_{L^2((0,T),H^1(\Omega))} \leq R, $$ where $R$ is constant?

Thank's in advance to the help.

Answer 1

This is a nice application for Young's inequality $$ a b \le \frac12 a^2 + \frac12 b^2 \qquad\forall a,b\ge 0.$$ (Note that this can be proven by expanding $(a-b)^2$).

Now, by replacing $a$ with $a \, \sqrt\varepsilon$ and $b$ with $b/\sqrt\varepsilon$, we find $$ a b \le \frac{\varepsilon}2 a^2 + \frac1{2\,\varepsilon} b \qquad\forall a,b \ge 0, \varepsilon > 0.$$ This is very useful, since the coefficient in front of $a^2$ can be made arbitrarily small.

You just need to apply this inequality with $a = \|w\|_{L^2(L^2)}$, $b = \|f\|_{L^2(L^2)}$ and $\varepsilon = 1/2$.