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I'm reading through Terry Tao's 'Why are solitons stable?' and I don't understand one of the bounds he's constructed on the $H^{1}$ norm of the solution $u(x,t)$ to the gKdV, $u_{t} + u_{xxx} + (u^{k})_{x} = 0$.

At the start of section 4 he bounds $||u(t)||^{2}_{H^{1}_{x}(\mathbb{R})}$ using this integral inequality: $$ \int_{\mathbb{R}}v^{k+1} \leq C(p)\left(\int_{\mathbb{R}}v^{2}\right)^{\frac{k+3}{4}}\left(\int_{\mathbb{R}}v_{x}^{2}\right)^{\frac{k-1}{4}}.$$ He calls this the Gagliardo-Nirenberg inequality. Given this inequality the $H^{1}$ norm of the solution is bounded by its mass and energy - great.

However, I thought that the Gagliardo-Nirenberg inequality was $$ ||u||_{L^{p*}(\mathbb{R}^{n})} \leq C(n,p) ||Du||_{L^{p}(\mathbb{R}^{n})} $$ with $1\leq p < n$ and $p* = \frac{pn}{n-p} > p$, which I don't think can be used here since the gKdV has one spatial dimension, i.e. $n = 1$ precluding any use of this result. How do we use Galiardo-Nirenberg here?

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up vote 4 down vote accepted

Both of the inequalities you cite are special cases of a family of interpolation inequalities in $R^n$ that Gagliardo and Nirenberg proved, which have the form $$ \|D^j u\|_{L^p} \le C \|D^m u\|_{L^r}^a \|u\|_{L^q}^{1-a}, $$ for $0\le j<m$ and $\frac jm\le a\le1$, with $$ \frac1p = \frac jn+ a\left(\frac1r-\frac mn\right)+(1-a)\frac1q, $$ except if $1<r<\infty$ and $m-j-n/r$ is an integer then $a=1$ is disallowed. Avner Friedman's PDE book, I believe, contains a full proof.

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Thanks, I'll have a look at Friedman. – in_wolframAlpha_we_trust Mar 23 '12 at 6:09

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