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Dimensional analysis tells you which norms of functional spaces must be compared. Often, this results into useful inequalities. A basic example is the Gagliardo-Nirenberg-Sobolev inequality

$$ \|f\|{L^q}\leq C{p,n}\|\nabla |f\|_{L^q}\leq C_{p,n}\|\nabla f\|_{L^p},\qquad\forall f\in{\mathcal D}(\mathbb R}^nf\in\mathcal{D}(\mathbb{R}^n), $$

which is valid only if $1\le p< n$ and (dimensional analysis) $$\frac1q=\frac1p-\frac1n.$$

A more complicated situation is that of Moser's inequalities, which involve more than two norms. Among them is the Ladyzhenskaia inequality

$$ \|f\|{L^4}\leq C\|f\|{L^2}^{1/2}\|\nabla |f\|_{L^4}\leq C\|f\|_{L^2}^{1/2}\|\nabla f\|_{L^2}^{1/2},\qquad\forall f\in{\mathcal D}(\mathbb R}^2f\in\mathcal{D}(\mathbb{R}^2). $$

This one was fundamental in the proof by C. Foias and G. Prodi of the global regularity of the Navier-Stokes solutions in two space dimension. This led me to mention that a modern attempt to prove the same result in three space dimension (1M dollar problem) is to study the Navier-Stokes equation within functional spaces that are scaling invariant, again a concept that comes from dimensional analysis.

Finally, an even more advanced situation arises when you deal with norms involving two variables of different nature, typically a time variable and space variables. Read about the Strichartz estimates. Their validity depends upon an equality between the parameters that reflects again dimensional analysis. But then you assume that the Fourier transform of the functions is supported by a submanifold with non-zero curvature.
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Dimensional analysis tells you which norms of functional spaces must be compared. Often, this results into useful inequalities. A basic example is the Gagliardo-Nirenberg-Sobolev inequality $$\|f\|$ \|f\|{L^q}\leq C{p,n}\|\nabla f\|_{L^p},\qquad\forall f\in{\mathcal D}(\mathbb R}^n,$$ R}^n, $$

which is valid only if $1\le p< n$ and (dimensional analysis) $$\frac1q=\frac1p-\frac1n.$$

A more complicated situation is that of Moser's inequalities, which involve more than two norms. Among them is the Ladyzhenskaia inequality $$\|f\|$ \|f\|{L^4}\leq C\|f\|{L^2}^{1/2}\|\nabla f\|_{L^2}^{1/2},\qquad\forall f\in{\mathcal D}(\mathbb R}^2.$$ R}^2. $$

This one was fundamental in the proof by C. Foias and G. Prodi of the global regularity of the Navier-Stokes solutions in two space dimension. This led me to mention that a modern attempt to prove the same result in three space dimension (1M dollar problem) is to study the Navier-Stokes equation within functional spaces that are scaling invariant, again a concept that comes from dimensional analysis.

Finally, an even more advanced situation arises when you deal with norms involving two variables of different nature, typically a time variable and space variables. Read about the Strichartz estimates(http://wiki.math.toronto.edu/DispersiveWiki/index.php/Strichartz_estimates). . Their validity depends upon an equality between the parameters that reflects again dimensional analysis. But then you assume that the Fourier transform of the functions is supported by a submanifold with non-zero curvature.
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Dimensional analysis tells you which norms of functional spaces must be compared. Often, this results into useful inequalities. A basic example is the Gagliardo-Nirenberg-Sobolev inequality $$\|f\|{L^q}\leq C{p,n}\|\nabla f\|_{L^p},\qquad\forall f\in{\mathcal D}(\mathbb R}^n,$$ which is valid only if $1\le p

A more complicated situation is that of Moser's inequalities, which involve more than two norms. Among them is the Ladyzhenskaia inequality $$\|f\|{L^4}\leq C\|f\|{L^2}^{1/2}\|\nabla f\|_{L^2}^{1/2},\qquad\forall f\in{\mathcal D}(\mathbb R}^2.$$ This one was fundamental in the proof by C. Foias and G. Prodi of the global regularity of the Navier-Stokes solutions in two space dimension. This led me to mention that a modern attempt to prove the same result in three space dimension (1M dollar problem) is to study the Navier-Stokes equation within functional spaces that are scaling invariant, again a concept that comes from dimensional analysis.

Finally, an even more advanced situation arises when you deal with norms involving two variables of different nature, typically a time variable and space variables. Read about the Strichartz estimates (http://wiki.math.toronto.edu/DispersiveWiki/index.php/Strichartz_estimates). Their validity depends upon an equality between the parameters that reflects again dimensional analysis. But then you assume that the Fourier transform of the functions is supported by a submanifold with non-zero curvature.