Many equations of mathematical physics are governed by a scaling symmetry: if $u$ solves the equation, then so does some rescaled version $u_\lambda$. The heat equation $\partial_t u = \Delta u$ on $\mathbb{R}^n$, for instance, is invariant under $(t,x) \mapsto (\lambda^2 t, \lambda x)$ in the sense that if $u(t,x)$ is a solution, then so is $u(\lambda^2 t, \lambda x)$ for every $\lambda > 0$. Understanding the long-time behaviour of solutions, or the threshold between global existence and finite-time blowup for nonlinear variants, requires working in a function space whose norm is itself invariant under the same family of dilations. Such a space is called a critical space for the equation.

The inhomogeneous Sobolev norm $\|f\|_{H^s}$ involves the weight $(1+|\xi|^2)^s$, which mixes the behaviour of $f$ at low frequencies (where it contributes control of the $L^2$ mass) and high frequencies (where it controls derivative information). Under dilation $f_\lambda(x) = \lambda^\alpha f(\lambda x)$, the factor $(1+|\lambda\xi|^2)^s$ does not simplify to a pure power of $\lambda$, and the $H^s$ norm cannot be invariant for any choice of $\alpha$. Replacing $(1+|\xi|^2)^s$ with the pure weight $|\xi|^{2s}$ removes the low-frequency contribution entirely and restores the scaling symmetry. This is the content of the homogeneous Sobolev spaces $\dot{H}^s(\mathbb{R}^n)$.

From Inhomogeneous to Homogeneous Spaces

The inhomogeneous Sobolev space $H^s(\mathbb{R}^n)$ uses the weight $(1 + |\xi|^2)^s$, which controls both the function itself (via the "$+1$" at low frequencies) and its derivatives (via $|\xi|^{2s}$ at high frequencies). However, this weight is not scaling invariant: under the dilation $f \mapsto f(\lambda\,\cdot\,)$, the Fourier transform rescales as $\hat{f}(\lambda\,\cdot\,)(\xi) = \lambda^{-n}\hat{f}(\xi/\lambda)$, and the weight $(1 + |\xi/\lambda|^2)^s$ does not simplify — the "$+1$" term breaks the homogeneity.

To recover scaling invariance, one replaces $(1 + |\xi|^2)^s$ with the homogeneous weight $|\xi|^{2s}$. This weight scales cleanly: $|\xi/\lambda|^{2s} = \lambda^{-2s}|\xi|^{2s}$, producing the invariance $\|f(\lambda\,\cdot\,)\|_{\dot{H}^s} = \lambda^{s - n/2}\|f\|_{\dot{H}^s}$. But removing the "$+1$" creates two problems:

1. Seminorm, not norm. The weight $|\xi|^{2s}$ vanishes at $\xi = 0$ for $s > 0$. Any tempered distribution whose Fourier transform is supported at $\{0\}$ — which corresponds to a polynomial in physical space — has zero $\dot{H}^s$ "norm." To obtain a genuine norm, one must quotient by polynomials: the space is defined on $\mathcal{S}'(\mathbb{R}^n)/T_{\mathcal{P}}$.

2. The weight $|\xi|^s$ is not smooth at $\xi = 0$. Unlike $(1 + |\xi|^2)^{s/2} \in \mathcal{O}_M(\mathbb{R}^n)$, the weight $|\xi|^s$ fails to be smooth at the origin (for non-integer $s$) and blows up there (for $s < 0$). The distributional product $|\xi|^s\hat{f}$ cannot be formed on all of $\mathbb{R}^n$. Instead, one works on $\mathbb{R}^n_0 = \mathbb{R}^n \setminus \{0\}$, where $|\xi|^s$ is smooth, and requires that $\hat{f}$ restricted to $\mathbb{R}^n_0$ is a regular distribution with a $T$-representative.

Both issues are resolved by the construction in the following section.

The Polynomial Quotient

Before defining the spaces, two structural points demand attention.

The first is algebraic. For $s > 0$, the weight $|\xi|^s$ vanishes at the origin, so any tempered distribution $f$ whose Fourier transform is supported at $\{0\}$ will contribute nothing to any $L^2$-based norm involving $|\xi|^s$. By the structure theorem for distributions supported at a point, the tempered distributions supported at $\{0\}$ are precisely the finite linear combinations of the Dirac delta and its distributional derivatives $\partial^\alpha \delta_0$ for multi-indices $\alpha \in \mathbb{N}^n$. Taking inverse Fourier transforms, these correspond exactly to the polynomials: $f(x) = \sum_\alpha c_\alpha x^\alpha$. Thus the space of polynomials $\mathcal{P}$ is in the kernel of any putative "$|\xi|^s$-weighted norm," and to obtain a genuine normed space one must pass to a quotient.

[definition: Space of Tempered Distributions Modulo Polynomials]
Let $\mathcal{P}$ denote the set of all polynomials on $\mathbb{R}^n$. Each polynomial $p \in \mathcal{P}$ has at most polynomial growth, so the regular distribution $T_p(\varphi) := \int_{\mathbb{R}^n} p(x)\,\varphi(x)\,d\mathcal{L}^n(x)$ is a well-defined element of $\mathcal{S}'(\mathbb{R}^n)$ (the integral converges because the rapid decay of $\varphi \in \mathcal{S}(\mathbb{R}^n)$ dominates the polynomial growth of $p$). Define

\begin{align*} T_{\mathcal{P}} := \{T_p \in \mathcal{S}'(\mathbb{R}^n) : p \in \mathcal{P}\}, \end{align*}

which is a linear subspace of $\mathcal{S}'(\mathbb{R}^n)$.

The space of tempered distributions modulo polynomials, denoted $\mathcal{S}'(\mathbb{R}^n)/T_{\mathcal{P}}$, is the quotient of $\mathcal{S}'(\mathbb{R}^n)$ by $T_{\mathcal{P}}$. Two tempered distributions $f, g \in \mathcal{S}'(\mathbb{R}^n)$ represent the same equivalence class in this quotient if and only if $f - g \in T_{\mathcal{P}}$, i.e. $f - g = T_p$ for some polynomial $p \in \mathcal{P}$.
[/definition]

The second point is analytic. The weight $|\xi|^s$ is not in $\mathcal{O}_M(\mathbb{R}^n)$: for non-integer $s$ it fails to be smooth at $\xi = 0$, and for $s < 0$ it blows up there. This means we cannot form the product $|\xi|^s \hat{f}$ as a tempered distribution on all of $\mathbb{R}^n$ — unlike the inhomogeneous weight $(1+|\xi|^2)^{s/2}$, which is smooth and polynomially bounded everywhere. Instead, we work on $\mathbb{R}^n_0$, where $|\xi|^s$ is smooth.

The quotient and the distributional restriction to $\mathbb{R}^n_0$ are perfectly compatible: if $f - f' \in \mathcal{P}$, then $\widehat{f - f'}$ is a distribution supported at $\{0\}$, so $\hat{f}$ and $\hat{f'}$ agree as elements of $\mathcal{D}'(\mathbb{R}^n_0)$. The restriction $\hat{f}\big|_{\mathcal{D}'(\mathbb{R}^n_0)}$ therefore depends only on the equivalence class $[f]$.

Definition

The definition requires that $\hat{f}$, viewed as an element of $\mathcal{D}'(\mathbb{R}^n_0)$ via the distributional restriction (i.e. $\hat{f}$ acting only on test functions $\phi \in C_c^\infty(\mathbb{R}^n_0)$), is a regular distribution. We write $(\hat{f})_{T\text{-rep}}$ for the $T$-representative of $\hat{f}$ on $\mathbb{R}^n_0$: the function $(\hat{f})_{T\text{-rep}} \in L^1_{\mathrm{loc}}(\mathbb{R}^n_0)$ satisfying $\hat{f}(\phi) = \int (\hat{f})_{T\text{-rep}}(\xi)\,\phi(\xi)\,d\mathcal{L}^n(\xi)$ for every $\phi \in C_c^\infty(\mathbb{R}^n_0)$. The notation $(\hat{f})_{T\text{-rep}}$ emphasises that this is the representative of $\hat{f}$ (the distributional Fourier transform), not the Fourier transform of some function $f_{T\text{-rep}}$. (This is the same convention used on the Inhomogeneous Sobolev Space page.)

[definition: Homogeneous Sobolev Space]
Let $s \in \mathbb{R}$ and $n \ge 1$. The homogeneous Sobolev space of order $s$ is
\begin{align*} \dot{H}^s(\mathbb{R}^n) &:= \left\{[f] \in \mathcal{S}'(\mathbb{R}^n)/T_{\mathcal{P}} \;\middle|\; \exists\, v \in L^1_{\mathrm{loc}}(\mathbb{R}^n_0) \text{ such that } \hat{f}\big|_{\mathcal{D}'(\mathbb{R}^n_0)} = T_v \text{ and } |\xi|^s v \in L^2(\mathbb{R}^n) \right\}, \end{align*}
where $T_v: C_c^\infty(\mathbb{R}^n_0) \to \mathbb{C}$ is the regular distribution defined by
\begin{align*} T_v(\phi) := \int_{\mathbb{R}^n_0} v(\xi)\,\phi(\xi)\, d\mathcal{L}^n(\xi). \end{align*}
We write $v = (\hat{f})_{T\text{-rep}}$ and call $v$ the $T$-representative of $\hat{f}$ on $\mathbb{R}^n_0$. The notation $(\hat{f})_{T\text{-rep}}$ emphasises that this is the representative of the distribution $\hat{f}$, not the Fourier transform of some function $f_{T\text{-rep}}$.

The norm is $\|[f]\|_{\dot{H}^s} := \| |\xi|^s(\hat{f})_{T\text{-rep}} \|_{L^2(\mathbb{R}^n)}$ and the inner product is
\begin{align*} \langle [f], [g] \rangle_{\dot{H}^s} &:= \int_{\mathbb{R}^n} |\xi|^{2s}\,(\hat{f})_{T\text{-rep}}(\xi)\,\overline{(\hat{g})_{T\text{-rep}}(\xi)} \, d\mathcal{L}^n(\xi). \end{align*}
[/definition]

There are three things to verify before the definition can be used: that $(\hat{f})_{T\text{-rep}}$ is unique and depends only on the equivalence class $[f]$ (so the norm is well-defined); that every integrand is a genuine pointwise product of measurable functions (no distributional products are involved, since $|\xi|^s$ is smooth and positive on $\mathbb{R}^n_0$ and $\{0\}$ has $\mathcal{L}^n$-measure zero); and that the resulting normed space is complete. All three are established by the following theorem, whose proof shows that $\dot{H}^s$ is isometrically isomorphic to $L^2$ via the map $[f] \mapsto |\xi|^s(\hat{f})_{T\text{-rep}}$.

[quotetheorem:470]

[remark: The Integral Formula for $L^2$ Functions]
When $f \in L^2(\mathbb{R}^n)$, the equivalence class $[T_f] \in \mathcal{S}'(\mathbb{R}^n)/T_{\mathcal{P}}$ belongs to $\dot{H}^s(\mathbb{R}^n)$ whenever $|\xi|^s \hat{f} \in L^2(\mathbb{R}^n)$. By the Plancherel theorem, $\widehat{T_f} = T_{\hat{f}}$ where $\hat{f} \in L^2(\mathbb{R}^n)$ is the $L^2$ Fourier transform. Since $T_{\hat{f}}$ is a regular distribution on all of $\mathbb{R}^n$ (and hence on $\mathbb{R}^n_0$), the $T$-representative is $(\widehat{T_f})_{T\text{-rep}} = \hat{f}$, and the norm becomes
\begin{align*} \|[T_f]\|_{\dot{H}^s}^2 = \int_{\mathbb{R}^n} |\xi|^{2s}\, |\hat{f}(\xi)|^2\, d\mathcal{L}^n(\xi), \end{align*}
where $\hat{f}$ is a genuine $L^2$ function — no distributional restrictions or $T$-representatives are needed. This is the formula used throughout PDE theory whenever the function in question is known to be in $L^2$. Compare with the inhomogeneous version, where the weight $|\xi|^{2s}$ is replaced by $(1 + |\xi|^2)^s$.
[/remark]

Special Cases

For $s = 0$, the membership condition reduces to $\hat{f}_{T\text{-rep}} \in L^2(\mathbb{R}^n)$. The following theorem makes the resulting identification precise.

[quotetheorem:471]

For positive integer $s = k$, elements of $\dot{H}^k(\mathbb{R}^n)$ are those tempered distributions (modulo polynomials) whose $k$-th order distributional partial derivatives all belong to $L^2(\mathbb{R}^n)$ — only the top-order derivatives are controlled.

[quotetheorem:472]

When the Quotient Is Superfluous

The quotient by $T_{\mathcal{P}}$ is genuinely necessary when $s \ge n/2$, since in that range the weight $|\xi|^{-s}$ is not locally integrable near the origin, and the semi-norm cannot distinguish two distributions differing by a polynomial. For $s < n/2$, however, the situation is simpler: the $T$-representative $\hat{f}_{T\text{-rep}}$ extends from $\mathbb{R}^n_0$ to a function in $L^1_{\mathrm{loc}}(\mathbb{R}^n)$, so $\hat{f}$ is a regular distribution on all of $\mathbb{R}^n$, and $f$ itself is a well-defined element of $\mathcal{S}'(\mathbb{R}^n)$ — not merely a class modulo polynomials.

[quotetheorem:224]

Scaling Invariance

The key feature of $\dot{H}^s(\mathbb{R}^n)$ — and the reason it is preferred over $H^s$ in the study of scaling-critical problems — is that its norm scales with a pure power of the dilation parameter.

[example: Dilation Invariance of the Homogeneous Norm]
For $\lambda > 0$ and $[f] \in \dot{H}^s(\mathbb{R}^n)$, define the rescaled function:
\begin{align*} f_\lambda: \mathbb{R}^n &\to \mathbb{R} \\ x &\mapsto \lambda^{n/2 - s} f(\lambda x). \end{align*}
The prefactor $\lambda^{n/2-s}$ is the unique power making the $\dot{H}^s$ norm invariant. To verify this, note that the $T$-representative of $\widehat{f_\lambda}$ satisfies $(\widehat{f_\lambda})_{T\text{-rep}}(\xi) = \lambda^{n/2-s}\lambda^{-n}\hat{f}_{T\text{-rep}}(\lambda^{-1}\xi)$ (from the standard Fourier scaling relation). Substituting $\eta = \lambda^{-1}\xi$, so that $d\mathcal{L}^n(\xi) = \lambda^n\,d\mathcal{L}^n(\eta)$ and $|\xi| = \lambda|\eta|$:
\begin{align*} \|f_\lambda\|_{\dot{H}^s}^2 &= \int_{\mathbb{R}^n} |\xi|^{2s} \left|\lambda^{n/2-s}\lambda^{-n}\hat{f}_{T\text{-rep}}(\lambda^{-1}\xi)\right|^2 d\mathcal{L}^n(\xi) \\ &= \lambda^{-n-2s}\int_{\mathbb{R}^n}|\lambda\eta|^{2s}|\hat{f}_{T\text{-rep}}(\eta)|^2\lambda^n\,d\mathcal{L}^n(\eta) \\ &= \lambda^{-n-2s+2s+n}\int_{\mathbb{R}^n}|\eta|^{2s}|\hat{f}_{T\text{-rep}}(\eta)|^2\,d\mathcal{L}^n(\eta) \\ &= \|f\|_{\dot{H}^s}^2. \end{align*}
So $\|f_\lambda\|_{\dot{H}^s} = \|f\|_{\dot{H}^s}$ for every $\lambda > 0$: the norm is perfectly scale-invariant. By contrast, the same computation for the inhomogeneous norm (using the $T$-representative $\hat{f}_{T\text{-rep}}$ from the Integral Formula) shows that $(1 + \lambda^2|\eta|^2)^s/|\eta|^{2s}$ depends on $\lambda$ in a way that cannot be factored out of the integral, confirming that $\|f_\lambda\|_{H^s}$ cannot equal $\|f\|_{H^s}$ for all $f$ and all $\lambda \neq 1$.
[/example]

The Sobolev Embedding

The critical exponent in the Sobolev embedding can be read off from the scaling invariance alone, before any analysis is done. The embedding $\dot{H}^s(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$, if it holds, must be an inequality $\|f\|_{L^p} \le C\|f\|_{\dot{H}^s}$ that is invariant under $f \mapsto f_\lambda$. Rescaling the left side:
\begin{align*} \|f_\lambda\|_{L^p}^p = \int_{\mathbb{R}^n}|\lambda^{n/2-s}f(\lambda x)|^p\,d\mathcal{L}^n(x) = \lambda^{p(n/2-s)-n}\|f\|_{L^p}^p, \end{align*}
so $\|f_\lambda\|_{L^p} = \lambda^{n/2-s-n/p}\|f\|_{L^p}$. For this to match $\|f_\lambda\|_{\dot{H}^s} = \|f\|_{\dot{H}^s}$, the exponent on the left must vanish:
\begin{align*} \frac{n}{2} - s - \frac{n}{p} = 0 \implies \frac{1}{p} = \frac{1}{2} - \frac{s}{n}. \end{align*}
This scaling argument determines the unique possible embedding exponent. The embedding itself holds whenever $0 \le s < n/2$, so that $p = \frac{2n}{n-2s} \in [2, \infty)$.

[quotetheorem:225]

Applications to PDEs

The real power of homogeneous Sobolev spaces is in the analysis of nonlinear PDEs with scaling symmetries, where identifying the critical Sobolev space is the first step toward understanding global versus blowup behaviour. In the examples below, we work with solutions whose Fourier transforms are functions (typically because the initial data lies in $L^2$ or $\mathcal{S}$), so the $T$-representative $\hat{u}_{T\text{-rep}}$ coincides with the classical Fourier transform and we write $\hat{u}(\xi)$ without ambiguity.

Heat Equation

[example: Regularity Propagation For the Heat Equation]
Consider the heat equation $\partial_t u = \Delta u$ on $\mathbb{R}^n$ with initial data $u_0 \in \dot{H}^s(\mathbb{R}^n)$ for some $s \in \mathbb{R}$. Taking the Fourier transform in the spatial variable, the equation becomes the ODE $\partial_t\hat{u}(t,\xi) = -|\xi|^2\hat{u}(t,\xi)$, which has explicit solution:
\begin{align*} \hat{u}(t,\xi) &= e^{-|\xi|^2 t}\,(\hat{u}_0)_{T\text{-rep}}(\xi) \quad \text{for every } t > 0 \text{ and } \mathcal{L}^n\text{-a.e. } \xi \in \mathbb{R}^n. \end{align*}
The homogeneous Sobolev norm of the solution at time $t$ in a smoother space $\dot{H}^{s'}$ for any $s' \ge s$ satisfies:
\begin{align*} \|u(t)\|_{\dot{H}^{s'}}^2 &= \int_{\mathbb{R}^n}|\xi|^{2s'}e^{-2|\xi|^2 t}|(\hat{u}_0)_{T\text{-rep}}(\xi)|^2\,d\mathcal{L}^n(\xi) \\ &= \int_{\mathbb{R}^n}|\xi|^{2s}\cdot|\xi|^{2(s'-s)}e^{-2|\xi|^2 t}\cdot|(\hat{u}_0)_{T\text{-rep}}(\xi)|^2\,d\mathcal{L}^n(\xi). \end{align*}
Writing $\rho = 2|\xi|^2 t$ and using the elementary bound $\rho^{s'-s}e^{-\rho} \le \left(\frac{s'-s}{e}\right)^{s'-s}$ valid for all $\rho \ge 0$, the factor $|\xi|^{2(s'-s)}e^{-2|\xi|^2 t} = (2t)^{-(s'-s)}(2|\xi|^2 t)^{s'-s}e^{-2|\xi|^2 t}$ is bounded above by $C_{s'-s}\,t^{-(s'-s)}$, where $C_{s'-s}$ depends only on $s'-s$. Therefore:
\begin{align*} \|u(t)\|_{\dot{H}^{s'}} &\le C_{s'-s}\, t^{-(s'-s)/2}\|u_0\|_{\dot{H}^s}. \end{align*}
This is the smoothing estimate: the heat flow instantaneously promotes initial data in $\dot{H}^s$ to the smoother class $\dot{H}^{s'}$ for every $s' > s$, at the cost of a power of $t$ that blows up as $t \to 0^+$. Combining with the Sobolev embedding from Theorem 225 applied at time $t$ with regularity index $s'$ satisfying $\frac{1}{q} = \frac{1}{2} - \frac{s'}{n}$:
\begin{align*} \|u(t)\|_{L^q(\mathbb{R}^n)} &\le C\,t^{-n(1/2 - 1/q)/2}\|u_0\|_{L^2(\mathbb{R}^n)}, \end{align*}
which is the classical $L^2 \to L^q$ dispersive decay of the heat semigroup with rate $t^{-n(1/2-1/q)/2}$.

The homogeneous space $\dot{H}^s$ is precisely the right framework to track these estimates because the smoothing exponent $(s'-s)/2$ and the dispersive rate $n(1/2 - 1/q)/2$ both have a transparent scaling origin: they count how many derivative-weights are gained divided by 2, reflecting the parabolic scaling $t \sim |\xi|^{-2}$.
[/example]

Nonlinear Schrödinger Equation

[example: Critical Space For the Nonlinear Schrodinger Equation]
The defocusing nonlinear Schrödinger equation on $\mathbb{R}^n$ with power-type nonlinearity is:
\begin{align*} i\partial_t u + \Delta u &= |u|^{p-1}u, \end{align*}
for an exponent $p > 1$ and a complex-valued solution $u: \mathbb{R}_{t} \times \mathbb{R}^n_x \to \mathbb{C}$. This equation has a two-parameter family of symmetries, one of which is the dilation symmetry: if $u(t,x)$ is a solution, then so is:
\begin{align*} u_\lambda: \mathbb{R} \times \mathbb{R}^n &\to \mathbb{C} \\ (t,x) &\mapsto \lambda^{2/(p-1)}u(\lambda^2 t,\lambda x) \end{align*}
for every $\lambda > 0$. To verify this, observe that $i\partial_t u_\lambda + \Delta u_\lambda = \lambda^{2/(p-1)+2}(i\partial_t u + \Delta u)(\lambda^2 t, \lambda x)$, while $|u_\lambda|^{p-1}u_\lambda = \lambda^{2p/(p-1)}|u|^{p-1}u(\lambda^2 t, \lambda x)$. Equating the powers $2/(p-1) + 2 = 2p/(p-1)$ confirms the symmetry. The initial data transforms as $u_{0,\lambda}(x) = \lambda^{2/(p-1)}u_0(\lambda x)$.

Applying the scaling computation from the Dilation Invariance example with $\alpha = 2/(p-1)$:
\begin{align*} \|u_{0,\lambda}\|_{\dot{H}^s}^2 &= \lambda^{4/(p-1) + 2s - n}\|u_0\|_{\dot{H}^s}^2. \end{align*}
The $\dot{H}^s$ norm of the initial data is preserved by the symmetry of the equation if and only if the exponent vanishes:
\begin{align*} \frac{4}{p-1} + 2s - n = 0 \implies s_c = \frac{n}{2} - \frac{2}{p-1}. \end{align*}
The space $\dot{H}^{s_c}(\mathbb{R}^n)$ is called the critical Sobolev space for this equation. It is the unique space of the form $\dot{H}^s$ in which the scaling symmetry acts isometrically on initial data.

The sign of $s_c$ partitions the nonlinearity into three regimes. When $p > 1 + 4/n$ (so $s_c > 0$), the equation is energy-supercritical and global well-posedness in $\dot{H}^{s_c}$ remains largely open. When $p = 1 + 4/n$ (so $s_c = 0$, the critical space is $\dot{H}^0 \cong L^2$), the equation is mass-critical. When $p = 1 + 4/(n-2)$ for $n \ge 3$ (so $s_c = 1$ and the critical space is $\dot{H}^1$, the energy space), the equation is energy-critical; global well-posedness and scattering in $\dot{H}^1$ for radial data in three dimensions was established by Bourgain (1999), and in the non-radial case by Colliander-Keel-Staffilani-Takaoka-Tao (2008).
[/example]

Navier-Stokes Equations

[example: Critical Space For the Navier Stokes Equations]
The incompressible Navier-Stokes equations on $\mathbb{R}^n$ for a velocity field $u: \mathbb{R}_t \times \mathbb{R}^n_x \to \mathbb{R}^n$ and pressure $p: \mathbb{R}_t \times \mathbb{R}^n_x \to \mathbb{R}$ are:
\begin{align*} \partial_t u + (u \cdot \nabla)u - \Delta u &= -\nabla p, \\ \nabla \cdot u &= 0. \end{align*}
This system admits the scaling symmetry: if $(u,p)$ is a solution, then so is $(u_\lambda, p_\lambda)$ defined by:
\begin{align*} u_\lambda(t,x) &= \lambda u(\lambda^2 t, \lambda x), \\ p_\lambda(t,x) &= \lambda^2 p(\lambda^2 t, \lambda x), \end{align*}
for every $\lambda > 0$. To verify: the inertial term rescales as $(u_\lambda\cdot\nabla)u_\lambda = \lambda^3(u\cdot\nabla)u(\lambda^2 t,\lambda x)$, and similarly $\partial_t u_\lambda = \lambda^3\partial_t u(\lambda^2 t,\lambda x)$, $\Delta u_\lambda = \lambda^3\Delta u(\lambda^2 t,\lambda x)$, and $\nabla p_\lambda = \lambda^3\nabla p(\lambda^2 t,\lambda x)$, so the equation is preserved.

The initial data transforms as $u_{0,\lambda}(x) = \lambda u_0(\lambda x)$, corresponding to $\alpha = 1$ in the scaling computation. Applying the formula with this value:
\begin{align*} \|u_{0,\lambda}\|_{\dot{H}^s}^2 &= \lambda^{2+2s-n}\|u_0\|_{\dot{H}^s}^2. \end{align*}
The norm is invariant under this symmetry if and only if $2 + 2s - n = 0$, giving the critical regularity:
\begin{align*} s_c = \frac{n}{2} - 1. \end{align*}
The space $\dot{H}^{n/2-1}(\mathbb{R}^n)$ is therefore the critical Sobolev space for the Navier-Stokes equations. In the physically relevant dimension $n = 3$ this is $\dot{H}^{1/2}(\mathbb{R}^3)$, while for $n = 2$ one obtains $s_c = 0$, so $\dot{H}^0 \cong L^2(\mathbb{R}^2)$ is critical — reflecting the fact that the 2D equations are globally well-posed. The Sobolev embedding gives $\dot{H}^{n/2-1}(\mathbb{R}^n) \hookrightarrow L^n(\mathbb{R}^n)$ (taking $s = n/2 - 1$ in the embedding exponent $1/p = 1/2 - s/n = 1/n$), so $L^n(\mathbb{R}^n)$ is another critical space for Navier-Stokes, a fact first exploited by Fujita and Kato (1964) in their foundational work on mild solutions.

A striking consequence of the critical scaling is the following: if $u_0 \in \dot{H}^{n/2-1}(\mathbb{R}^n)$ and $\|u_0\|_{\dot{H}^{n/2-1}}$ is sufficiently small, then the Navier-Stokes equations have a unique global mild solution. The smallness condition is scale-invariant and cannot be relaxed to a condition on the $H^s$ norm for any $s \neq n/2-1$, since the $H^s$ norm for $s \neq n/2-1$ is not preserved by the symmetry and would grow or decay under the natural rescaling of the problem.
[/example]

Problems

[problem]
Let $n \ge 3$ and let $u$ solve the linear wave equation $\partial_{tt}u - \Delta u = 0$ on $\mathbb{R}^{1+n}$. The scaling symmetry of this equation is $u_\lambda(t,x) = u(\lambda t, \lambda x)$, so the initial data transforms as $(u_0,u_1) \mapsto (u_0(\lambda\cdot), \lambda u_1(\lambda\cdot))$.

1. Compute $\|(u_0)_\lambda\|_{\dot{H}^s}$ in terms of $\|u_0\|_{\dot{H}^s}$ and $\lambda$, where $(u_0)_\lambda(x) = u_0(\lambda x)$.
2. Find the critical regularity $s_c$ for the wave equation by requiring $\|(u_0)_\lambda\|_{\dot{H}^{s_c}} = \|u_0\|_{\dot{H}^{s_c}}$ for all $\lambda > 0$.
3. For the nonlinear wave equation $\partial_{tt}u - \Delta u = |u|^{p-1}u$, the scaling is $u_\lambda(t,x) = \lambda^\alpha u(\lambda t, \lambda x)$. Determine $\alpha$ from the equation and find the critical Sobolev regularity $s_c(n,p)$.
   [/problem]

[solution]
Part 1. For $(u_0)_\lambda(x) = u_0(\lambda x)$, the $T$-representative of the Fourier transform satisfies $(\widehat{(u_0)_\lambda})_{T\text{-rep}}(\xi) = \lambda^{-n}(\hat{u}_0)_{T\text{-rep}}(\lambda^{-1}\xi)$. Substituting $\eta = \lambda^{-1}\xi$ with $d\mathcal{L}^n(\xi) = \lambda^n\,d\mathcal{L}^n(\eta)$:
\begin{align*} \|(u_0)_\lambda\|_{\dot{H}^s}^2 &= \int_{\mathbb{R}^n}|\xi|^{2s}\lambda^{-2n}|(\hat{u}_0)_{T\text{-rep}}(\lambda^{-1}\xi)|^2\,d\mathcal{L}^n(\xi) \\ &= \lambda^{-2n}\int_{\mathbb{R}^n}|\lambda\eta|^{2s}|(\hat{u}_0)_{T\text{-rep}}(\eta)|^2\lambda^n\,d\mathcal{L}^n(\eta) \\ &= \lambda^{2s-n}\|u_0\|_{\dot{H}^s}^2. \end{align*}
Therefore $\|(u_0)_\lambda\|_{\dot{H}^s} = \lambda^{s - n/2}\|u_0\|_{\dot{H}^s}$.

Part 2. For scale-invariance we need $s - n/2 = 0$, giving $s_c = n/2$. The critical space for the linear wave equation is $\dot{H}^{n/2}(\mathbb{R}^n)$.

Part 3. Substituting $u_\lambda(t,x) = \lambda^\alpha u(\lambda t, \lambda x)$ into $\partial_{tt}u - \Delta u = |u|^{p-1}u$: the left side acquires a factor $\lambda^{\alpha+2}$ and the right side acquires $\lambda^{\alpha p}$. For the equation to be preserved: $\alpha + 2 = \alpha p$, giving $\alpha = 2/(p-1)$. The initial data transforms as $(u_0)_\lambda(x) = \lambda^{2/(p-1)}u_0(\lambda x)$, so by Part 1 with this prefactor (the extra $\lambda^{2/(p-1)}$ factor contributes $\lambda^{4/(p-1)}$ to the squared norm):
\begin{align*} \|{(u_0)_\lambda}\|_{\dot{H}^s}^2 &= \lambda^{4/(p-1)+2s-n}\|u_0\|_{\dot{H}^s}^2. \end{align*}
Scale-invariance requires $4/(p-1) + 2s - n = 0$, so the critical regularity is:
\begin{align*} s_c(n,p) &= \frac{n}{2} - \frac{2}{p-1}. \end{align*}
[/solution]

References

1. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2010).
2. T. Cazenave, Semilinear Schrödinger Equations (2003).
3. P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem (2002).
4. F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations (2015).
5. E. M. Stein, Singular Integrals and Differentiability Properties of Functions (1970).