Fréchet–Kolmogorov theorem

For a subset ${\displaystyle B}$ of ${\displaystyle L^{p}(\Omega )}$, where ${\displaystyle \Omega }$ is a bounded subset of ${\displaystyle \mathbb {R} ^{n}}$, the condition of equitightness is not needed. Hence, a necessary and sufficient condition for ${\displaystyle B}$ to be relatively compact is that the property of equicontinuity holds. However, this property must be interpreted with care as the below example shows.

Let ${\displaystyle (u_{\epsilon })_{\epsilon }}$ be a sequence of solutions of the viscous Burgers equation posed in ${\displaystyle \mathbb {R} \times (0,T)}$:

${\displaystyle {\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial u^{2}}{\partial x}}=\epsilon \Delta u,\quad u(x,0)=u_{0}(x),}$

with ${\displaystyle u_{0}}$ smooth enough. If the solutions ${\displaystyle (u_{\epsilon })_{\epsilon }}$ enjoy the ${\displaystyle L^{1}}$-contraction and ${\displaystyle L^{\infty }}$-bound properties,[2] we will show existence of solutions of the inviscid Burgers equation

${\displaystyle {\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial u^{2}}{\partial x}}=0,\quad u(x,0)=u_{0}(x).}$

The first property can be stated as follows: If ${\displaystyle u,v}$ are solutions of the Burgers equation with ${\displaystyle u_{0},v_{0}}$ as initial data, then

${\displaystyle \int _{\mathbb {R} }|u(x,t)-v(x,t)|dx\leq \int _{\mathbb {R} }|u_{0}(x)-v_{0}(x)|dx.}$

The second property simply means that ${\displaystyle \Vert u(\cdot ,t)\Vert _{L^{\infty }(\mathbb {R} )}\leq \Vert u_{0}\Vert _{L^{\infty }(\mathbb {R} )}}$.

Now, let ${\displaystyle K\subset \mathbb {R} \times (0,T)}$ be any compact set, and define

${\displaystyle w_{\epsilon }(x,t):=u_{\epsilon }(x,t)\mathbf {1} _{K}(x,t),}$

where ${\displaystyle \mathbf {1} _{K}}$ is ${\displaystyle 1}$ on the set ${\displaystyle K}$ and 0 otherwise. Automatically, ${\displaystyle B:=\{(w_{\epsilon })_{\epsilon }\}\subset L^{1}(\mathbb {R} ^{2})}$ since

${\displaystyle \int _{\mathbb {R} ^{2}}|w_{\epsilon }(x,t)|dxdt=\int _{\mathbb {R} ^{2}}|u_{\epsilon }(x,t)\mathbf {1} _{K}(x,t)|dxdt\leq \Vert u_{0}\Vert _{L^{\infty }(\mathbb {R} )}|K|<\infty .}$

Equicontinuity is a consequence of the ${\displaystyle L^{1}}$-contraction since ${\displaystyle u_{\epsilon }(x-h,t)}$ is a solution of the Burgers equation with ${\displaystyle u_{0}(x-h)}$ as initial data and since the ${\displaystyle L^{\infty }}$-bound holds: We have that

${\displaystyle \Vert w_{\epsilon }(\cdot -h,\cdot -h)-w_{\epsilon }\Vert _{L^{1}(\mathbb {R} ^{2})}\leq \Vert w_{\epsilon }(\cdot -h,\cdot -h)-w_{\epsilon }(\cdot ,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}+\Vert w_{\epsilon }(\cdot ,\cdot -h)-w_{\epsilon }\Vert _{L^{1}(\mathbb {R} ^{2})}.}$

We continue by considering

${\displaystyle {\begin{aligned}&\Vert w_{\epsilon }(\cdot -h,\cdot -h)-w_{\epsilon }(\cdot ,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}\\&\leq \Vert (u_{\epsilon }(\cdot -h,\cdot -h)-u_{\epsilon }(\cdot ,\cdot -h))\mathbf {1} _{K}(\cdot -h,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}+\Vert u_{\epsilon }(\cdot ,\cdot -h)(\mathbf {1} _{K}(\cdot -h,\cdot -h)-\mathbf {1} _{K}(\cdot ,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}.\end{aligned}}}$

The first term on the right-hand side satisfies

${\displaystyle \Vert (u_{\epsilon }(\cdot -h,\cdot -h)-u_{\epsilon }(\cdot ,\cdot -h))\mathbf {1} _{K}(\cdot -h,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}\leq T\Vert u_{0}(\cdot -h)-u_{0}\Vert _{L^{1}(\mathbb {R} )}}$

by a change of variable and the ${\displaystyle L^{1}}$-contraction. The second term satisfies

${\displaystyle \Vert u_{\epsilon }(\cdot ,\cdot -h)(\mathbf {1} _{K}(\cdot -h,\cdot -h)-\mathbf {1} _{K}(\cdot ,\cdot -h))\Vert _{L^{1}(\mathbb {R} ^{2})}\leq \Vert u_{0}\Vert _{L^{\infty }(\mathbb {R} )}\Vert \mathbf {1} _{K}(\cdot -h,\cdot )-\mathbf {1} _{K}\Vert _{L^{1}(\mathbb {R} ^{2})}}$

by a change of variable and the ${\displaystyle L^{\infty }}$-bound. Moreover,

${\displaystyle \Vert w_{\epsilon }(\cdot ,\cdot -h)-w_{\epsilon }\Vert _{L^{1}(\mathbb {R} ^{2})}\leq \Vert (u_{\epsilon }(\cdot ,\cdot -h)-u_{\epsilon })\mathbf {1} _{K}(\cdot ,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}+\Vert u_{\epsilon }(\mathbf {1} _{K}(\cdot ,\cdot -h)-\mathbf {1} _{K})\Vert _{L^{1}(\mathbb {R} ^{2})}.}$

Both terms can be estimated as before when noticing that the time equicontinuity follows again by the ${\displaystyle L^{1}}$-contraction.[3] The continuity of the translation mapping in ${\displaystyle L^{1}}$ then gives equicontinuity uniformly on ${\displaystyle B}$.

Equitightness holds by definition of ${\displaystyle (w_{\epsilon })_{\epsilon }}$ by taking ${\displaystyle r}$ big enough.

Hence, ${\displaystyle B}$ is relatively compact in ${\displaystyle L^{1}(\mathbb {R} ^{2})}$, and then there is a convergent subsequence of ${\displaystyle (u_{\epsilon })_{\epsilon }}$ in ${\displaystyle L^{1}(K)}$. By a covering argument, the last convergence is in ${\displaystyle L_{loc}^{1}(\mathbb {R} \times (0,T))}$.

To conclude existence, it remains to check that the limit function, as ${\displaystyle \epsilon \to 0^{+}}$, of a subsequence of ${\displaystyle (u_{\epsilon })_{\epsilon }}$ satisfies

${\displaystyle {\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial u^{2}}{\partial x}}=0,\quad u(x,0)=u_{0}(x).}$