Fréchet–Kolmogorov theorem

Statement

Let B be a subset of L^p(\mathbb{R}^n) with p\in1,\infty), and let \tau_h f denote the translation of f by h, that is, \tau_h f(x)=f(x-h) .

The subset B is [relatively compact if and only if the following properties hold:

1. (Equicontinuous) \lim_{|h|\to 0}\Vert\tau_h f-f\Vert_{L^p(\mathbb{R}^n)} = 0 uniformly on B.
2. (Equitight) \lim_{r\to\infty}\int_{|x|r}\left|f\right|^p=0 uniformly on B.

The first property can be stated as \forall \varepsilon 0 , , \exists \delta 0 such that \Vert\tau_h f-f\Vert_{L^p(\mathbb{R}^n)} with |h|

Usually, the Fréchet–Kolmogorov theorem is formulated with the extra assumption that B is bounded (i.e., \Vert f\Vert_{L^p(\mathbb{R}^n)} uniformly on B). However, it has been shown that equitightness and equicontinuity imply this property.

Special case

For a subset B of L^p(\Omega), where \Omega is a bounded subset of \mathbb{R}^n, the condition of equitightness is not needed. Hence, a necessary and sufficient condition for 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.

Examples

Existence of solutions of a PDE

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

:\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 u_0 smooth enough. If the solutions (u_\epsilon)_\epsilon enjoy the L^1-contraction and L^\infty-bound properties, we will show existence of solutions of the inviscid Burgers equation

:\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 u,v are solutions of the Burgers equation with u_0,v_0 as initial data, then

:\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 \Vert u(\cdot,t)\Vert_{L^\infty(\mathbb{R})}\leq \Vert u_0\Vert_{L^\infty(\mathbb{R})}.

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

:w_\epsilon(x,t):=u_\epsilon(x,t)\mathbf{1}_K(x,t),

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

:\int_{\mathbb{R}^2}|w_\epsilon(x,t)|dx dt= \int_{\mathbb{R}^2}|u_\epsilon(x,t)\mathbf{1}K(x,t)|dx dt\leq \Vert u_0\Vert{L^\infty(\mathbb{R})}|K|

Equicontinuity is a consequence of the L^1-contraction since u_\epsilon(x-h,t) is a solution of the Burgers equation with u_0(x-h) as initial data and since the L^\infty-bound holds: We have that

:\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

:\begin{align} &\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{align}

The first term on the right-hand side satisfies

:\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 L^1-contraction. The second term satisfies

:\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 L^\infty-bound. Moreover,

:\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 L^1-contraction. The continuity of the translation mapping in L^1 then gives equicontinuity uniformly on B.

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

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

To conclude existence, it remains to check that the limit function, as \epsilon\to0^+, of a subsequence of (u_\epsilon)_\epsilon satisfies

:\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial u^2}{\partial x} = 0, \quad u(x,0) = u_0(x).

References

Literature

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References

1. (1957). "Criteria of compactness in function spaces". Upsekhi Math. Nauk..
2. (1996). "Weak and Measure-Valued Solutions to Evolutionary PDEs". Chapman and Hall/CRC.
3. Kruzhkov, S. N.. (1970). "First order quasi-linear equations in several independent variables". Math. USSR Sbornik.