Using periodic boundary conditions to approximate the Navier-Stokes equations on $\mathbb{R}^3$ and the transfer of regularity

This paper considers solutions $u_\alpha$ of the three-dimensional Navier--Stokes equations on the periodic domains $Q_\alpha:=(-\alpha,\alpha)^3$ as the domain size $\alpha\to\infty$, and compares them to solutions of the same equations on the whole space. For compactly-supported initial data $u_\alpha^0\in H^1(Q_\alpha)$, an appropriate extension of $u_\alpha$ converges to a solution $u$ of the equations on ${\mathbb R}^3$, strongly in $L^r(0,T;H^1({\mathbb R}^3))$, $r\in[1,\infty)$. The same also holds when $u_\alpha^0$ is the velocity corresponding to a fixed, compactly-supported vorticity. A consequence is that if an initial compactly-supported velocity $u_0\in H^1({\mathbb R}^3)$ or an initial compactly-supported vorticity $\omega_0\in H^1({\mathbb R}^3)$ gives rise to a smooth solution on $[0,T^*]$ for the equations posed on ${\mathbb R}^3$, a smooth solution will also exist on $[0,T^*]$ for the same initial data for the periodic problem posed on ${Q_\alpha}$ for $\alpha$ sufficiently large; this illustrates a `transfer of regularity' from the whole space to the periodic case.


Introduction
The aim of this paper is to compare solutions of the Navier-Stokes equations ∂ t u − ∆u + (u · ∇)u + ∇p = 0, ∇ · u = 0, (1.1) posed on 'large' periodic domains Q α := (−α, α) 3 and on the whole space R 3 . One would expect, when the initial velocity is sufficiently localised, that the solutions on a 'large enough' domain should mimic those on R 3 , and this approach is the basis of many numerical experiments. Indeed, discussions with Robert Kerr about his numerical investigations (Kerr, 2018) of the trefoil configurations of vorticity from the experiments of Scheeler et al. (2014) were the original motivation for this paper, which gives a rigorous justification of this intuition.
Section 3 contains an analysis of the velocity fields that arise from such compactly-supported vorticities. The results there both provide a natural family of initial data to consider on the domains Q α , and also serve to illustrate of some of the arguments that follow in a relatively simple setting.
It is shown that given a fixed compactly-supported vorticity ω ∈ H 1 (R 3 ), the corresponding velocities u α on Q α have extensions to R 3 ,ũ α , that converge strongly in H 1 (R 3 ) to the velocity on R 3 reconstructed from ω using the Biot-Savart Law. Obtaining strong convergence in H 1 (R 3 ) requires uniform bounds on the 'tails' x∈Qα: |x|≥R |∇u α | 2 , a technique also employed later for solutions of the Navier-Stokes equations, and which goes back at least to Leray (1934).
After recalling some basic existence results for weak and strong solutions of the Navier-Stokes equations in Section 4, it is shown that a subsequence of weak solutions on Q α (solutions bounded in L 2 that satisfy the energy inequality) will converge to a weak solution on R 3 , given weak convergence of the initial data in L 2 (R 3 ). This result goes back at least to Heywood (1988), who used it as a way of proving the existence of weak solutions on the whole space.
The main result of the paper concerns the convergence of strong solutions (i.e. solutions that remain bounded in H 1 ) given convergence of the initial data in H 1 (R 3 ); due to uniqueness of the limiting solution this convergence now occurs without the need to extract a subsequence. By bounding the 'tails' of |u α | 2 at infinity it is shown thatũ α converges to u strongly in L p (0, T ; L 2 (R 3 )) for all p ∈ [1, ∞), and then, via interpolation of the H 1 norm between L 2 and H 2 , the boundedness of u α in L 2 (0, T ; H 2 (R 3 )) shows thatũ α converges strongly to u in L r (0, T ; H 1 (R 3 )), r ∈ [1, 4).
Finally, using this strong convergence, comes what is perhaps the most striking result of the paper: if u 0 ∈ H 1 (R 3 ) with compact support (or ω 0 ∈ H 1 (R 3 ) with compact support) gives rise to a strong solution on [0, T * ] and u α 0 ∈ H 1 (Q α ) converges to u 0 in H 1 (R 3 ), then for large enough α the equations on Q α with initial data u 0 α give rise to a unique strong solution on the same interval, andũ α → u as α → ∞ in L r (0, T ; H 1 (R 3 )), r ∈ [1, 4). This shows that the existence of a regular solution on the whole space implies the existence of a regular solution on a large enough periodic domain.
The relationship between the existence of smooth solutions for the equations in various settings (peiodic boundary conditions, Schwartz solutions on R 3 , homogeneous and inhomogeneous problems) has also been considered, from a different point of view, by Tao (2013).
There are other 'transfer of regularity' results for the Navier-Stokes equations in different contexts. Constantin (1988) showed that if u 0 ∈ H s+2 , s ≥ 3, gives rise to a solution in L ∞ (0, T * ; H s+2 ) of the Euler equations, then for the Navier-Stokes equations with dissipative term −ν∆u, one can take ν sufficiently small to ensure that the same initial condition produces an H s -bounded solution of the Navier-Stokes equations on [0, T * ]. A variant of this approach in Chernyshenko et al. (2007) shows that if u 0 gives rise to a regular solution of the Navier-Stokes equations on [0, T * ] then a sufficiently 'good' numerical scheme will have a similarly smooth solution that will also exist on [0, T * ]. Other results that 'transfer regularity' start with two-dimensional flows: Raugel & Sell (1993) considered the problem posed on thin three-dimensional domains, and Gallagher (1997) considered flows with initial data that are 'close to two dimensional'.
There is, of course, another way to view solving the equations on Q α , α ≥ α 0 , with fixed initial data u 0 of compact support. Here, rather than keeping u 0 fixed and increasing α, one could keep the domain fixed and rescale u 0 : taking α 0 = 1 for simplicity, the problem on Q α becomes a problem posed on Ω 1 by setting A solution (u(x, t), p(x, t)) on Q α becomes the rescaled solution (αu(αx, α 2 t), α 2 p(αx, α 2 t)) on Q 1 . However, if the solution on Q α exists for t ∈ [0, T ], then the rescaled solution on Q 1 exists only for t ∈ [0, T /α 2 ]. It follows that such a rescaling is not a useful tool for considering the behaviour of solutions as α → ∞ in the sense proposed here. Nevertheless, related scaling ideas are used here to check that various inequalities hold with constants independent of the domain parameter α.

Preliminaries
The expression L 2 (Q α ) denotes the space of functions that are 2α-periodic in every direction, with Qα |u| 2 < ∞, where Q α = (−α, α) 3 . Throughout the paper, a dot over a space denotes that the functions have zero average: so, for example,L 2 (Q α ) denotes that subset of L 2 (Q α ) consisting of those functions that also satisfy the condition Qα u = 0. (2.1) The notation f, g L 2 (Qα) = Qα f (x)g(x) dx is used for the inner product in L 2 (Q α ).
The space of 2α-periodic functions with weak derivatives up to order s in L 2 (Q α ), again satisfying (2.1), is denoted byḢ s (Q α ). Due to the zeroaverage condition, theḢ s (Q α ) norm defined by setting is equivalent to the full H s (Q α ) norm. Indeed, for all r ≥ s ≥ 0 the generalised Poincaré inequality holds, from which the equivalence follows.
Note also for later use that if ∆u ∈ L 2 (Q α ) then u ∈ H 2 (Q α ) with The notationĊ ∞ (Q α ) denotes the space of all C ∞ 2α-periodic functions satisfying the same zero average condition, . Throughout, the σ subscript indicates that the functions are divergence free.
. The second of these two is less obvious so the proof is given here. Heywood (1976): so given any u ∈ H 1 σ (R 3 ) and ε > 0, there At various points it is important that the constants in inequalities valid on Q α do not depend on α, i.e. on the size of the domain. To ensure this, inequalities are shown on Q 1 and then rescaled: given a function f α defined on Q α , the rescaled function f (x) = f α (αx) is defined on Q 1 . The L p norms of derivatives of order k then scale according to 3 Convergence of velocities corresponding to compactly-supported vorticity 3.1 Reconstruction of u from ω One of the issues for the convergence results considered here is to identify a class of initial data that is 'localised' in a reasonable way. One possible choice (although Theorem 6.2 is more general) is to take a compactly supported vorticity ω and to consider the corresponding velocity fields obtained by 'inverting' the curl operator on the corresponding domain. This amounts to solving the equations curl u = ω, ∇ · u = 0; (3.1) by taking the curl of both equations and using the vector identity the weak form of this system is: given ω ∈L 2 σ (Ω), 2) for Ω = Q α , and replacingḢ 1 σ with in H 1 σ (i.e. relaxing the zero average condition) on R 3 . Note the integration by parts in the right-hand side from curl ω, φ , which allows for ω ∈ L 2 and not only ω ∈ H 1 .
On the whole space, an expression for u can be obtained using the fundamental solution of the Laplacian and an integration by parts, namely the Biot-Savart Law [In the case of R 2 modified versions of the equivalent to the Biot-Savart Law are available which do not require decay of ω and u at infinity, see Serfati (1995) and Ambrose et al. (2015), for example. For bounded domains see Enciso, García-Ferrero, & Peralta-Salas (2018), for example.] On periodic domains, while u α = curl −1 α ω can be written explicitly in terms of the Fourier expansionit will be more useful here to observe that u α is still the solution of the equation −∆u α = curl ω.
On the periodic domain Q 1 , if Q 1 g = 0, then the equation −∆u = g, Q 1 u = 0, has a solution given in the form where φ and S are smooth and φ(z) = 1 for |z| < 1/10 and φ(z) = 0 for |z| > 1/4, see Theorem C.5 in Robinson et al. (2016), for example. Then, when ω has compact support in Q 1 ,

Bounds on u from bounds on ω
The following result is extremely useful; it is valid on Q α for every α and on R 3 . While a similar inequality could be obtained using the Calderón-Zygmund Theorem and (3.3), equality follows here from a much simpler argument (see equation (1.4.20) in Doering & Gibbon, 1995).
Proof. Assume first that u is smooth and ω ∈ L 2 . Then, since integrating by parts twice in the final term and using the fact that u is divergence free. Now if u ∈ H 1 , ω ∈ L 2 and mollifying u produces a smooth u ε with ∇ × u ε ∈ L 2 ; the same argument shows that since ω ε → ω, ∂ i (u ε ) j → ∂ i u j for every i, j, yielding the same equality for these more general u.
The Biot-Savart Law and Young's inequality provide L q estimates on u given L p bounds on ω.
The same estimate also holds when ω ∈L p σ (Q α ): where C p is independent of α.
Proof. On the whole space u is given by (3.3). So u is given by the convolution of ω with a kernel of order |x| −2 ; in three dimensions this belongs to the weak Lebesgue space L 3/2,∞ , and (3.5) follows using the weak-Lebesgue space version of Young's inequality, For the same bound on Q 1 , consider the expression in (3.4), The kernel in the first term is once again in L 3/2,∞ (Q 1 ) and the kernel in the second term is in L 3/2 (Q 1 ); these two terms are thus bounded in L q (Q 1 ) using Young's inequality. For the final term u 3 (x), Minkowski's inequality yields Noting that S is smooth and that only x, y ∈ Q 1 are relevant, the bound curl y S α (·, y) L q (Q 1 ) ≤ M holds, and hence using Hölder's inequality and the fact that Q 1 is bounded.
These three upper bounds combine to yield (3.6) on Q 1 . The fact that the same inequality holds with a constant independent of α follows since both norms in (3.6) behave the same way under the rescaling x → αx, see (2.3).

Extension of functions from
Given ω ∈L 2 σ (R 3 ) with support contained in Q α 0 , Lemma 3.2 gives a family {u α } α≥α 0 of velocity fields defined on Q α (α ≥ α 0 ). In order to be able to take a meaningful limit on the whole of R 3 , each u α will be extended to the whole of R 3 in such a way that the support ofũ α is contained in a domain only slightly larger than Q α .
Later a similar extension will be used for time-dependent functions u α (x, t); in this caseũ , with the cut-off function ψ α being independent of t. This means, in particular, that so that bounds on ∂ tũα can be deduced from bounds on ∂ t u α as done forũ α above.
3.4 Convergence of curl −1 α ω to curl −1 ω as α → ∞ Theorem 3.4 will show that the fieldsũ α from Lemma 3.2 converge to u strongly in H 1 (R 3 ) whenever ω ∈ H 1 (R 3 ). The following lemma (see Leray, 1934, or Lemma 6.34 in Ożański & Pooley, 2018 can be used to improve the L 2 -convergence ofũ α to u on compact subsets of R 3 to convergence on the whole of R 3 by bounding the 'tails' of u α uniformly. The argument that follows obtains bounds on the 'tail' of a sequence u α ∈ L 2 (Q α ); in order to apply Lemma 3.3 the corresponding bounds onũ α will be needed. Therefore note here that if u α ∈ L 2 (Q α ) and R < α − 1 then i.e. the integral on the left-hand side of (3.8) can at most include the 'tails' from the periodic cells immediately adjacent to Q α , see Figure 3.4 for an illustration of this in the two-dimensional case, where the corresponding constant is 9. [In 2D this can be improved to 4; following a similar idea the constant in the 3D case can be improved to 10.] Figure 1: The support ofũ α is contained in the large central square in the left-hand figure, and |ũ α | ≤ |u α | everywhere. Periodised circles of radius R are shown in white. Clearly |x|≥R |ũ α | 2 ≤ 9 x∈Qα: |x|≥R |u α | 2 . However, with portions of this darker square moved using periodicity (on the right) this can be improved to |x|≥R |ũ α | 2 ≤ 4 x∈Qα: |x|≥R |u α | 2 .
Proof. If ω ∈ L 2 then the uniform estimates for u α in (3.9) follow from Lemmas 3.1 and 3.2. Now extend each u α to a functionũ α defined on all of R 3 as outlined above, and in this way obtain a set of functions withũ α uniformly bounded (with respect to α) in H 1 (R 3 ). Since H 1 (R 3 ) is reflexive, it follows from reflexive weak sequential compactness that there exists an element v ∈ H 1 (R 3 ) such thatũ α j ⇀ v weakly in H 1 (R 3 ), which in turn implies the strong convergence in L 2 (K) for every compact subset K of R 3 .
It remains to show that v = u := curl −1 ω and that the convergence takes place as α → ∞ and not just for a subsequence.
for every ϕ ∈Ċ ∞ c,σ (R 3 ); the equality then holds for every ϕ ∈ H 1 σ (R 3 ) by density (see Lemma 2.1). Since u ∈ H 1 σ (R 3 ) it follows that u is the unique H 1 solution of −∆u = curl ω, which is precisely curl −1 ω. This also shows that the limit of any convergent subsequence must be the same, and it follows that u α → u as claimed in the statement of the theorem.
If in addition ω ∈ H 1 (R 3 ) then standard elliptic regularity results (see Evans, 2010, for example) gives uniform estimates onũ α in H 2 (R 3 ), since then ∆u α L 2 (Qα) = curl ω L 2 (Qα) and this yields a bound on the other second derivatives, see (2.2). The weak convergence in H 2 (R 3 ) now follows since H 2 is reflexive, which implies the strong convergence in H 1 (K) for every compact subset K of R 3 .
To improve this to strong convergence in H 1 (R 3 ), take φ = u α ̺ α as the test function in to Q α , where we take 0 < r < R < α; note that and taking r sufficiently large that supp(ω) ⊂ B(0, r) yields Lemma 3.3 now guarantees that ∇ũ α → ∇u in L 2 (R 3 ).

Weak and strong solutions of the Navier-Stokes equations
For Ω = Q α or R 3 , denote by D σ (Ω) the space of all test functions on Ω × [0, ∞) given by Definition 4.1. A function u is a weak solution of the Navier-Stokes equations corresponding to the initial condition for all test functions φ ∈ D σ (Ω).
The following theorem combines the basic existence result for weak solutions (Leray, 1934;Hopf, 1951) with the property that at least one solution exists that satisfies the strong energy inequality (Leray, 1934;Ladyzhenskaya, 1969): see Theorems 4.4, 4.6, 4.10, and 14.4 in Robinson et al. (2016).
Theorem 4.2. For every initial condition u 0 ∈L 2 σ (Ω) there exists at least one global-in-time weak solution u of the Navier-Stokes equations on Ω that satisfies the strong energy inequality for all t > s Note that it follows from this definition that any weak solution u has a weak time derivative ∂ t u with with c independent of α; see Lemma 3.7 in Robinson et al. (2016).
Key to later results in this paper is the notion of a strong solution.
The following theorem on the existence of strong solutions is again valid on Q α and R 3 ; the constant c is the same for all these domains. The result as stated combines Theorems 6.4, 6.8, 6.15, and 7.5 in Robinson et al. (2016).

Convergence of weak solutions
Convergence of weak solutions as α → ∞ is relatively straightforward; indeed, a similar method has been used by Heywood (1988; see also Theorem 4.10 in Robinson et al., 2016) to prove the existence of weak solutions on the whole space, although with that aim it is probably more natural to consider the equations with Dirichlet boundary conditions on the domains B(0, α), which can easily be extended by zero to all of R 3 .
Proposition 5.1. Suppose that u 0 α ∈L 2 σ (Q α ) withũ 0 α ⇀ u 0 in L 2 (R 3 ). Let u α be weak solutions of the equations on Q α with initial conditions u 0 α that satisfy the energy inequality for almost every t > 0. Then there exists a weak solution u of the equations on R 3 , and a subsequence u α j such that, for every T > 0,ũ α j converges to u weakly in L 2 (0, T ; H 1 ) and strongly in L 2 (0, T ; L 2 (K)) for every compact subset K of R 3 .
It remains only to show that u is a solution of the equations on the whole space.
To do this, take any test function φ ∈ D σ (R 3 ) and let M and T be large enough that the support of φ is contained in Q M ×[0, T ). Then for all α ≥ M it follows from Definition 4.1, sinceũ α = u α on Q α , that Passing to the limit as j → ∞ -using the weak convergence of gradients, the strong convergence in L 2 (0, T ; L 2 (Ω M )), and the fact thatũ 0 α j ⇀ u 0 -shows that u is a weak solution of the equations on R 3 with initial condition u 0 , as required.
Note that the above proof does not show that the solution u on R 3 satisfies the energy inequality; this is why the limiting procedure here is not the ideal way to generate solutions of the equations on R 3 .

Convergence of strong solutions
The main result of this paper, Theorem 6.2, will show that given a suitably convergent family of initial data u 0 α ∈ H 1 (Q α ), the 'solutions'ũ α converge strongly to u in L 2 (0, T ; H 1 (R 3 )).

Uniform inequalities
Key to obtaining uniform estimates for strong solutions on expanding domains are the following inequalities.
Proof. The validity of the estimate (6.1) for a fixed value of α is standard, and follows by splitting the Fourier series expansion of u into 'low modes' and 'high modes' (see Exercise 1.10 in Robinson et al. (2016), for example): The rescalings in (2.3) now show that this inequality is valid with the same constant on Q α . Inequality (6.2) in the case α = 1 is a consequence of the embedding H 1 (Q 1 ) ⊂ L 6 (Q 1 ) valid for three-dimensional domains, and the Poincaré inequality u L 2 (Q 1 ) ≤ C P ∇u L 2 (Q 1 ) which holds when Q 1 u = 0. A similar rescaling argument shows that the same constant works for every α.

Convergence in
, the following theorem shows that the corresponding strong solutions converge in L 2 (0, T ; H 1 (R 3 )). One particular example of such a family is provided by Theorem 3.4: take a fixed compactly-supported vorticity, and set u 0 α = curl −1 α ω and u 0 = curl −1 ω. Alternatively, simply take a compactly-supported initial condition u 0 ∈ H 1 σ (R 3 ) and let u 0 α = u 0 | Qα once α is sufficiently large.
There is a uniform time for which the existence of a smooth solution u α (on Q α ) and u (on R 3 ) can be guaranteed, starting with this initial condition. The following theorem shows that the extended solutionsũ α must converge to u. That there is weak convergence [as in Proposition 5.1] is fairly standard and follows directly from uniform bounds on u α ; that the convergence is strong in L 2 (0, T ; H 1 (R 3 )) is more surprising, and requires a more careful analysis. This strong convergence is crucial for the 'transference of regularity' result that follows in Section 7.
Set T = T (M) from Theorem 4.4. Denote by u α the strong solution of the Navier-Stokes equations on Q α with initial data u 0 α , and by u the solution on R 3 with initial data u 0 ; all of these solutions exist on [0, T ]. Then for all 1 ≤ s < 2 Proof. Since the solution u α is smooth on [0, T ] it is admissible to take the inner product with u α in L 2 (Q α ) to obtain This gives bounds on u α in L ∞ (0, T ; L 2 (Q α )) and L 2 (0, T ; H 1 (Q α )) that are uniform with respect to α.
Equation (6.6) shows that the solutions u α satisfy the energy inequality (5.1), so Proposition 5.1 already guarantees that a subsequence (at least) converges to a weak solution on R 3 with initial data u 0 . However, although u 0 gives rise to a strong solution, weak-strong uniqueness (see Theorem 6.10 in Robinson et al. (2016), for example) cannot be used here, since the limiting solution u from Proposition 5.1 does not necessarily satisfy the energy inequality (which is required in the proof of weak-strong uniqueness).
Better convergence ofũ α to u can be obtained via bounds on u α in H 1 and bounds on u α in H 2 . Take the inner product of the equation with −∆u α in L 2 (Q α ) to obtain where the constant C A does not depend on α (see Lemma 6.1). It follows that and therefore . (6.8) and, integrating (6.7) from 0 to T and using the bound in (6.8), that Therefore u α is bounded uniformly in L ∞ (0, T ; H 1 (Q α )) and in L 2 (0, T ; H 2 (Q α )).
To obtain bounds on the time derivative, since the equation holds as an equality in L 2 (0, T ; L 2 (Q α )) it follows that The Helmholtz decomposition provides a bound on ∇p α in L 2 (Q α ): write These two spaces are orthogonal: for any v ∈ H(Q α ) and ∇ψ ∈ G(Q α ) v, ∇ψ L 2 (Qα) = 0.
We know from before that u is at least a weak solution on [0, T ): these bounds now show that u has the required regularity to be a strong solution.

The transfer of regularity result
The following theorem shows that if u 0 gives rise to a smooth solution on [0, T * ] on the whole space, the corresponding periodic problems will have smooth solutions on the same time interval once the size of the periodic domain is sufficiently large. Note that T * does not need to be a 'guaranteed local existence time' from the proof of the existence of strong solutions, but could be significantly longer.
Continue in this way, noting that at each step the interval of existence of the solutions on Q α (for α ≥ α n ) increases by at least τ /2. After N steps the entire interval [0, T * ] has been covered, showing that the solution on Q α starting at u 0 α is strong on [0, T * ] for all α ≥ α N .
Note that this result does not say that if the equations are regular on R 3 -i.e. if any smooth (compactly-supported) initial condition gives rise to a smooth solution for all t > 0 -then they are regular on Q α for α large enough (which would then imply regularity on Q α for any α). Rather, for a fixed (compactly-supported) initial condition, regularity on R 3 on a given time interval carries over to Q α for α sufficiently large.
A full 'transfer of regularity' from one problem to another would require a convergence result in which the distance between solutions on R 3 and Q α could be bounded in terms of the H 1 norm of the initial data, which appears to require much more sophisticated methods that the compactness-based arguments employed here. [For results in this direction for the Ginzburg-Landau equation see Mielke (1997) and for the two-dimensional Navier-Stokes equations see Zelik (2013).]

Conclusion
Given fixed sufficiently regular initial data with compact support, solutions of the Navier-Stokes equations on expanding periodic domains converge to the corresponding solution on the whole space; and this can to some extent be 'reversed', in that a compactly-supported initial condition that leads to a strong solution on a time interval [0, T * ] (which could be significantly longer than what is guaranteed by standard existence theorems) will give rise to a strong solution on the same time interval on a sufficiently large periodic domain.
It is natural to conjecture that a similar result holds given any choice of smooth, simply-connected, bounded subset Ω of R 3 , replacing (−α, α) 3 by αΩ and imposing no-slip (Dirichlet) boundary conditions on the boundary of αΩ. However, the estimates on the pressure required in the proof given here become much more delicate in the case of a bounded domain (see Sohr & von Wahl, 1986, for example).
While the results here demonstrate convergence, they give no error estimates; this appears to be a significantly harder problem, but a particularly interesting one if one is to view solving the equations on a periodic domain as a 'numerical approximation' to the solution of the equations on the whole space.