Sharp bounds on enstrophy growth for viscous scalar conservation laws

We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the L ∞ and total variation bounds and viscosity. This answers a conjecture by Ayala and Protas (2011 Physica D 240 1553–63), based on numerical evidence, for the viscous Burgers equation.


Introduction
We consider the initial-value problem for the one-dimensional viscous Burgers equation Original Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
where ν > 0, T := R/Z is the unit circle equipped with periodic boundary conditions, and u 0 has zero average.Solutions to (1.1) exhibit 'steepening of gradients', caused by the advection term, which is subsequently arrested by viscous dissipation when the diffusion begins to dominate.
We are interested in studying the maximum amplification achieved by the 'enstrophy' E(t) := ∥∂ x u ν (t, •)∥ 2 L 2 .This problem has two variants.Lu and Doering [3], who were originally motivated by connections between enstrophy growth in the 3D Navier-Stokes equations and potential finite-time singularity, proposed to maximize the instantaneous enstrophy growth rate, namely, subject to the constraint E = E 0 .The maximizers are explicit and saturate the analytical upper bound thereby resolving the problem.However, Ayala and Protas [1] observed numerically that the growth of Lu and Doering's maximizers is not sustained.They proposed to maximize sup t>0 E(t) subject to the constraint E(0) = E 0 .As a proxy, the authors numerically maximize the finite-time enstrophy E(T) and observe the scaling for the maximal enstrophy under O(1) viscosity at various times T. Subsequently, Pelinovsky exhibited solutions with enstrophy satisfying the scaling (1.4) with T ∼ E −1/2 0 through the Hopf-Cole transformation ([6, theorem 1.1] and [5]). 3otably, the analytical upper bound deduced in [1, appendix A], does not match the numerical evidence (1.4).In this paper, we clarify this discrepancy by proving the sharp upper bound.
It will be convenient to non-dimensionalize time so that either ν = 1 and E 0 → +∞, as in [1], or E 0 = 1 and ν → 0 + , which is the convention we follow.This is accomplished via the rescaling ).Then we anticipate that the most advantageous configuration for enstrophy growth is to develop the steepest viscous shock possible from initial data with O(1) enstrophy.This process is primarily restricted by the total variation ∥∂ x u ν (t, •)∥ L 1 , which is monotonically decreasing and, on the torus, controlled by the initial enstrophy.We leverage this observation, not exploited in [1], to prove the sharp upper bound.

Theorem 1.1 (upper bound on the enstrophy growth for the viscous Burgers equation).
There exists an absolute constant C 0 > 0 such that the following holds.Let u 0 ∈ H 1 (T) with ∥∂ x u 0 ∥ L 2 ⩽ 1 and zero mean.Let ν > 0 and u ν be the solution of the viscous Burgers equation (1.1) on the unit torus T with initial data u 0 .Then We further provide an alternative proof (see [6]) of the corresponding lower bound.

Theorem 1.2 (lower bound on the enstrophy growth for the viscous Burgers equation).
There exists u 0 : T → R satisfying the conditions of theorem 1.1 and a constant c 0 > 0 such that Finally, we apply the same strategy as in theorem 1.1 to prove sharp upper bounds on the enstrophy growth for solutions to the Cauchy problem for a multi-dimensional viscous scalar conservation law with locally Lipschitz continuous flux f : R → R n on the domain M := R n or M := T n L := (R/LZ) n with L ⩾ 1.The key point is to leverage the monotonicity of the L ∞ -norm and the total variation.

Theorem 1.3 (multi-dimensional conservation laws with Lipschitz continuous flux).
There exists an absolute constant C 0 > 0 depending only on the Lipschitz norm of f | [−1,1] and the dimension n ⩾ 1 such that the following holds.Let (1.12) Let ν > 0 and u ν be the solution of the viscous scalar conservation law (1.11) on M with initial data u 0 .Then The constant C 0 is independent of the domain.We view theorem 1.1 as an immediate corollary of theorem 1.3.

Proofs
The Cauchy problem (1.11) is globally well-posed in the subcritical space L ∞ in any dimension.The finiteness of the enstrophy, Ẇ1,1 -seminorm, and various other quantities can be proven a posteriori by considering the equation (2.6) satisfied by ∇u ν with f ′ (u) viewed as a known bounded function.We refer to [4, theorem 2.9 and lemma 2.16] for well-posedness in Our strategy is based on L ∞ and TV-bounds for the viscous conservation law and two subsequent estimates.First, we propagate the initial enstrophy bound on the time interval (0, ν]; after time O(ν), the estimate degenerates-see (2.9).Then we view u ν as solving the heat equation with data and right-hand side controlled only by the monotone quantities, and we rely on smoothing, which is effective after time O(ν), to bound the enstrophy-see (2.12) and (2.13).The smoothing estimate depends only on the monotone quantities and controls the solution on (ν, +∞).
We collect the necessary linear estimates in the following lemma.

Lemma 2.1 (heat estimates)
Then, for all t > 0, we have All implied constants are allowed to depend on the dimension n.
Step 1. Conserved quantities.By the maximum principle for the PDE ∂ t u ν + f ′ (u) • ∇u ν = ν∆u ν and using the transport-diffusion equation for ∇u ν , we deduce respectively; see [4, theorem 2.29] for further details.
Step 2. Propagation estimates for the enstrophy in the time interval t ∈ (0, ν].We multiply (2.6) by ∇u ν and integrate by parts to obtain a differential inequality for the enstrophy: where we used Young's inequality ab ⩽ a 2 / 2ν + νb 2 / 2 in the second line and the L ∞ -bound in the third line.By Gronwall's inequality, (2.8) yields (2.9) In particular, (2.10) Step 3. Duhamel's formula and smoothing effect: t ∈ (ν, ∞).By Duhamel's formula, we write We bound the right-hand side in terms of the conserved quantities.First, we have ≲ (ντ ) (2.12) Then, moving the divergence onto the heat kernel and applying (2.2), we have (2.13) We can optimize the above inequalities by choosing τ = ν.Then Step 4. Conclusion of the proof.Combining steps 2 and 3, we conclude the proof: we use the propagation estimate (2.10) to handle times t ∈ (0, ν]; and the smoothing estimate (2.14) (with t 0 = t − ν) for t ∈ (ν, +∞).

Remark 2.2 (viscous Burgers equation in 1D
).In the setting of the Burgers equation (1.1), we can avoid using the heat estimates from lemma 2.1.Indeed, since Ẇ1,1 and L ∞ have roughly the same 'strength' in dimension one, we may substitute (2.15) for (2.12) and ˆt0+τ for (2.13).On the other hand, the interpolation argument from lemma 2.1 is necessary when n ⩾ 2 to maximally utilize the L ∞ -bound.
Remark 2.3 (less restrictive assumption).From (2.9), we observe that theorem 1.3 remains valid under the assumption Finally, we present the proof of theorem 1.2.
Proof of theorem 1.2.First, we prescribe initial data v 0 for the inviscid problem whose entropy solution v shocks only at the origin.Let v 0 ∈ C ∞ (T) be odd on the fundamental domain , and decreasing on [− 1 6 , 0].Concavity ensures that the solution is described by the method of characteristics on [− 1 2 , 0): indeed, given a particle label α, the 'local turnover time' t * (α) = −1/v ′ 0 (α), at which the derivative of the flow map η(α) = α + v 0 (α)t vanishes, will be at least the time t s (α) = v 0 (α)/α at which the characteristic enters the origin.For t > t * (0), v has the desired shock at the origin.
Let u 0 = Uv 0 where U = 1/∥∂ x v 0 ∥ L 2 > 0 is a normalizing factor to ensure that the conditions of the above theorem are satisfied.This amounts to a time rescaling: u(t, x) = Uv(Ut, x).The whole family of smooth solutions {u ν } ν>0 to the viscous equation (1.1) converges to the unique entropy solution u of the inviscid problem ∂ t u + u∂ x u = 0, t > 0, x ∈ T , u (0, x) = u 0 (x) , x ∈ T .
(2.17) Indeed, compactness in L p , with p < +∞, is ensured by the uniform bounds in L ∞ and W 1,1 and the convergence along the whole sequence is guaranteed by the uniqueness of the limit entropy solution (owing to Urysohn's subsequence principle); for further details on the compactness argument and on the entropy-admissibility of the limit point, we refer to [4,  The right-hand side of (2.18) is uniformly bounded in L 1 and converges in the weak- * sense of finite measures.On the time interval I := ( 1 6U + ε, 1 3U − ε), for 0 < ε ≪ 1, we have that u = U(1 (−Uε,0) − 1 (0,Uε) ) in the neighbourhood O := (−Uε, Uε) and, therefore, the left-hand side of (2.18) is given by − theorems 4.62, 4.71, and 5.1].Notably, the energy density measure u 2 /2 satisfies 23 U 3 δ {x=0} in O. Hence, we conclude