Biharmonic wave maps: local wellposedness in high regularity

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.


Introduction
Let (N, g) be a smooth Riemannian manifold which we assume to be isometrically embedded into some Euclidean space R L . Biharmonic wave maps are critical points u : R n × [0, T) → N of the (extrinsic) action functional These maps model the movement of a thin, stiff, elastic object within the target manifold N. The Euler-Lagrange equation of Φ has been calculated in [6] (in the case N = S l ⊂ R l+1 ) and in [13] (for arbitrary N) and it is given by In particular, if the manifold N has non-vanishing curvature, the condition (1.2) is rewritten as a nonlinear partial differential equation (1.3) where N is a nonlinear expression of the indicated derivatives of u. It is explicitely given in (2.1). We also note that the following energy is (formally) conserved up to the existence time.
In the flat case N = R L (or any affine subspace), the condition (1.2) reduces to the free evolution of a system of decoupled (or linearly coupled) biharmonic wave equations which appear in the elasticity theory of vibrating plates. Here, requiring the parametrization of a thin plate, the bending energy of the elastic plate involves integrated curvature terms of the plate's surface. Hence, in the case of sufficiently stiff material, the potential energy in (1.1) is a reasonable approximation of the elastic energy. We refer to the classical book of Courant and Hilbert [2, chapters 4.10, 5.6] for more information.
A well-studied hyperbolic geometric evolution problem is the wave maps equation (1.6) which arises as the Euler-Lagrange equation of a first order analogue of the action (1.1) with constraint u ∈ N . Here, = ∂ 2 t − ∆ is the d'Alembert operator, m is the Minkowski metric and A(u) is the second fundamental form of the embedded manifold N. The Cauchy problem for (1.6) has been studied intensively as a model for the subtle interplay of nonlinear dispersion, gauge invariance and singularity formation. In particular, we refer to the global regularity theory achieved by novel renormalization techniques of Tao in [17] and [18], see also the survey article by Tataru [19]. In the energy-critical dimension n = 2 a proof of the threshold conjecture on the question of blow-up versus global regularity and scattering is given by Sterbenz and Tataru in [16].
A different but related model is the Schrödinger maps problem for a map u : R n × R → N into a Kähler manifold N. This is the Hamiltonian flow for the Dirichlet energy of u induced by the (symplectic) Kähler form on N. For N = S 2 the Hamiltonian equation reads as ∂ t u = u × ∆u on R n × R, (1.7) and attracted a lot of attention in the past decades. We refer to the global regularity results for N = S 2 and n 2 by Bejenaru, Ionescu, Kenig and Tataru in [1] and for homogeneous spaces N and large dimension by Nahmod, Stefanov and Uhlenbeck in [11]. While different, the methods in both cases exploit the geometric nature of the Schrödinger maps flow by the choice of a suitable frame system along a solution.
In sharp contrast, there is very little literature on the bi-harmonic wave maps (1.2), as discussed below. The main goal of this paper is the proof of the following local wellposedness result for the Cauchy problem corresponding to (1.2) in Sobolev spaces with sufficiently high regularity. We stress that it is difficult to employ the energy method for high regularity solutions of (1.2) since N explicitly depends on the third order derivatives ∇ 3 x u and the energy contains only ∆u. We will overcome this difficulty by exploiting the geometric constraints of solutions. From now on, let N be a compact Riemannian manifold, isometrically embedded into R L . Theorem 1.1. Let u 0 , u 1 : R n → R L satisfy u 0 (x) ∈ N and u 1 (x) ∈ T u0(x) N for a.e. x ∈ R n as well as for some k ∈ N with k > n 2 + 2. Then the following assertions hold: (a) There exists a maximal existence time and a unique solution u : R n × [0, T m ) → N of (1.2) with u(0) = u 0 , ∂ t u(0) = u 1 , and (b) For T 0 ∈ (0, T m ) there exists a (sufficiently small) radius R 0 > 0 such that for all initial data (v 0 , v 1 ) as above that satisfy (1.8) In particular, for smooth initial data u 0 , u 1 : It is worthwhile to remark that both u 0 and u(t) do not necessarily belong to L 2 (R n ) and it is only the difference of these two functions which is contained in this space. We further mention that the lower bound k > n 2 + 2 ensures the existence of L ∞ bounds for ∂ t u ∈ H k−2 (R n ) from Sobolev's embedding. This is necessary in order to establish our energy estimates in the following sections.
The first, second and fourth author have recently shown in [6] that there exists a global weak solution of (1.2) for initial data in the energy space H 2 × L 2 in the case N = S l ⊂ R l+1 . In [6] a crucial ingredient is a conservation law which allows to construct the desired solution as a weak limit of a sequence of solutions of suitably regularized problems. The derivation of this conservation law relies on the fact that the action functional Φ is invariant under rotations in the highly symmetric setting N = S l , and this argument does not apply to arbitrary target manifolds N.
Moreover, the third author has shown energy estimates for biharmonic wave maps in low dimensions n = 1, 2 in [13]. When combining this result with the above blow-up criterion (1.8), he then obtained the existence of a unique global smooth solution of (1.2) for smooth and compactly supported initial data. This results extends earlier work of Fan and Ozawa [5] for spherical target manifolds.
A local well-posedness result as in theorem 1.1 is standard for second-order wave equations with derivative nonlinearities such as wave maps. It can be found for example in the books of Shatah and Struwe [14] and Sogge [15]. In contrast to this case, our nonlinearity N (u) depends on the third spatial derivative of u which cannot directly be controlled by the energy of (2.1) that only contains second order spatial derivatives. In our proof we use the geometric nature of the equation in several crucial steps in order to be able to rewrite this expression in terms of derivatives of lower order.
Concerning the continuous dependence of the solution on the initial data, as the nonlinearity N (u) depends on third spatial derivatives, no Lipschitz estimate in the norm H k × H k−2 is expected from the energy method (as we observe e.g. from the a priori estimates in section 6) and we cannot apply a fixed point argument. In comparison to semi-linear wave equations with derivative nonlinearities (such as wave maps), this makes the well-posedness problem for (1.2) more involved.
We briefly note that our result applies to an intrinsic version of a biharmonic wave map. The functional Φ has an intrinsic analogue Ψ defined by where (∆u) T = P u (∆u) is the tension field of a smooth function u : R n × [0, T) → N . In contrast to Φ, the functional Ψ is independent of the embedding of N → R L . Since the Euler-Lagrange equation differs only by lower order terms (see (2.2) in section 2 below), we can prove the existence of local unique intrinsic biharmonic wave maps with initial data as in theorem 1.1. However, we do not have a result for initial data with (only) covariant derivatives in L 2 .
In the following, we briefly outline the structure of the paper. In section 3, we use a vanishing viscosity approximation and solve the corresponding Cauchy problem for the damped problem In order to obtain a limiting solution for (1.2) as ε 0, we prove a priori energy estimates which are uniform in ε in section 4. As a by-product we obtain the blow-up criterion in theorem 1.1. The existence part in theorem 1.1 is then shown in section 5, and in section 6 we prove that the solutions are unique. Finally we establish the continuity of the flow map in section 7.

Notation and preliminaries
In this section and in the following we will write C for a generic constant only depending on N, n and k, and often also . . . instead of C (· · · ). In order to obtain the explicit form of (1.2), we use the fact that there exists some δ 0 > 0 and a smooth family of linear maps is an orthogonal projection onto the tangent space T p N. The Euler-Lagrange equation (1.2) can thus be written as Exploiting that u takes values in N, we have where the tensors d j P are explicitly described below. We briefly remark that, compared with the right hand side of (2.1), the Euler-Lagrange equation for the intrinsic biharmonic wave maps problem (1.9) differs by P u (dP u (∇u, ∇u) · d 2 P u (∇u, ∇u, ·)) + P u (div(dP u (∇u, ∇u) · dP u (∇u, ·))). (2. 2) The projectors P p are derivatives of the metric distance (with respect to N) in R L , i.e.
Moreover, if p ∈ R L is sufficiently close to N, then π has the nearest point property, i.e. |π( p) − p| = inf q∈N |q − p|, and hence Therefore P p : R L → T π( p) N is well-defined. Using cut-off functions we extend the identity (2.3), and thus also the equation P p = d p π( p), to all of R L . Moreover, all derivatives of P p are bounded on R n . In this way one can investigate (2.1) without restricting the coefficients a priori. Further, for l ∈ N 0 we denote by d l P p the derivative of order l of the map P p , which is a (l + 1)-linear form on R L . For the coefficients in the standard coordinates in R L we write We now derive (2.1) from the condition (1.2) for smooth solutions u : R m × [0, T) → N . Note that we use the sum convention, i.e. the same indices in super-/subscript means summation.
Since ∂ t u ∈ T u N, we infer the identity for k = 1, . . . , L. Because of ∇u ∈ T u N , we also obtain The symmetry of the indices then implies We briefly state the expressions from (2.2) in coordinates, i.e.
for l = 1, . . . , L. In the following we use the shorthand ∇ k1 u ∇ k2 u for (linear combinations of) products of partial derivatives of the components u l of u for l = 1, . . . , L. Here the partial derivatives are of order k 1 ∈ N and k 2 ∈ N, respectively. With this notation we can rewrite equation (2.1) as The Leibniz formula implies the following identity.
In order to include the case m = 0 in the lemma, we will use m j=min{1,m} for the sum in (2.4) or similar formulas. The calculation of derivatives ∇ m (N (u)) and ∇ m (N (u) − N (v)) for sufficiently regular u, v : R n × [0, T] → R L and m ∈ N 0 has been included in appendix A, employing the -convention. The results from appendix A will be used frequently throughout the paper. In the following sections, we also need a version of the classical Moser estimate, see e.g. [20, chapter 13].
In particular,

Existence for the parabolic approximation
Since N (u) = N (u, u t , ∇u, ∇ 2 u, ∇ 3 u), energy estimates for the operator ∂ 2 t + ∆ 2 are not sufficient to show the existence of a solution of (2.1). Instead, we use the damped plate operator with ε ∈ (0, 1] fixed, as a regularization. More precisely, we prove the existence of a solution u ε : R n × [0, T ε ] → N of the Cauchy problem In the following we mostly drop the super-/subscript ε and write (u, T) instead of (u ε , T ε ). We note that the condition in (3.1) reads as 3) We thus study the regularized problem
Lemma 3.1. Let ε ∈ (0, 1) and take u 0 , u 1 : In addition, and there exists a constant C < ∞ such that (3.7) Before we prove lemma 3.1, we reduce the problem to functions in , Since the operators A k extend each other we drop the subscript k. It is well known that −A generates an analytic Then there exists a unique solution U of the linear equation We remark that the solution of (3.11) is given by ds. (3.12) We quantify the above result by the following higher-order energy estimates.
(3.14) To control the second summand with ε in (3.13), we test the differentiated version of (3.15) by ε∇ l ∆ 2 u. Here we proceed similarly as before, where we integrate the term by parts in t and x before aborbing it. It remains to estimate the L 2 -norm of v t (t) and the H 2 -norm of v(t). These inequalities follow by testing the equation with u t and using the fact that , which is insufficient for an application of lemmas 3.2 and 3.3 in a fixed point argument for v. We thus approximate u 0 by u δ which acts on the space for parameters R > 0 and T ∈ (0, 1) fixed below and the metric given by Let ε ∈ (0, 1) be fixed. We will show that the map for a constant Ĉ depending only on N, n, and k. To show this statement, we have to prove the estimates To employ the inequality (3.13) for r = k − 3, we need to bound the norms respectively. This is done by means of lemma A.1 and corollary A.4 combined with a careful application of the Moser estimate in lemma 2.2. We give the relevant details below in section 4 in the proof of the a priori estimate and in section 6 for the uniqueness since these parts require more thought. In this way we obtain in the fixed point We next define R 0 , R and T > 0 in the same way as R 0,δ , R δ and T δ using u 0 instead of u δ

Similar to the proof of the Lipschitz estimate (3.21), lemma 3.3 then yields the bound
where the limits exist in these spaces. In particular, (v, v t ) is a solution of (3.8) and u = v + u 0 solves (3.4). Moreover, by (3.13) the function u δ = v δ + u δ 0 satisfies inequality (3.7), and therefore this estimate also holds for u since u δ t → u t strongly For the uniqueness of v, we note that, for a second solution ṽ, the functions w = v −ṽ and We next show that the above solution actually takes values in the target manifold.
Proof. Fix ε ∈ (0, 1). Let u : R n × [0, T] → R L be the solution of (3.4) constructed in lemma 3.1. We first show that u(x, t) ∈ N for x ∈ R n and t > 0 small enough. Since and u 0 ∈ N a.e. on R n , there exists a time T ∈ (0, T] such that for t ∈ [0,T] the distance We then let w =ū − u and we note that we conclude that Next, we note that By testing the above equation for w by w t , it follows This fact implies that w t = 0 and hence w = 0, which means that u ∈ N . The claimed uniqueness follows similarly to the end of the proof of lemma 3.1. Finally, we let T ε,m T be the supremum of times T > 0 such that we have a solution u : and ∇u ∈ L 2 (0, T ; H k (R n )) which satisfies (3.7) on [0, T ]. □ Remark 3.5. We remark that up to now we fixed ε ∈ (0, 1). Since the constants in the upper bound in estimates such as (3.22) are of order O ε −1 , we have to prove ε independent estimates in the next section.

The a priori estimate
We now prove an a priori estimate for the solution u ε : R n × [0, T ε,m ) → N of the equation given by proposition 3.4 with ε ∈ (0, 1) and initial data u 0 , u 1 : R n → R L such that u 0 (x) ∈ N and u 1 (x) ∈ T u0(x) N for a.e. x ∈ R n as well as for some k ∈ N with k > n 2 + 2. As before we write u instead of u ε , and we fix a number T < T ε,m . Moreover, (3.7) says that for t ∈ [0, T]. We recall that the summand with ε on the right-hand side is well defined because of (3.3).
In the following, we often make use of the relations N (u) ⊥ T u N and u t ∈ T u N which hold since u(x, t) ∈ N for a.e. (x, t) ∈ R n × [0, T]. In particular, N (u) = (I − P u )N (u). Using this fact, we first write where the second equality follows from the Leibniz formula (4.5) We start by estimating Lemma 2.1 yields the identity which implies the pointwise inequality On the other hand, lemma A.1 allows us to bound |∇ m2 (N (u))| pointwise (up to a constant) by terms of the form where i = 1, . . . , m 2 and m 1 + · · · +m i + k 1 + · · · = m 2 − i are as in lemma A.1. Moreover, in the case i = 0 (where no derivatives fall on the coefficients) the terms are of the form where k j ∈ N 0 and k 1 + k 2 + · · · = m 2 . Note that m 2 k − 3 since m 1 > 0. In the following we use the notation (4.8)-(4.10) for all five cases, setting i = 0 for the latter three.
Combining the above considerations with lemma 2.2, we can now estimate the norm where we distinguish five cases according to the terms in the brackets in (4.8)-(4.10).
and derivatives of order Employing also Young's inequality, it follows |∇k 1 +1 u| · · · |∇k j+1 u ∇m 1 +1 u| · · · |∇m i +1 u ∇ k1 u t ∇ k2 u t The other cases will be treated similarly. Note that here and in the following the L ∞ norms and especially u t L ∞ are bounded by our choice of k.
Here it is exploited that m 1 > 0 in I 1 due to the cancellation from (4.4). This time lemma 2.2 is applied with f 1 = · · · = f j+i+2 = ∇u and derivatives of order (4.6). We estimate As in the previous case, C ( We apply lemma 2.2 to the functions f 1 = · · · = f j+i+3 = ∇u with derivatives of order leading to the bound We now use lemma 2.2 with f 1 = · · · = f j+i+4 = ∇u and derivatives of order Hence, we have Summing up the five cases, we infer Next, in I 2 from (4.5) we integrate by parts in order to conclude R n =: I 1 2 + I 2 2 . These terms are estimated by We control ∇ k−3 (N (u)) L 2 by terms of the form (4.8)-(4.10) in the L 2 norm, obtaining as above

(4.18)
Putting together (4.11), (4.14), (4.17) and (4.18), we arrive at the inequality Subtracting the last term on both sides of (4.2), for t ∈ [0, T] we conclude (4.19) It remains to bound the lower order terms. Testing (4.1) by u t ∈ T u N , we infer (4.20) Since also The other derivatives are treated via interpolation, more precisely Estimate (4.19) and the above inequalities lead to the core estimate for solutions of (3.1) and T < T ,m . Using Gronwall's lemma we also obtain (4.23) At least for small times we want to remove the dependence on u on the right-hand side of (4.22) and thus we introduce the quantity for t ∈ [0, T ,m ). We observe that α(t) is equivalent to the square of the Sobolev norms appearing in (4.22). Since the solutions to (3.1) are (locally) unique, our reasoning is also valid for any initial time t 0 ∈ (0, T ,m ). The estimates (4.19), (4.20) and (4.21) thus imply (1 + α(s) k )α(s) ds.
By the above arguments, the function α is differentiable a.e. so that for a.e. 0 t 0 t < T ε,m . We now proceed similarly to [7], where regularization by the (intrinsic) biharmonic energy has been applied in order to obtain the existence of local Schrödinger maps.
We now assume by contradiction that T ε,m T 0 for some (fixed) ε ∈ (0, 1). We apply the contraction argument in the proof of lemma 3.1 for the initial time t 0 ∈ [0, T ε,m ) and data (u(t 0 ), u t (t 0 )) in the fixed-point space B r (T) with radius Since t 0 < T 0 , estimate (4.26) yields the uniform bound

S Herr et al Nonlinearity 33 (2020) 2270
As a result, the time is less or equal than the time T δ for B r (T) in (3.19). Therefore, the solution can be uniquely extended to [0, t 0 + T] in the regularity class of proposition 3.4. For t 0 > T ε,m − T this fact contradicts the maximality of T ε,m , showing the result. □

Proof of the main theorem
We now combine the existence result from proposition 3.4 with lemma 4.1. Thus, there exists a solution u ε : R n × [0, T 0 ] → N of (3.1) for each ε ∈ (0, 1), where T 0 > 0 only depends on ∇u 0 H k−1 and u 1 H k−2 . From (4.26) and the inequality where 1 l 1 k and 0 l 2 k. (Here and below we do not indicate that we pass to subsequences.) In particular, and (∇u, ∂ t u) is weakly continuous in H k−1 × H k−2 . We first assume k 4 (which is no restriction if n 2). Estimating the nonlinearity similarly to section 4, we also deduce from (3.3) and (4.26) that ∂ 2 t u ε ∈ C 0 ([0, T 0 ], H k−4 ) is uniformly bounded as ε → 0 + . Compactness and Sobolev's embedding further yield More precisely for α ∈ (0, 1) and v ε = u ε − u 0 , our a priori estimates and [9, proposition 1.

1.4] imply uniform bounds (in ε) in the spaces
and k 3, we infer that ε∆∂ t u ε → 0 in L 2 t,x . Combining this fact with (5.1) and recalling (3.4), we conclude ). In the case n = 1 and k = 3 we obtain the convergence N ε (u ε ) → N (u) in the sense of the duality (H 1 , H −1 ) because we still have locally uniformly, as well as ∇ 3 u ε → ∇ 3 u and ∇∂ t u ε → ∇∂ t u in C 0 ([0, T 0 ], H −1 loc ) as ε → 0 + . Summing up, we have constructed a local solution u : [0, T 0 ] × R n → N of (2.1) with u(0) = u 0 and ∂ t u(0) = u 1 such that (∇u, ∂ t u) is bounded and weakly continuous in In lemma 6.1 it will be shown that such a solution is locally unique. We recall from the proof of proposition 4.1 that the solution u : Arguing as in section 4, we establish the energy equality (The integral is well-defined in view of the cancellation of one derivative in (4.3).) However, in contrast to the approximations u ε , the solution u has only k weak spatial derivatives (and ∂ t u has k − 2). For this reason, when deriving (5.4) we have to replace one spatial derivative by a difference quotient. The details are outlined in appendix C. We conclude that the highest derivatives ∇ k−2 u t , ∇ k u : [0, T m ) → L 2 are continuous, employing their weak continuity and that the right-hand side of (5.4) is continuous in t. The continuity of the lower order derivatives can be shown as in the next section, so that as asserted. Finally, following the proof of the a priori estimate in section 4 we can derive the blow-up criterion (1.8), see appendix C.
To show theorem 1.1 it thus remains to establish the uniqueness statement and the continuous dependence on the initial data, which is done in the next sections 6 and 7.

Uniqueness
Lemma 6.1. Let u, v : R n × [0, T] → N be two solutions of (1.2) with initial data u 0 : R n → N and u 1 : R n → R L such that u 1 ∈ T u0 N on R n and Proof of lemma 6.1. We derive the uniqueness statement from a Gronwall argument based on the equality d dt which is a consequence of (2.1). Setting for t ∈ [0, T]. We first estimate (6.1) in the case l = k − 3. Since u and v map into N, we have N (u) = (I − P u )(N (u)) and analogously for v. It follows and hence In this way, we can avoid that all derivatives fall on ∇ 3 w. We next write Observe that We then control ∇ k−3 N (u) L 2 using lemma 2.2 as above for the a priori estimate (4.22). Further, lemma A.2 implies that R n I 2 dx is bounded by terms of the form where m 1 , . . . , m j and h 1 , . . . , h j−1 are as in lemma A.2. In (6.3) we then estimate as above in the a priori estimate. For (6.4), it suffices to control terms of the form where |∇ k1 u t ∇ k2 u t | · · · is given as in the nonlinearity N (u) and the orders m 1 . . . , m j , m 1 , . . . ,m i , and k 1 , k 2 . . . are as used before. To apply lemma 2.2, as above we choose and f i+j +1 , f i+j+2 , . . ., according to the respective terms in N (u). We can thus estimate (6.5) in L 2 by We can similarly derive the estimate (integrating dP v (∇ 3 w ∇u) by parts) We then calculate (again similar to section 6) Using integration by parts and (7.2), the last term is rewritten as which is well defined by the higher regularity of u δ . Technically this has to be established by difference quotients as in appendix C, however we omit the details here. The advantage of estimating u δ − u is that the bad terms (with respect to the regularity of u) will be bounded by the regularized initial data from lemma B.1. Their norm will grow as δ → 0 + in a controlled way. Moreover, when estimating (7.3) and (7.4), these bad terms only appear in the products Here the decay of u δ − u L ∞ as δ → 0 + will compensate the growth in (7.5). We now carry out the details in several steps.
Step 3. In the case T 0 T 0 the proof is complete. Otherwise we repeat the same argument starting from , v t (T 0 )). Observe that (7.14) yields For a sufficiently small δ 2 ∈ (0, δ * ] and all δ ∈ (0, δ 2 ], we derive as in (7.6) and (7.8). Based on these bounds we can repeat the arguments of Steps 1 and 2 on the interval [T 0 , min{2T 0 , T 0 }] =: J 1 . However we have to replace the bound (7.1) involving R by (7.14) which yields As in (7.13) we then obtain : and for m = 1 Proof. The result follows from subtracting the expansion in lemma 2.
Proof. The assertions are consequences of the Leibniz rule and lemma A.2. □ Corollary A.4. We have for m ∈ N, m 2 and w = u − v that is a linear combination of terms of the form  Then, we use corollary A.3 for the first three terms in the sum above. For the latter three, we use lemma 2.1 and the Leibniz rule. □ Let ε ∈ (0, 1). We recall from (3.4) the definition with w = u − v and j, m 1 , . . . , m j , k 1 , k 2 , k 3 , h 1 , . . . , h j−1 similarly to corollary A.4.
The implicit constants may depend on ε here.

Appendix B. Approximation of the initial data
In this section we construct certain approximations of initial data in order to conclude continuous dependence of the solution on the initial data. As in the previous sections, take functions u 0 , u 1 : R n → R L with u 0 ∈ N , u 1 ∈ T u0 N a.e. on R n , and (∇u 0 , u 1 ) ∈ H k−1 (R n ) × H k−2 (R n ).
Recall that π is the nearest point map and that P u0 * η δ (u 1 * η δ ) ∈ T u δ 0 N by definition of the projector P and u δ 0 . Especially we have |u δ 0 (x) − u 0 * η δ (x)| = dist(u 0 * η δ (x), N) |u 0 (x) − u 0 * η δ (x)|, |u δ 0 (x) − u 0 (x)| 2|u 0 (x) − u 0 * η δ (x)| for x ∈ R n . We further note that u δ 0 and u δ 1 are smooth maps and that we have the uniform convergence u δ 0 → u 0 , u δ 1 → u 1 as δ → 0 + by construction of u δ 0 (and the mean value theorem for u δ 1 ). Assertion (B.1) follows from by Young's inequality for the convolution. Since ∇u δ 0 = P u0 * η δ ((∇u 0 ) * η δ ), we further have to treat the terms Let u be the solution of (2.1) obtained in section 5. We compute where the second identity follows from (I − P u )u t = 0. For a fixed time t ∈ [0, T m ), the regularity of u yields the limit Here we also used that (1 + ∇u(s) for t ∈ [0, T] and T < T m . In the limit h → 0 it follows by dominated convergence. The right-hand side is continuous in t, and hence the highest derivatives ∇ k u t , ∇ k−2 u : [0, T m ) → L 2 are continuous, since we already know their weak S Herr et al Nonlinearity 33 (2020) 2270