HYPERBOLIC NAVIER-STOKES EQUATIONS I: LOCAL WELL-POSEDNESS

We replace a Fourier type law by a Cattaneo type law in the derivation of the fundamental equations of fluid mechanics. This leads to hyperbolicly perturbed quasilinear Navier-Stokes equations. For this problem the standard approach by means of quasilinear symmetric hyperbolic systems seems to fail by the fact that finite propagation speed might not be expected. Therefore a somewhat different approach via viscosity solutions is developed in order to prove higher regularity energy estimates for the linearized system. Surprisingly, this method yields stronger results than previous methods, by the fact that we can relax the regularity assumptions on the coefficients to a minimum. This leads to a short and elegant proof of a local-in-time existence result for the corresponding first order quasilinear system, hence also for the original hyperbolicly perturbed Navier-Stokes equations.

1. Introduction. Let n ≥ 2 and T, τ > 0. The intention of this note is to examine the hyperbolicly perturbed Navier-Stokes equations        τ u tt − µ∆u + τ (u · ∇)∂ t u + ((τ ∂ t u + u) · ∇)u + u t = −∇π in (0, T ) × R n , div u = 0 in (0, T ) × R n , u| t=0 = u 0 in R n , where u : (0, T ) × R n → R n denotes the velocity of a fluid and p : (0, T ) × R n → R the related pressure. System (1) is obtained by replacing a Fourier type law by the law of Cattaneo. More precisely, we replace the constitutive law for the deformation tensor given by with viscosity coefficient µ > 0 by the relation which represents the first order Taylor approximation of the delayed deformation condition S(t + τ ) = µ 2 (∇u(t) + (∇u(t)) ′ ), t > 0, for small τ > 0. Relation (2) is a Fourier type law. It leads to the well-known paradox of infinite propagation speed for classical parabolic equations. There are applications, however, for that it is more reasonable to work with hyperbolic models, cf. [14] and the references therein. This is also underlined by experiments that document the existence of hyperbolic heat waves.
Recall that the classical Navier-Stokes equations, determined by Fourier's law, are represented by the system   where the deformation tensor is given by (2). In this situation the second line in (4) implies that div 2S(u) = µ∆u.
Consequently, by introducing the new pressure π = p + τ p t , under the assumption of Cattaneo's law the classical Navier-Stokes equations turn into the hyperbolicly perturbed system (1). The hyperbolic fluid model (1) was already derived in [3] and [4]. In these papers on an elementary level the authors discussed consequences and differences of (1) compared with the classical model.
In [11] Paicu and Raugel consider the classical Navier-Stokes equations including merely the hyperbolic perturbation τ u tt for small τ > 0. The global well-posedness for mild solutions in two dimensions for sufficiently small τ , and the global existence for small data and sufficiently small τ in three dimensions in analogy to the classical case are proved. In [11] also a number of justifications for their model are presented, see the references therein. By just adding the term τ u tt to (4) the resulting system remains semilinear and therefore methods for the construction of a mild solution can still be applied. This, however, is no longer possible for system (1), since due to the third term in the first line of (1) this system is a quasilinear one. So, from this point of view system (1) rather differs from the the system considered in [11].
We remark that our new Navier-Stokes system is related to the Oldroyd model which considers instead of (3) the more general model where E := 1 2 (∇u + ∇u T ), cf. de Araújo, de Menzenes and Marinho [2] and Joseph [6]; in comparison to our model we have ν = 0 (and µ = 1). If ν = 0 then, from the point of derivatives getting involved, S is on a similar level as E, as in the classical case (4).
In a first step towards the local-in-time existence result in order, as usually we transform (1) into a first order quasilinear system of the form with V := (u, ∂ 1 u, . . . , ∂ n u, ∂ t u) T . A standard approach, used for standard quasilinear symmetric hyperbolic systems, is to derive a priori estimates in Sobolev spaces of higher order for a linearized version by means of finite propagation speed and then to apply a fixed point iteration to the nonlinear problem. This method, however, seems to fail for the first order system resulting from (1). The crucial point here is the finite propagation speed. It seems not to be available (and this can be regarded as a conjecture of the authors) for equations (1) neither for the corresponding first order quasilinear system or for the associated linearization. The reason for this conjecture lies in the presence of the pressure gradient in equations (1). Of course, as in a standard way for Navier-Stokes equations, ∇p could be removed by applying the Leray-Helmholtz projector onto solenoidal fields to the first line of (1) and then dealing with the resulting system. But either way leads to nonlocal terms in the equations which indicates that finite propagation speed might not be expected.
(The authors, however, so far have not been able to prove this rigorously.) In case of dimension n = 2 or n = 3 we can obtain finite propagation speed for curl u, for instance. This observation is justified by applying curl to (1), since then gradient terms also vanish and (1) turns into an equation for the vorticity curl u (see Section 2). From this point of view, problem (1) and the resulting system (7) are somewhat different from standard quasilinear symmetric hyperbolic systems. By the just mentioned fact, in this note we developed a different approach to first order hyperbolic systems, which also covers equations of type (1). On a standard way by employing Kato's theory we first prove the existence of strong solutions for a linearized version of (7) (see Lemma 4.2). However, the essential step is to derive higher order a priori estimates for the linearized solution, which are required for the application of a fixed point iteration to (7). Here we choose an approach via viscosity solutions, i.e., we add a small viscous term to (7) such that the resulting system becomes parabolic. This method provides a smooth way to justify the formal calculations that lead to higher energy estimates for the solution of the linearized equations. A nice outcome of this method is that we can provide such estimates under minimal regularity assumptions on the coefficients of the linearized operators (see Theorem 4.5). In fact, the regularity assumptions to be made on the coefficients are weaker than the regularity of the obtained solution. Minimal in this context means that we only have to assume the regularity that is required to give sense to the natural energy estimates. Furthermore, these helpful energy estimates for the solution are also provided by the method.
This seems to be different and new in comparison to similar results for standard symmetric hyperbolic systems that are based on finite propagation speed of the displacement. In pertinent textbooks such as [10,Theorem 2.1] or [13,Theorem 5.1], for instance, always the assumed regularity for the coefficients is higher than the regularity obtained for the solutions, and it seems to be difficult or even impossible to improve this to our results by the methods used therein. In [5] an abstract approach to quasilinear evolution equations is developed generalizing results obtained in [7]. But also there the assumed regularity on the coefficients is higher than the obtained for the solution. Only for the approach developed in [8] this is not the case. There the coefficients are assumed to be elements of uniformly local Sobolev spaces. This assumption is enough by the fact that the standard Sobolev embedding and the required algebra properties are still valid. Thus the assumptions in [8] for the coefficients of the linearized system are comparable to ours. On the other hand, it is not so obvious whether the approach to quasilinear hyperbolic systems given in [8] applies to system (1) due to the presence of the presssure term ∇π or the Helmholtz projection respectively.
Based on the linear theory developed here the application of Majda's fixed point iteration, cf [10], in order to construct local-in-time strong solutions to (7) becomes rather short and elegant (see Theorem 5.1). This is due to the fact that by the quality of the linear results provided here no smoothing of the data, in particular of the coefficients, for the fixed point iteration is required anymore. By our energy estimates for the linearized solutions, here we also get immediately upper bounds for the approximate solutions of the fixed point iteration. This again is in contrast to [10] (or [13]). There upper bounds have to be derived by estimating the approximate solutions in an elaborate way employing the structure of the underlying quasilinear symmetric hyperbolic system. Also continuity (in time) of the solutions (as given in (41)) immediately follows from the linear results. This is also quite different from the approach performed in [10] or [13], where exhausting procedures via the strong convergence in weaker norms and the weak continuity in higher norms have to be applied in order to prove continuity. This seems to be a futher nice advantage of our approach in comparison to previous methods.
We want to emphasize that the approach developed in this note is by no means restricted to first order quasilinear systems arising from equations of type (1). In fact, it is quite generally applicable, in particular to standard quasilinear symmetric hyperbolic systems. Thus by our approach on a different (perhabs even more elegant) way we can handle, for example, quasilinear wave equations or systems arising in thermoelasticity such as treated in [10] or [14]. Moreover, the final results for the quasilinear systems are of the same quality as the results obtained by previous methods. On the other hand, obviously the approach presented here is more general, since we can deal as well with problems of type (1), which might not produce finite propagation speed. Furthermore, also Oldroyd models such as (6) can be covered by our approach which is different from the methods used e.g. in [6].
We proceed with the precise statement of our main results. By virtue of the second line in (1) we define the ground space as Here, as usual in the Navier-Stokes context, σ refers to the solenoidality (i.e. div u = 0) of the vector fields. Also note that the symbol C ∞ b (Ω) stands for smooth functions whose derivatives of each order k ∈ N 0 are also bounded on the set Ω.
there exists a time T * > 0 and a unique solution (u, π) of equations (1) satisfying The existence time T * can be estimated from below as with a constant C > 0 depending only on m and the dimension n.
As an immediate consequence we also have Corollary 1. In the situation of Theorem 1.1 additionally assume that Then the solution u, p is classical, i.e. we have . Remark 1. We remark that it seems to be anything but obvious how to extend the above results to domains with a boundary under the assumption of no-slip conditions. For instance, to the authors it is not clear, wether the methods used in the proof of higher regularity in Theorem 4.5 can be generalized to a half-space. Moreover, the fact that the Helmholtz projector P onto solenoidal fields commutes with derivatives of arbitrary order in R n is extensively used throughout the paper. This fact is no longer true if a boundary is present, which at least gives rise to further technicalities. Based on reflection arguments, an approach to (1) subject to tangential slip boundary conditions in the half-space R n + is given in [15]. The paper is organized as follows. We start in Section 2 with a remark on finite propagation speed. In Section 3 we perform the transformation of (1) into a first order quasilinear system. Section 4 represents the heart of this work and provides the linear theory. First we prove the existence of strong solutions to a linearized version of (7). As mentioned before, the essential point then is to derive higher regularity of this solution. This result is obtained by employing the method of viscosity solutions. In Section 5 we prove the local-in-time existence for the first order quasilinear system, which finally results in our main results Theorem 1.1 and Corollary 1 by the equivalence of systems (1) and (7). 2. Remark on finite propagation speed. For the local solution obtained in the previous section, we can prove the finite propagation speed for the vorticity v := curl u = ∇ × u. Namely, v satisfies the differential equation where J(∇u) denotes the Jacobi matrix of the first derivatives of u. The part in brackets {. . . } involves at most first-order derivatives of v. Therefore, the general energy estimates for hyperbolic equations of second order -after transformation to a first-order symmetric-hyperbolic system -apply as described in [13], and give the finite propagation speed. As mentioned before, note that this can still not be expected for u due to the presence of the pressure terms.
3. Transformation into a symmetric system. We start by introducing some notation. Note that we use standard notation throughout this note, for the appearing function spaces see e.g. [1]. Let X be a Banach space and Ω ⊂ R n be a set. Then L p (Ω, X) denotes the standard Lebesgue space of p-integrable X-valued functions for 1 ≤ p < ∞. For p = ∞, L ∞ (Ω, X) denotes the space of all (essentially) bounded functions equipped with the standard norm ess sup x∈Ω · X . Accordingly, In the case k = 0 we also write · p for the norm. Moreover, we set H k (Ω, X) := W k,2 (Ω, X). In this paper from the just introduced spaces only L 2 (Ω, X), H k (Ω, X), L ∞ (Ω, X) and W k,∞ (Ω, X) will appear. Also note that if X = C m or X = R m we write just L 2 (Ω), H k (Ω), etc. We will also make use of the homogeneous Sobolev space We also use standard notation for spaces of continuous functions. For k ∈ N 0 ∪ {∞}, C k (Ω, X) denotes the space of k-times continuously differentiable functions and we write C(Ω, X) if k = 0. If the functions in C k (Ω, X) are additionally bounded, we use the symbol C k b (Ω, X) and its subspace of compactly supported functions is denoted by C k 0 (Ω, X). The (X, X ′ ) dual pairing we denote by ·, · X,X ′ . To obtain consistency with the scalar product if X is a Hilbert space, observe that the second argument in ·, · X,X ′ is defined with complex conjugation, i.e., we have , if x ′ (x) denotes the standard dual pairing. If H is a Hilbert space we write ·, · H . From time to time we also omit the subscript and just write ·, · , if no confusion seems likely. The space of linear bounded operators from X to a Banach space Y is denoted by L (X, Y ).
Suppose (u, p) with u : R n+1 + → R n and p : R n+1 + → R is the solution of sytem (1). In this section we transform equations (1) into a first order quasilinear hyperbolic system for the vector As for the classical Navier-Stokes equations the pressure term ∇p will be eliminated by employing the Leray-Helmholtz projector onto solenoidal fields where π ∈ H 1 (R n ) is the unique solution of the weak Neumann poblem This leads to the well-known orthogonal decomposition where G 2 (R n ) := {∇π : π ∈ H 1 (R n )}. Applying P to the first line of (1), this system is formally reduced to considered in the space L 2 σ (R n ). For the development of the linear theory it will be convenient to get rid of the τ in front of u tt and µ in front of ∆u. For this purpose we introduce the dilated function Then u solves (9) if and only if v solves (10) will be the one which is considered in the sequel and which we transform into a first order system. For j = 1, . . . , n we define the symmetric matrices (11) with I n the identity in R n and where −I n represents the (j + 1, n + 2)-th and the (n + 2, j + 1)-th entry of A j (V ). The operator M j is defined as and corresponds to the quasilinear term in (10). We also define the (n × n) · ((n + 2) × (n + 2)) matrix operators Finally, we set Then, it is easily checked that (10) is equivalent to the first order quasilinear hyperbolic system Observe that the difference to standard quasilinear symmetric hyperbolic systems lies in the presence of the projector P. In the next two sections we will develop the required linear and quasilinear existence theory for systems of the form (13).
Here we consider a linearized version of system (13). To be precise, we assume that A j and B are matrices of the form given in (11) and (12), where M j (V ) and B j (V ) are replaced by a j I n and b j I n , respectively, with given functions a j , b j : Observe that it is well-known that in R n the Helmholtz projection is bounded on the entire scale of Sobolev spaces, that is, we have P ∈ L (H m (R n )) for every m ∈ Z. This, for instance, follows easily by its symbol representation and Plancherel's theorem, where F denotes the Fourier transformation. In this spirit the last expression in the definition of D(A) makes sense, due to n j=1 ∂ j V j+1 ∈ H −1 (R n ). In this section we aim for the well-posedness and higher regularity of the linear nonautonomous first order hyperbolic system For this purpose we start with the following result for the 'principal' linear part A. Proof. By the definition of A j we have that

This yields
Then the first n + 1 components in (15) imply that V n+2 ∈ H 1 (R n ) and that V n+2 By the last component in (15) this, in turn, yields that P n j=1 ∂ j V j+1 k converges in L 2 (R n ). By the fact that V k → V in H, we also obtain Since the convergence in L 2 is stronger as the convergence in H −1 we conclude that P Next, for V ∈ D(A) and U ∈ H we have By the symmetry of P on L 2 and since we use the same symbol for the Helmholtz projection on H m for different m, we also have P ′ = P if P is the projection on H m . For U ∈ D(A) we therefore can continue the above calculation as where we used the fact that div (a 1 , . . . , a n ) T = 0 in the second equality. This shows that A(t) is skew-symmetric and that D(A(t)) ⊂ D(A(t) ′ ).
For the converse inclusion we pick First we choose V ∈ D(A) such that V k = 0 except for k = ℓ + 1 with fixed ℓ ∈ {1, . . . , n} and such that V ℓ+1 ∈ C ∞ 0 (R n ). In view of (15) we then obtain This shows that ∂ ℓ U n+2 has a representant in L 2 (R n ) for every ℓ ∈ {1, . . . , n}. Thus Thanks to W n+2 , n j=1 P a j ∂ j U n+2 ∈ L 2 (R n ), this shows that also P n j=1 ∂ j U j+1 belongs to L 2 (R n ). Consequently, U ∈ D(A) and we conclude that D(A(t) ′ ) ⊂ D(A(t)). The assertion is therefore proved.
The full linear operator can now be handled by a perturbation argument. Thus, (A(t)) t∈[0,T ] is a CD-system. By [9, Section 1.2] (see also [12]) therefore A is the propagator of an evolution family on H. By the fact that M ∈ C([0, T ], L (H)), a standard abstract perturbation argument (cf. [9, Remark 1.1(c)] or [12]) implies that A + M is still the propagator of an evolution family on H as claimed in the lemma.
Let now B be defined as in the beginning of this section with coefficients b j ∈ C b ([0, T ] × R n ). Then for M = B Lemma 4.2 implies the well-posedness of the problem on H. In other words, for each V 0 ∈ D(A) we obtain a unique solution This follows from standard theory, cf. [9] or [12]. However, in order to prove a localin-time existence result for the full quasilinear system, higher regularity in Sobolev spaces for the linear problem is required. For this purpose we employ the method of viscosity solutions.
Proof. It is well-known that ε∆ is the generator of an analytic C 0 -semigroup on H q (R n ) ∩ H. Note that by our regularity assumptions on a, b the nonautonomous operator (A+B) represents a lower order perturbation of ε∆ regarded as a propagator on H q (R n )∩H. By standard abstract perturbation results (cf. [12]) we therefore obtain that −ε∆ + A + B is the propagator of an evolution family (U ε (t, s)) 0≤s≤t≤T on H q (R n ) ∩ H such that V (t) := U ε (t, 0)V 0 satisfies (18) and (19).
In the proof of the next Theorem we will also frequently make use of the following estimates, which are often quoted as "Moser-type inequalities". For a proof we refer to [13,Lemma 4.9].
Lemma 4.4. Let m ∈ N. There is a constant C = C(m, n) > 0 such that for all f, g ∈ W m,2 (R n ) ∩ L ∞ (R n ) and α ∈ N n 0 , |α| ≤ m, the following inequalities hold: where ∇ m u denotes the entirety of all m-th order derivatives of a function u.
The next result provides higher regularity of the solutions of (17) under, and this is essential, in a certain sense minimal regularity assumptions on the data and the coefficients. In particular, in Sobolev spaces of higher order these regularity assumptions are weaker as the obtained regularity for the solutions. This will be very helpful for the construction of time-local strong solutions for the full nonlinear problem in Section 5. and let the coefficients a = (a 1 , . . . , a n ) satisfy the assumptions of Lemma 4.2. Assume additionally that Then the unique solution V = U (t, 0)V 0 of problem (14) satisfies Furthermore, the evolution family U satisfies the estimates for all 0 ≤ s ≤ t ≤ T with constants C 1 , C 2 > 0 depending only on m and the dimension n, and where we put Proof. The proof is splitted into five steps.
Step 1: construction of suitable approximate solutions V k,ε . We denote by J x k f and J t k f the convolution of a function f with the Friedrichs mollifier in the variable x and t, respectively. We set for j = 1, . . . , n and k ∈ N, where E 0 denotes the trivial extension by 0 from [0, T ] to R. Then we readily obtain We fix q > m + 1 and denote by A k and B k the operators being defined as A and B with coefficients a k and b k , respectively. Due to Lemma 4.3 for every k ∈ N and ε > 0 there is a viscosity solution, denoted by V k,ε , of the system Step 2: uniform boundedness of V k,ε . Let α ∈ N n 0 such that |α| ≤ m + 1. Since m + 1 < q, we may apply ∂ α to (29) to the result where e i = (0, . . . , 0, I n , 0, . . . , 0) for i ∈ {1, . . . , n + 2} denotes the i-th unit matrix in (R n ) n+2 . Inequality (21) applied on the terms involving the a j,k 's and (20) on the terms involving the b j,k 's yields In view of the Sobolev embedding and by our assumption m > n/2 we can continue this calculation to the result Forming the dual pairing of (31) with ∂ α V k,ε implies Summing up over |α| ≤ m + 1 and integrating over t then yields Thus, applying Gronwall's lemma and taking into account (26)-(28), we end up with This shows that V k,ε is uniformly bounded in L ∞ ([0, T ], H m+1 (R n )) and that ε∇V k,ε is uniformly bounded in L 2 ([0, T ], H m+1 (R n )). Again by an application of (20) we therefore obtain that (A k + B k )V k,ε is uniformly bounded in L ∞ ([0, T ], H m (R n )). From that, the uniform boundedness of ε∆V k,ε in L 2 ([0, T ], H m (R n )), and equations (29) we infer that also ∂ t V k,ε is uniformly bounded in L 2 ([0, T ], H m (R n )). Thus, we have proved that V k,ε is uniformly bounded in the class Step 3: weak * convergence of V k,ε to the solution V of (14). The outcome of step 2 implies the existence of a subsequence of V k,ε , for simplicity also denoted by V k,ε , converging weakly * in the class (34) for k → ∞ and ε → 0. Denote by U its limit. Then U also belongs to (34). Thanks to the Sobolev embedding we also have Next, we show that U solves (17). In fact, multiplying , div ϕ n+2 = 0 to (29) and integrating by parts gives us Due to (27), (28), and m > n/2 we have This shows that as k → ∞ and ε → 0. The boundedness of V k,ε in L ∞ ([0, T ], H) also yields Thus, letting k → ∞ and ε → 0 implies Thanks to the fact that U belongs to (34) and in view of (35), we can reverse the integration by parts to the result .
This, in turn, implies that U (0) = V 0 , hence that U solves (14). By virtue of (35) and by the assumptions on a, b, the fact that U solves (14) also yields Since we assumed that n ≥ 2, hence that m > n/2 ≥ 2, we obtain that U is a strong solution of (14). Consequently, U is unique and therefore coincides with V = U (·, ·)V 0 , where U is the evolution family given by Lemma 4.2.
Step 4: proof of estimates (24) and (25). Note that by (33) and the fact that U = V , we obtain . By the just proved facts for the solution of this system we deduce t+s s (a(r), b(r)) m+1 + 1 dr , hence (24). The estimate for the time derivative of U now easily follows by where we applied once more Lemma 4.4.
Step 5: continuity of V in time. From step 4 and our assumptions on a, b we immediately see that It remains to show that in (37) W 1,∞ and L ∞ can be replaced by C 1 and C, respectively. To this end, we will employ the variation of constant formula. Thanks to (35) and (36) we have for arbitrary V 0 ∈ H m+1 (R n ). In view of m ≥ 2, we may apply ∂ α for |α| ≤ 1 to (14). This leads to with Very similar to the calculations that lead to (32) we can derive By virtue of our assumptions on a, b and since On the other hand, by applying the Hölder inequality we can also estimate as Since m − 1 ≥ m/2 > n/4 for m ≥ 2, the Sobolev embedding implies that H m−1 (R n ) ֒→ L 4 (R n ). Hence the above inequality gives us F (V ) ∈ L ∞ ((0, T ), H). By our asumptions on a and b and in view of (38), F (V ) is even continuous in time.
So, altogether we obtain According to H 1 (R n ) ∩ H֒→D(A), [9, Remark 1.3] therefore implies that ∂ α V is the unique strong solution of (39) given by the variation of constant formula Here U still denotes the evolution system generated by the propagator A + B. From our assumptions (22) on a, b and step 4 we know that U satisfies the estimate for some C 1 > 0. Since U is an evolution system on H we also have for some C 2 > 0. Interpolating these two inequalities yields with C = max(C 1 , C 2 ) and where [·, ·] θ denotes the complex interpolation space for θ ∈ (0, 1). By the fact that H is complementary in L 2 (R n ), [17,Theorem 1.17.1.1] implies that Consequently, for θ = m/(m + 1) we deduce From this we immediately gain the estimate Inserting this into (40) while taking the H m -norm and keeping in mind continuity relation (38) and that F (V ) ∈ L 1 ((0, T ), H m (R n )) then gives us This shows that t → U (t, 0) is strongly continuous in t = 0 w.r.t. the H m+1 -norm. The fact that U is an evolution family then implies the continuity on [0, T ]. So, we have proved The assertion that V ∈ C 1 ([0, T ], H m (R n )) then follows again by ∂ t V = −(A + B)V and by our assumption a, b ∈ C([0, T ], H m (R n )) on the coefficients. The result is therefore proved.

5.
Quasilinear local existence. Based on a fixed point iteration here we construct local-in-time solutions to the first order quasilinear system (13). The idea of this fixed point iteration goes back to Majda [10]. However, by the strength of our linear result Theorem 4.5 the proof of the quasilinear local-in-time existence performed here becomes much more elegant compared to the methods used in [10] or [13].
Theorem 5.1. Let m ∈ N 0 , m > n/2, and let V 0 ∈ H ∩ H m+1 (R n ). Then, there exists a T > 0 and a unique solution of system (13). The existence time T can be estimated from below as with a constant C > 0 depending only on m and the dimension n. Proof.
and for k ∈ N 0 let V k+1 be inductively defined as the solution of the initial value problem By the fact that we see that Theorem 4.5 (i.p. (22) and (23)) implies that every solution belongs to the class of the coefficients for the next step. Hence, V k+1 is well-defined for every k ∈ N 0 . Next, we will prove the following uniform bounds.
Lemma 5.2. There exist R, L, T * > 0 such that for all k ∈ N 0 we have Proof. We use induction over k ∈ N 0 . For k = 0 we have which is to understand as a first condition on the size of R. In view of ∂ t V 0 = 0 we see that L is still arbitrary. Now, assume that the assertion holds for k ∈ N 0 . Estimate (24) in combination with (22) and the induction hypothesis imply

Then for
This leads to estimate (42) for the size of the existence time.
Similarly, for the time derivative of V k+1 we employ estimate (25) in combination with (22) to the result Thus, again for T * ≤ 1/(R + 1) we deduce This fixes L and the lemma is proved.
The just obtained uniform boundedness of (V k ) k∈N was the essential step in proving suitable convergence of (V k ) k∈N such that we may pass to the limit in (43). In fact, first Lemma 5.2 implies convergence in C([0, T ], H). This can be seen by considering Multiplying by W k+1 , integrating over R n , and utilizing the structure of A, B and the Sobolev embedding we obtain Integrating with respect to time implies where we have used the boundedness of the sequence (V k ) k∈N in L ∞ ([0, T * ], H m+1 ) proved in Lemma 5.2. Applying once more Gronwall's lemma gives us Hence, if not already small enough, we choose T * e C2T * < C/2 to achieve W k → 0 and therefore that V k → V strongly in C([0, T * ], H) for some V ∈ C([0, T * ], H) as k → ∞. Thanks to the interpolation inequality strongly in C([0, T * ], H). Thus we can pass to the limit in (43) which yields that V is a solution of (13).
To see that V satisfies (41) we argue as follows. By the boundedness of (V k ) k∈N there is a weak* limit V * in Next, observe that we have Obviously U is a strong solution of (44). On the other hand, by the discussion above we know that Thus, V is a strong solution of (44) as well. By the uniqueness of strong solutions of the linear system (44) we obtain V = U , hence (41).
Forming the dual pairing of (45) with W gives us Consequently, W = 0 by Gronwall's lemma. This completes the proof of Theorem 5.1.
We conclude with the proof of our main result Theorem 1.1.
This yields that (u, π) is the unique solution of (1) with the claimed regularity.
Corollary 1 now is easily obtained as follows Proof. Assuming u 0 , u 1 ∈ ∞ k=0 H k (R n ) implies that u ∈ C 2 ([0, T * ], H m (R n )) for every m ∈ N. By applying ∂ t iteratively to equations (9) and taking into account the boundedness of P on every H m (R n ), we even obtain that u ∈ C ∞ ([0, T * ], H m (R n )) for every m ∈ N. From representation (46) we then deduce the same regularity for ∇π. The Sobolev embedding finally yields the assertion.