Well-posedness of hyperbolic systems with multiplicities and smooth coefficients

We study hyperbolic systems with multiplicities and smooth coefficients. In the case of non-analytic, smooth coefficients, we prove well-posedness in any Gevrey class and when the coefficients are analytic, we prove C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document} well-posedness. The proof is based on a transformation to block Sylvester form introduced by D’Ancona and Spagnolo (Boll UMI 8(1B):169–185, 1998) which increases the system size but does not change the eigenvalues. This reduction introduces lower order terms for which appropriate Levi-type conditions are found. These translate then into conditions on the original coefficient matrix. This paper can be considered as a generalisation of Garetto and Ruzhansky (Math Ann 357(2):401–440, 2013), where weakly hyperbolic higher order equations with lower order terms were considered.


Introduction
We consider the Cauchy problem where D t = −i∂ t , D x = −i∂ x , and A(t, D x ) is an m × m matrix of first-order differential operators with time-dependent coefficients and u is a column vector with components u 1 , . . . , u m . We assume that (1) is hyperbolic, whereby we mean that the matrix A(t, ξ) has only real eigenvalues. These eigenvalues, rescaled to order 0 by multiplying by ξ −1 , will be denoted by λ 1 (t, ξ), . . . , λ m (t, ξ). Following Kinoshita and Spagnolo in [22], we assume throughout this paper that there exists a positive constant C such that for all 1 ≤ i < j ≤ m. As observed in [14] combining the well-posedness results in [21,25] we already know that the Cauchy problem (1) is well-posed in the Gevrey class γ s , with 1 ≤ s < 1 + 1 m as well as in the corresponding spaces of (Gevrey-Beurling) ultradistributions. In this paper we want to prove that when A(t, D x ) has smooth coefficients and the condition (2) on the eigenvalues holds, then the Gevrey well-posedness result above can be extended to any s ≥ 1. Since, by the results of Kajitani and Yuzawa when s ≥ 1 + 1 m at least an ultradistributional solution to the Cauchy problem (1) exists, we will prove that this solution does actually belong to the Gevrey class γ s . In the case of analytic coefficients, we will prove instead that the Cauchy problem (1) is C ∞ well-posed.
In this paper we assume that the Gevrey classes γ s (R n ) are well-known: these are spaces of all f ∈ C ∞ (R n ) such that for every compact set K ⊂ R n there exists a constant C > 0 such that for all β ∈ N n 0 we have the estimate For s = 1, we obtain the class of analytic functions. We refer to [11] for a detailed discussion and Fourier characterisations of Gevrey spaces of different types and the definition of the corresponding spaces of ultradistributions.
The well-posedness of hyperbolic equations and systems with multiplicities has been a challenging problem for a long time. In the last decades several results have been obtained for scalar equations with t-dependent coefficients ( [2][3][4]6,7,[11][12][13]22], to quote a few) but the research on hyperbolic systems with multiplicities has not been as successful. We mention here the work of D'Ancona, Kinoshita and Spagnolo [8] on weakly hyperbolic systems (i.e. systems with multiplicities) of size 2 × 2 and 3 × 3 with Hölder dependent coefficients later generalised to any matrix size by Yuzawa in [25] and to (t, x)-dependent coefficients by Kajitani and Yuzawa in [21]. In all these papers, well-posedness is obtained in Gevrey classes of a certain order depending on the regularity of the coefficients and the system size. Systems of this type have recently also been investigated in [10,14].
It is a natural question to ask if under stronger assumptions on the regularity of the coefficients, for instance smooth or analytic coefficients, the well-posedness of the corresponding Cauchy problem could be improved, in the sense if one could get well-posedness in every Gevrey class or C ∞ -well-posedness. It is known that this is possible for scalar equations under suitable assumptions on the multiple roots and Levi conditions on the lower order terms, see [12,22] for C k and C ∞ coefficients and [12,17,22] for analytic coefficients. This paper gives a positive answer to this question by extending the results for scalar equations in [12,22] to systems with multiplicities. This will require a transformation of the system in (1) into block-diagonal form with Sylvester blocks which increases the system size from m × m to m 2 × m 2 but does not change the eigenvalues, in the sense that every block will have the same eigenvalues as A(t, ξ). Such a transformation, introduced by D'Ancona and Spagnolo in [9], has the side effect to generate a matrix of lower order terms even when the original system is homogeneous, i.e., (1) will be transformed into a Cauchy problem of the type It becomes therefore crucial to understand how the lower order terms in B(t, ξ) are related to the matrix A(t, ξ), which is in turn related to A(t, ξ), and which Levi-type conditions have to be formulated on them to get the desired well-posedness. These Levi-type conditions will then be expressed in terms of the matrix A(t, ξ). In the next subsection we collect our main results and we give a scheme of the proof.
x ∈ R n , be an m × m matrix of first order differential operators with C ∞ -coefficients. Let A(t, ξ) have real eigenvalues satisfying condition (2). Assume that the Cauchy problem for l = 1, . . . , m −1 and j = 1, . . . , m. Hence, for all s ≥ 1 and for all u 0 ∈ γ s (R n ) m there exists a unique solution u ∈ C 1 ([0, T ], γ s (R n )) m of the Cauchy problem (1).
The formulation of the Levi-type conditions given above requires a precise knowledge of the matrix B(t, ξ). For that see the Sect. 3.4. It is possible to state the previous well-posedness result completely in terms of the matrix A(t, ξ) and the Cauchy problem (1). This means to introduce an additional hypothesis on the coefficients of A(t, ξ) which implies the Levi-type conditions on B(t, ξ). In the final section of the paper we will prove that in some cases, for instance when m = 2, this second formulation is equivalent to the one given in Theorem 1.1.
x ∈ R n , be an m × m matrix of first order differential operators with C ∞ -coefficients. Let A(t, ξ) have real eigenvalues satisfying condition (2) and let Q = (q i j ) be the symmetriser of Since the entries of the symmetriser are polynomials depending on the eigenvalues of A(t, ξ), we require in Theorem 1.2 that the t-derivatives of A(t, ξ) up to order m −1 are bounded by suitable polynomials of the eigenvalues λ 1 (t, ξ), . . . , λ m (t, ξ). Note that, as observed already in the appendix of [12], these polynomials can be expressed in terms of the entries of A(t, ξ).
When the entries of A(t, ξ) are analytic, then we prove that the Cauchy problem (1) is C ∞ well-posed. The precise statements can be obtained by replacing γ s with C ∞ in Theorems 1.1 and 1.2.
We conclude this subsection by presenting the scheme of the proof of Theorem 1.1 which combines ideas from [9,12] .
Step 1 Compute the adjunct matrix adj(I m τ − A(t, ξ)) = cof(I m τ − A T (t, ξ)), where I m is the identity matrix of size m × m. We thus have the relation where the c h (t, ξ) are homogeneous polynomials of order h in ξ and are given by the coefficients of the characteristic polynomial of A(t, ξ). See Appendix.
Step 2 Apply the operator adj(I m D t − A(t, D x )), associated to the symbol adj(I m τ − A(t, ξ)), to the system (1) and obtain a set of scalar equations for u 1 to u m , where the operator acting on these is associated to det(I m τ − A(t, ξ)). Additionally, one gets some lower order terms which can be computed explicitly.
Step 3 Convert the resulting set of equations to Sylvester block diagonal form following the method of Taylor in [23], i.e by setting for k = 1, . . . , m. This transformation maps each equation to a system in Sylvester form and glues those systems in block diagonal form together. Hence, we achieve a block diagonal form with Sylvester blocks associated to the characteristic polynomial of (1). This means that each block will have the same eigenvalues as A(t, ξ). The initial data will be transformed in the same way to obtain a new set of initial data U 0 for the new system.
Step 4 Consider the resulting system where A(t, D x ) and B(t, D x ) are matrices of size m 2 × m 2 with a special structure. As explained above, A(t, D x ) is a block diagonal matrix with m identical blocks of Silvester matrices having the same eigenvalues as A(t, ξ) and B(t, D x ) is composed of m × m 2 blocks with only the last row not identically zero. Since the original homogeneous system has been transformed into a system with lower order terms, to get well-posedness of the corresponding Cauchy problem (6), we need to find some Levi-type conditions. These are obtained by following the ideas for scalar equations in [12].
Step 5 We apply the partial Fourier transform with respect to x to (6) and we prove an energy estimate from which the assertions of the well-posedness theorems follow in a standard way. A key point is the construction of the quasi-symmetriser of the matrix A(t, ξ).
The remainder of the paper is organised as follows. In Sect. 2, we present a short survey on the quasi-symmetriser which will be employed to formulate and prove the energy estimate. The core of Sect. 3 is the transformation of A(t, ξ) from (1) to block Sylvester form. An explicit description of adj(I m D t − A(t, D x )) and the resulting lower order terms is also given in Sect. 3, together with a detailed scheme of the proof in the cases m = 2 and m = 3. Section 4 is devoted to the energy estimate and Sect. 5 to the estimates for the lower order terms. The paper ends with the well-posedness results in Sect. 6 and the Appendix, where we collect some algebraic results concerning adj(I m τ − A(t, ξ)).
To construct the quasi-symmetriser, we follow [22] and define P (m) (λ) inductively by P (1) (λ) = 1 and Further, we set, for ε ∈ (0, 1], where H (m) ε = diag{ε m−1 , . . . , ε, 1}. We remark that P (m) (λ) depends only on λ . Finally, the quasi-symmetriser is the Hermitian matrix To describe the properties of Q where the matrices Q We finally recall that a family {Q α } of non-negative Hermitian matrices is called nearly diagonal if there exists a positive constant c 0 such that 11 , . . . , q α,mm }. The following linear algebra result is proven in [22,Lemma1]. Lemma 2.2 is employed to prove that the family Q (m) ε (λ) of quasi-symmetrisers defined above is nearly diagonal when λ belongs to a suitable set. The following statement is proven in [22,Proposition 3].

Proposition 2.3 For any M > 0 define the set
Then the family of matrices {Q We conclude this section with a result on nearly diagonal matrices depending on three parameters, ε, t, and ξ which will be crucial in Sect. 4. Note that this is a straightforward extension of Lemma 2 in [22] valid for matrices depending on two parameters, ε and t.
be a nearly diagonal family of coercive Hermitian matrices of class C k in t, k ≥ 1. Then, there exists a constant C T > 0 such that for any continuous function V : for all ξ ∈ R n .

Remark 2.5 All results of this section hold true in the when
The corresponding block diagonal matrix with W m (λ) blocks is denoted by W (m) (λ). Proofs follow from a block-wise treatment and application of the results above.

The quasi-symmetriser in the case m = 2 and m = 3
For the advantage of the reader, we conclude this section by computing the quasisymmetrisers Q (2) ε and Q (3) ε . For m = 2, we obtain Similarly, for m = 3, we obtain

Sylvester block diagonal reduction
This section is devoted to the Sylvester block diagonal reduction that will be employed on the system (1). This transformation has been introduced by D'Ancona and Spagnolo in [9]. Here we give a detailed description of how this reduction works on the system I m D t − A(t, D x ) and we present explicit formulas for the matrix of lower order terms generated by the procedure. Note that these results are obtained from general linear algebra statements that are collected in the appendix at the end of the paper. We will refer to Appendix throughout this section. The subsections refer to the steps of the proof outlined in Sect. 1.1.

Step 1: The adjunct adj(I m D t − A(t, D x ))
A straightforward application of Lemma 7.4 leads us to the following proposition. (1). Then,

Proposition 3.1 Let I m D t − A(t, D x ) be the operator in
λ = (λ 1 , . . . , λ m ) and σ (m) h (λ) as defined in (7). The differential operator adj(I m D t − A(t, D x )) is of order m−1 with respect to D t and every differential operator Proposition 3.1 completes Step 1 of our proof as outlined in the scheme. We can therefore proceed to Step 2.

Step 2: Computation of the lower order terms Proposition The lower order terms that arise after applying the adjunct
where A h (t, D x ) is defined in (8) and Proof From Proposition 3.1 and Leibniz rule, we have Now we write the second summand in the last equation in (11) as Xu + Y u where Xu contains all terms with h = 0 and and then by (10) By (8), we obtain and, thus, .
Using that A m = 0 [thanks to the Cayley-Hamilton theorem, see (58)] and A 0 = I m , we obtain (13) which concludes the proof.
It will be convenient for the description of some important matrices in this paper to rewrite the lower order terms in a different way. More precisely, we have the following corollary. (9) as

Corollary 3.3 We can write the lower order term in
where and A h (t, D x ) is given by (8).
Proof Formula (14) follows from (9) by interchanging the order of the sums appropriately. Indeed, we have, using (8) and (10), that with Note that in computing B h+1 in the last line of (16), we use the binomial identity and reorder the summation. This completes the proof after relabelling summation indices.
Note that by rewriting the lower order terms as in Corollary 3.3 we clearly see that   (14) has only one term. We have Example 3.5 Consider m = 3. The sum in (14) has two terms. We have Here we used the fact that

Corollary 3.3 completes
Step 2 of our proof and allows us to transform (1) into where δ(t, D t , D x ) has symbol det(I m τ − A(t, ξ)) and B(t, D t , D x ) is given by (14).
with c h (t, ξ) homogeneous polynomial of order h with respect to ξ and therefore δ(t, D t , D x )I m is a decoupled system of m identical scalar differential operators of order m while B(t, D t , D x ) is a system of differential operators of order m − 1. As mentioned before, the c h (t, ξ) are the coefficients of the characteristic polynomial of A(t, ξ), see Appendix.

Step 3: Reduction to a first order system of pseudodifferential equations
We now transform the system in (17) into a system of pseudodifferential equations by following Taylor in [23]. More precisely, we transform each m-th order scalar equation in δ(t, D t , D x )I m into a first order pseudodifferential system in Sylvester form. In this way we obtain m systems with identical Sylvester matrix which can be put together in block-diagonal form obtaining a block-diagonal m 2 × m 2 matrix with m identical Sylvester blocks. The precise structure of the lower order terms will be worked out in the next subsection. To carry out this transformation, we set where the u i are the components of the original vector u in (1). We can rewrite the Cauchy problem for (17) as where the components U 0,i of the m 2 -column vector U 0 are given by and u is the solution of the Cauchy problem (1) with u(0, x) = u 0 . Passing now to analyse the matrices and the matrix B(t, D x ) is composed of m matrices of size m × m 2 as follows: Note that the entries of the matrices A(t, D x ) and B(t, D x ) are pseudodifferential operators of order 1 and 0, respectively.

Step 4: Structure of the matrix B(t, D x ) of the lower order terms
To analyse the structure of the (17) via the transformation (18). From Corollary 3.3 we have that (14). By the previously described transform (18), we obtain that and, thus, see that the coefficients b (22) will be associated to . . , m and so forth. In particular, we get that l i,m+( j−1)m (t, D x ) ≡ 0 for j = 1, . . . , m which is due to the fact that (1) is homogeneous. As a general formula for the non-zero elements of B(t, D x ), we can write To avoid further complication of the notation, we consider the b (l) i j (t, ξ) from now on as the by ξ l−m scaled elements in (23) if referenced as elements of B(t, ξ).
For the convenience of the reader, we conclude this section by illustrating the Steps 1-4 in the case m = 2 and m = 3. For simplicity, we take x ∈ R.

Steps 1-4 for m = 2
We consider the system Applying the corresponding operator to (24), we obtain and, thus, get the system (20), with the block Note that the entries of the matrix B i (t, D x ) can be obtained from (23) by setting h = 0 and j = 1, 2.

Steps 1-4 for m = 3
We consider and therefore Applying this operator to the original system, we obtain where we used the fact that adj(A) = A 2 + c 1 A + c 2 I 3 (see example Example 7.6) and set corresponding to (14). Now we introduce Thus, we obtain where A(t, D x ) is a block diagonal matrix with three blocks of the type Indeed, since .
Finally, the matrix B(t, D x ) is made of three blocks of 3 × 9 matrices for k = 1, 2, 3 and j = 1, 2. The elements b (1) k j and b (2) k j can are the scaled (k, j)elements of the matrices B 1 (t, D x ) and B 2 (t, D x ) from (26) respectively.

Energy estimate
Now we apply the Fourier transform with respect to x to the Cauchy problem in (19) and set F x→ξ (U )(t, ξ) =: V (t, ξ). We then obtain where V 0 = U 0 . From now on, we will concentrate on (28) and the matrix Note that by construction of A(t, ξ), the matrix A 0 (t, ξ) is made of m identical Sylvester type blocks with eigenvalues λ l (t, ξ), l = 1, . . . , m, where λ l (t, ξ) ξ , l = 1, . . . , m are the rescaled eigenvalues of the original matrix A(t, ξ) in (1).

Step 5: Computing the energy estimate
Let Q (m) ε (t, ξ) be the quasi-symmetriser of the matrix A 0 (t, ξ). By Remark 2.5 it will be a m 2 × m 2 block diagonal matrix with m identical blocks given by the quasisymmetriser Q (m) ε (t, ξ) of the defining block of A 0 (t, ξ) (see Sect. 2 for definition and properties). Hence, we define the energy where (·|·) denotes the scalar product in R m 2 . To improve the readability, we drop the dependencies on t and ξ in the following unless we find it important to stress. By direct computations we have It follows that By Proposition 2.1 it follows that Q (m) ε (t, ξ) is a family of C ∞ , non-negative Hermitian matrices such that In addition, by the same proposition, there exists a constant C m > 0 such that for all t ∈ [0, T ], ξ ∈ R n and ε ∈ (0, 1] the following estimates hold uniformly in V ∈ R m 2 : Finally, the hypothesis (2) on the eigenvalues and Proposition 2.3 ensure that the family Note that since the entries of the matrix A(t, ξ) in (1) are C ∞ with respect to t, the matrices A(t, ξ) and B(t, ξ) as well as the quasi-symmetriser have the same regularity properties.
We now proceed by estimating the three summands in the right-hand side of (29). Due to the block diagonal structure of the matrices involved we can make use of the proof strategy adopted for the scalar case in [12, Subsections 4.1, 4.2, 4.3].

First term
A block-wise application of Lemma 2.4 yields the estimate

Second term
From the property (31) we immediately have that

Third term
In this subsection, we treat the third term on the right-hand side of (29). By Proposition 2.1(iv) and the definition of the matrix B(t, ξ) we have that for all i = 1, . . . , m, due to the structure of zeros in B and in Q (m−1) ε (π i λ) . Thus, Since from Proposition 2.1(i) the quasi-symmetriser is made of non-negative matrices we have that It is purpose of the next section to find suitable Levi conditions on B(t, ξ) such that holds for some constant C 3 > 0 independent of t ∈ [0, T ], ξ ∈ R n and V ∈ C m 2 . We will then formulate these Levi-type conditions in terms of the matrix A in (1).

Estimates for the lower order terms
We remind the reader of the fact that the b (l) i j (t, ξ), if referenced as elements of B(t, ξ), are the by ξ l−m scaled (i, j)-elements of B l (t, ξ) in (14). See also Sect. 3.4 for details.
To start, we rewrite ((Q (m) It follows that we have that if holds true for some constant C > 0, independent of t, ξ and V , then estimate (32) will hold true as well.
In the sequel, for the sake of simplicity we will make use of the following notation: given f and g two real valued functions in the variable y, f (y) ≺ g(y) if there exists a constant C > 0 such that f (y) ≤ Cg(y) for all y. More precisely, we will set y = (t, ξ) or y = (t, ξ, V ). Thus, (33) can be rewritten as In analogy to the scalar case in [12] we will now focus on (33). Before proceeding with our general result, for advantage of the reader we will illustrate the main ideas leading to the Levi-type conditions on B in the case m = 2 and m = 3.

The case m = 2
For simplicity we take n = 1. From Sects. 3.5 and 2.1 we have that respectively. We have Thus, we obtain that |W (2) We now estimate the left-hand side of (34) from above and the right-hand side from below. We get and, by using the inequality |z 1 | 2 + |z 2 | 2 ≥ 1 2 |z 1 − z 2 | 2 , z 1 , z 2 ∈ C, and the condition (2) on the eigenvalues, Combining the last two inequalities, we finally obtain that |W (2) This is a Levi-type condition on the matrix of the lower order terms B written in terms of the entries of the original matrix A in (1). Note that by adopting the notations introduced in Sect. 3.6 for the matrix B in the case m = 2 as well, i.e., the Levi-type conditions above can be written as |b (1) where λ 2 1 + λ 2 2 is the entry q 11 of the symmetriser of the matrix A 0 = A ξ −1 .

The case m = 3
We begin by recalling that from Sect. 3.6 the 9 × 9 matrix B(t, ξ) is given by the 3 × 9 matrices B k (t, ξ), k = 1, 2, 3, as follows: and Note that W (3) B is a 9 × 9 matrix with three blocks of three identical rows and W (3) V is a 9 × 1 matrix with three blocks of rows having the same structure in λ 1 , λ 2 and λ 3 .
From (36), we deduce that Taking inspiration from the Levi conditions in [12] and in analogy with the case m = 2 we set |b (1) 2 are the entries q 11 and q 22 of the symmetriser of A 0 = ξ −1 A, respectively. By imposing these conditions on the lower order terms we have that Making a comparison with [12], we observe that V 1 , V 4 , and V 7 play the role of V 1 in [12] and V 2 , V 5 and V 8 play the role of V 2 in [12]. Finally, from (37), we obtain that It is our aim to prove that |W (3) BV | 2 ≺ |W (3) V | 2 . We do this by estimating |W (3) BV | 2 and |W (3) V | 2 in different zones. More precisely, inspired by [12] we decompose R 9 as for some δ 1 > 0. Estimate on δ 1 1 . By definition of the zone, we obtain from (39) Thanks to the hypothesis (2) on the eigenvalues, we have the following estimates 1 Note that in the previous bound from below we have taken in considerations only the terms with V 1 , V 2 and V 3 . Repeating the same arguments for the groups of terms with V 4 , V 5 , V 6 and V 7 , V 8 , V 9 , respectively, we get that Hence, Thus, combining the last estimate with (39), we obtain |W (3) BV | ≺ |W (3) V | for all V ∈ δ 1 1 . No assumptions have been made on δ 1 > 0. 1 Using Estimate on ( δ 1 1 ) c . By definition of the zone ( δ 1 1 ) c , we obtain from (39) that Further, by taking into considerations only the terms with V 1 , for some constant γ 1 , γ 2 > 0 suitably chosen. 2 The hypothesis (2) implies Applying the last inequality to (41), we obtain Now, repeating the same argument for the terms involving V 4 , V 5 , V 6 and V 7 , V 8 , V 9 , respectively, we get It follows that for all V ∈ ( δ 1 1 ) c the bound from below holds, provided that δ 1 is chosen large enough. Combining this with (40), we get |W (3) BV | ≺ |W V | on ( δ 1 1 ) c and, thus, on R 9 .

The general case
Recall from Sect. 3.4 that the m 2 × m 2 matrix B(t, ξ) is made up of m matrices of dimension m × m 2 that contain only in the last line non-zero elements, see (21). To not further complicate the notation, we will in what follows denote W (m) simply by W and will also assume that the b (l) i j (t, ξ) in B(t, ξ) are properly scaled by ξ l−m . For that see Sect. 3.4, specifically formula (23). Thus, we have The B i (t, ξ) are then given by We are now ready to prove the following theorem.
Note that if m = 2 no zone argument is needed to prove the theorem above (see Sect. 5.1) and when m = 3 just one zone is needed (see Sect. 5.2). The proof of Theorem 5.1 has the same structure as the proof of Theorem 5 in [12] and requires some auxiliary lemmas. all i and j with 1 ≤ i, j ≤ m and k = 1, ..., m − 1

Lemma 5.2 For
Proof The proof can be found in [12,Lemma3].
Proof The proof of this lemma follows by induction by applying Lemma 5.2 and can also be obtained by repeated application of Lemma 4 in [12] to the respective groups of V i .
Proof of Theorem 5.1. By the definition of B, we have that |WBV | 2 ≺ |W V | 2 is equivalent to Making use of the Levi-type conditions (43), we obtain On δ 1 1 , we further obtain the estimate Lemma 5.3 gives, setting k = 1 in (46) that This proves inequality (47) in δ 1 1 . Now, we assume that From the definition of the zones for 1 ≤ k ≤ h − 1 and δ k ≥ 1, we obtain as well as Continuing these estimates recursively, we obtain that h we get the following estimate of the left-hand side of (47): Now, we have to estimate the right-hand side of (47) on δ 1 We make use of Lemma 5.3 and of the bound (49). We obtain where the second inequality follows from |V j+lm | 2 which follows from |z 1 + · · · + z k | ≤ k k i=1 |z i | 2 . This yields estimate (47) on the zone The last step is assuming that V ∈ More precisely from the previous estimate we obtain m − 2 inequalities starting with (where we put h = 1 in (50)) and ending with (50)]. Using now the second of the inequalities, i.e. h = 2 in (50), on the right hand side of (51), we get Then using the remaining estimates for h = 3 to h = m − 2 recursively, we finally arrive at for any 1 ≤ j ≤ m − 2, δ h ≥ 1. From (52) and the Levi-type conditions we deduce that Using Lemma 5.3, we get The second term on the right-hand side of the last inequality can be estimated with (52) and we obtain

Well-posedness results
In this section we prove our main result: the well-posedness of the Cauchy problem (1). We formulate the following theorem by adopting the language and the notations of the previous sections concerning the lower order terms. A different formulation will be given in Theorem 6.2. Note that Theorems 6.1 and 6.2 correspond to Theorems 1.1 and 1.2, respectively.
x ∈ R n , be an m × m matrix of first order differential operators with C ∞ -coefficients. Let A(t, ξ) have real eigenvalues satisfying condition (2). Let be the Cauchy problem (1). Assume that the Cauchy problem (19), Proof We assume s > 1 since the case s = 1 is known thanks to see [18,20]. By the finite propagation speed for hyperbolic equations it is not restrictive to take compactly supported initial data and, therefore, to have the solution u compactly supported in x. Note that if u 0 ∈ γ s c (R n ) m then by deriving the system in (1) with respect to t we immediately have that D j t u(0, x) ∈ γ s c (R n ) m for j = 1, . . . , m − 1. It follows that if u solves (1) then U defined in (18) solves the Cauchy problem (19) with initial data U 0 ∈ γ s c (R n ) m 2 . We now prove that U ∈ C 1 ([0, T ], γ s (R n )) m 2 . This will allow us to conclude that u ∈ C 1 ([0, T ], γ s (R n )) m . We recall that the Cauchy problem (19) is given by the system where A(t, ξ) is a block Sylvester matrix with m identical blocks having the same eigenvalues of A(t, ξ). We make use of the energy E ε defined via the quasisymmetriser in Sect. 4. Combining the energy estimate (29) with the estimates of the first, second and third term in Sects. 4.2, 4.3 and 4.4, respectively, we get where K ε (t, ξ) is defined in Sect. 4.2, the bound from above holds for all k ≥ 1 and C 1 , C 2 , C 3 are positive constants. Note that in the estimate (53) we have used both the condition (2) on the eigenvalues and the Levi-type conditions (43). Thanks to the reduction to block Sylvester form that we have applied to obtain the Cauchy problem (19), we deal here with the same kind of energy employed in [12] for the scalar weakly hyperbolic equations of order m. The proof therefore continues as the proof of Theorem 6 in [12] with the only difference that k can be taken arbitrary. This is due to the fact that the coefficients of the matrix A(t, ξ) are C ∞ with respect to t. It follows, by working on the Fourier transform level, that U ∈ C 1 ([0, T ], γ s (R n )) m 2 and therefore u ∈ C 1 ([0, T ], γ s (R n )) m .
We now formulate Theorem 6.1 with an additional condition on the matrix A(t, ξ) which implies the Levi-type conditions (43). A(t, D x ), t ∈ [0, T ], x ∈ R n , be an m × m matrix of first order differential operators with C ∞ -coefficients. Let A have real eigenvalues satisfying condition (2) and let Q = (q i j ) be the symmetriser of A 0 = ξ −1 A. Assume that max k=1,...,m−1 for k = 1, 2, 3 and j = 1, 2. We can easily see on the matrix

Theorem 6.2 Let
that the conditions above on the entries of A entail for all t ∈ [0, T ] and k = 1, 2, i.e. condition (54) .
We now assume that the coefficients of the matrix A(t, ξ) are analytic with respect to t. We will prove that in this case the Cauchy problem (1) with the same Levi-type conditions employed above is C ∞ well-posed.
The proof of the C ∞ well-posedness follows very closely the arguments in [12]. Thus, we will only give a sketch with the differences and refer the reader to the cited work for more details. We begin by recalling a lemma on analytic functions whose proof can be found in [12] (see Lemma 5 in [12]). Lemma 6.6 Let f (t, ξ) be an analytic function in t ∈ [0, T ], continuous and homogeneous of order 0 in ξ ∈ R n . Then, Proof Thanks to the finite propagation speed property it is not restrictive to assume that the initial data have compact support. By Remark 2.5, the entries of the quasisymmetriser Q (m) ε (t, ξ) are analytic in t ∈ [0, T ] and, using Proposition 2.1, can be written as q ε,i j (t, ξ) = q 0,i j (t, ξ) + ε 2 q 1,i j (t, ξ) + · · · + ε 2(m−1) q m−1,i j (t, ξ).
We note that q ε,(i+hm)( j+hm) = q ε,i j , h = 0, . . . , m − 1 due to the block-diagonal structure of Q (m) ε (t, ξ). Since all functions on the right hand side of (55) are analytic, we can use Lemma 6.6 on each of them. Note that the partition (τ h(ξ ) ) in Lemma 6.6 can be chosen independent from ε. Now, following [12,22], we use a Kovalevskayan-type energy near the points τ h(ξ ) and a hyperbolic-type energy on the rest of the interval [0, T ] (see also [19]). We start with the interval [0, τ 1 ] (τ 1 = τ 1(ξ ) ), setting The estimate on [0, ε] ∪ [τ 1 − ε, τ 1 ] is standard and the details are left to the reader. We obtain, as in [12], where we used (31) [see (iii) in Proposition 2.1] and the Levi-type conditions (43) for |ξ | ≥ R to ensure that we have Using Proposition 2.1 and the Cauchy-Schwarz inequality, we obtain Together with Lemma 6.6, using the last two inequalities, we conclude that |∂ t q i j (t, ξ)| |q i j (t, ξ)| dt ≤ C log T ε for a certain positive constant C not depending on t and ξ . Thanks to the block diagonal form of the quasi-symmetriser, the proof now continues as the proof of Theorem 7 in [12]. This leads to the inequality |V (t, ξ)| ≤ c ξ N (ξ )(m−1) e N (ξ )C T ξ N (ξ )C T , obtained by setting ε = ξ −1 . Lemma 6.6 guarantees that the function N (ξ ) is bounded in ξ . Therefore, we can conclude that there exists a κ ∈ N, depending only on n, m, and T as well as a positive constant C > 0 such that for all t ∈ [0, T ] and |ξ | ≥ R. Clearly this estimate implies the C ∞ well-posedness of the Cauchy problem (1).

Remark 6.8
Since the entries of the matrix A are at least C ∞ with respect to t in both Theorems 6.1 and 6.7, from the system itself in (1) we obtain that the dependence in t of the solution u is actually not only C 1 but C ∞ .

Remark 6.9
In this paper we have studied homogeneous systems. Our method, described in the previous sections, can be generalised to non-homogeneous systems with some technical work on the lower order terms. Key point is to investigate the relation of the matrix of the lower order terms in the original system with the matrix B obtained after reduction to block Sylvester form.
Further information about the adjunct may be found in [16]. By a straightforward application of the Laplace expansion formula for determinants [16], one can prove the following proposition.

Remark 7.3
We note that the adjunct/cofactor of a matrix is not uniquely determined if the matrix is singular. Since we use only the relation (i), we mean by adj(A) a matrix associated to A that satisfies (i), specified by (59). For further details we refer to [1,24].
We recall that the elementary symmetric polynomials σ Note that formula (58) is just the well known Cayley-Hamilton theorem (see for instance [16]). The other two formulas follow from a variant of its proof. This result coincides with our computations in Sect. 3.6.