On \(C^\infty \) wellposedness of hyperbolic systems with multiplicities
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Abstract
In this paper, we study firstorder hyperbolic systems of any order with multiple characteristics (weakly hyperbolic) and timedependent analytic coefficients. The main question is when the Cauchy problem for such systems is wellposed in \(C^{\infty }\) and in \({\mathcal {D}}'\). We prove that the analyticity of the coefficients combined with suitable hypotheses on the eigenvalues guarantees the \(C^\infty \) wellposedness of the corresponding Cauchy problem.
Keywords
Hyperbolic equations \(C^\infty \) wellposedness Analytic coefficientsMathematics Subject Classification
Primary 35L25 35L40 Secondary 46F051 Introduction
At the same time, if we do not assume that all the eigenvalues are distinct, much less is known even if \(A(t,\xi )\) is analytic in t. For example, if we assume that the characteristics (even xdependent) are smooth and satisfy certain transversality relations, the \(C^{\infty }\)wellposedness was shown in [21]. However, in the case of only timedependent coefficients these transversality conditions are not satisfied.
Recently, different authors have studied weakly hyperbolic scalar equations with analytic coefficients (see, for instance [18] and [13]), but systems have not been fully investigated from this point of view. For a discussion on the \(C^\infty \) wellposedness of hyperbolic \(2\times 2\) systems and hyperbolic systems with nondegenerate characteristics, we refer the reader to Nishitani’s recent book [24].
Here, for the first time, we consider \(m\times m\) firstorder hyperbolic systems with analytic coefficients and multiple eigenvalues and we prove that under suitable conditions on the matrix A, formulated in terms of its eigenvalues, they are \(C^\infty \)wellposed, in the sense that given initial data in \(C^\infty \) the Cauchy problem (1) have a unique solution in \((C^1([0,T];C^\infty (\mathbb {R}^n))^m\).
Thus, it is the purpose of this paper to investigate under which conditions on the matrix A the solution u does actually belong to the space \(C^1([0,T];C^\infty (\mathbb {R}^n))^m\). The main idea is an extension to systems of the previous works on higherorder equations with analytic coefficients and lowerorder terms after a reduction to block Sylvester form.

First, we make an observation (Theorem 2.2) that the results of Yuzawa [29], and Kajitani and Yuzawa [20], can be extended to produce the existence of some (ultradistributional) solution to the Cauchy problem (1). It is then our task to improve its regularity to \(C^{\infty }\) or to \({\mathcal {D}}'\) depending on the regularity of the Cauchy data. This step is done in Sect. 2.1.

Second, we consider matrices \(A(t,D_{x})\) in Sylvester form and prove (in Theorem 2.5) that in this case the Cauchy problem (1) is wellposed in \(C^{\infty }\). This step is done in Sect. 2.2.
 Third, we extend the above to any weakly hyperbolic matrix A or, in other words, we show that we can drop the assumption of Sylvester form for the matrix A. This is done by transforming a general \(m\times m\) systeminto the \(m^{2}\,\times \,m^{2}\) block Sylvester system, which is a key idea of the paper, so that we could use the established result in that case. This extended system will be still hyperbolic (in fact, the principal part will have the same eigenvalues), but such reduction will (unfortunately) produce some lowerorder terms. Therefore, we carry out a careful analysis of the appearing matrix of the lowerorder terms by considering the suitable Kovalevskian and hyperbolic energies in different frequency domains. This will yield the desired \(C^\infty \)wellposedness as well as the distributional wellposedness for the original Cauchy problem (1) in Theorem 3.3. This analysis will be carried out in Sects. 3 and 4.$$\begin{aligned} D_tA(t,D_x) \end{aligned}$$
Finally, we note that in problems concerning systems, it is often important whether the system can be diagonalised or whether it contains Jordan blocks, see, e.g., [21] or [15], for some respective results and further references. However, this is not an issue for the present paper since we are able to obtain the wellposedness results avoiding such assumptions. We also note that ideas similar to those in this paper can be also applied in other situations for less regular coefficients, see, e.g., [14] and [26].
2 Preliminary results
In this section, we discuss several preliminary results needed for our analysis. First, we make an observation that the results of Yuzawa [29], and Kajitani and Yuzawa [20], can be extended to produce the existence of an ultradistributional solution, thus enabling our further reductions. Then, we look at systems in the Sylvester form.
2.1 Ultradistributional wellposedness
Lemma 2.1
 (i)
For any \(v\in {\mathcal {E}}'_{(s)}(\mathbb {R}^n)\) and \(l\in \mathbb {R}\) there exists \(\rho >0\) such that \(v\in H^l_{\Lambda (\rho ,s)}\).
 (ii)
If \(v\in H^l_{\Lambda (\rho ,s)}\) is compactly supported then \(v\in {\mathcal {E}}'_{(s)}(\mathbb {R}^n)\).
Proof
We can now recall the precise form of Kajitani–Yuzawa result described earlier.
Theorem 2.2
As a consequence of Lemma 2.1 and Theorem 2.2, we obtain the following ultradistributional wellposedness result which will be the starting point for our analysis.
Theorem 2.3
Under the hypotheses of Theorem 2.2 for any initial data g with entries in \({\mathcal {E}}'_{(s)}(\mathbb {R}^n)\), the Cauchy problem (1) has a unique ultradistributional solution \(u\in C^1([0,T];{\mathcal {D}}'_{(s)}(\mathbb {R}^n))^m\).
We now turn to a preliminary setting of Sylvester matrices.
2.2 Systems in Sylvester form
First, we briefly collect some preliminaries, for more details we refer the reader to [13, 18].
Let \(Q_j\) be the principal \(j\times j\) minor of Q obtained by removing the first \(mj\) rows and columns of Q and let \(\Delta _j\) be its determinant. When \(j=m\), we use the notations Q and \(\Delta \) instead of \(Q_m\) and \(\Delta _m\). The following proposition shows how the hyperbolicity of \(A_0\) (or equivalently of A) can be seen at the level of the symmetriser Q and of its minors (see [17]).
Proposition 2.4
 (i)
A is strictly hyperbolic if and only if \(\Delta _j>0\) for all \(j=1,\ldots ,m\).
 (ii)A is weakly hyperbolic if and only if there exists \(r<m\) such thatand \(\Delta _r>0,\ldots ,\Delta _1>0\). (In this case there are exactly r distinct roots).$$\begin{aligned} \Delta =\Delta _{m1}=\cdots =\Delta _{r+1}=0 \end{aligned}$$
 (i)
A is a matrix of pseudodifferential operators of order 1,
 (ii)
A is in Sylvester form.
Theorem 2.5
For simplicity, we will refer to the wellposedness above as \(C^\infty \) wellposedness and distributional wellposedness in the interval [0, T]. Note that by the energy estimates we obtain first that the solution is \(C^1\) with respect to \(t\in [0,T]\) and then, by iterated differentiation in the original system, we conclude that the dependence in t is actually \(C^\infty \).
3 Main result
Lemma 3.1
Proof
The representation formula in Lemma 3.1 implies the following estimate.
Proposition 3.2
We can now state our main result, which extends Theorem 2.5 to a general hyperbolic matrix A.
Theorem 3.3
Before proceeding with the energy estimate which will allow us to prove Theorem 3.3 we focus on the case \(m=2\). The following explanatory example will help the reader to better understand the meaning of the hypotheses (18) and (19).
3.1 Example: the case \(m=2\)
Some results on the \(C^\infty \) wellposedness for \(2\times 2\) hyperbolic systems with analytic coefficients have been obtained in [8]. Although not directly comparable, both types of conditions have their advantages from different points of view (for instance the formulation for any matrix size in our case).
4 Proof of the main theorem
We begin by recalling some technical lemmas which have been proved in [13] which will be useful for our analysis of systems as well.
Lemma 4.1
Lemma 4.2
Remark 4.3
It is clear that Lemma 4.1 and Lemma 4.2 are valid also for the block diagonal matrix \({\mathcal {A}}\) and the corresponding symmetriser \({\mathcal {Q}}\) as defined at the beginning of this section.
Lemma 4.4
 (i)
there exists \(X\subset \mathbb {S}^{n1}\) such that \(\Delta (t,\xi )\not \equiv 0\) in \((\delta , T'+\delta )\) for any \({\xi }\in X\) and the set \(\mathbb {S}^{n1}\setminus X\) is negligible with respect to the Hausdorff \((n1)\)measure;
 (ii)for any \([0,T]\subset (\delta , T'+\delta )\) there exist \(c_1,c_2>0\) and \(p,q\in {\mathbb N}_0\) such that for any \(\xi \in X\) and any \(\varepsilon \in (0,\mathrm {e}^{1}]\) there exists \(A_{\xi ,\varepsilon }\subset [a,b]\) such that:

\(A_{\xi ,\varepsilon }\) is a union of at most p disjoint intervals,

\(meas(A_{\xi ,\varepsilon })\le \varepsilon \),

\(\min _{t\in [0,T]\setminus A_{\xi ,\varepsilon }} \Delta (t,\xi )\ge c_1\varepsilon ^{2q}\Vert \Delta (\cdot ,\xi )\Vert _{L^\infty ([0,T])}\),
 $$\begin{aligned} \int _{t\in [0,T]\setminus A_{\xi ,\varepsilon }}\frac{\partial _t\Delta (t,\xi )}{\Delta (t,\xi )}\, dt\le c_2\log \frac{1}{\varepsilon }. \end{aligned}$$

We now define a Kovalevskian energy on \(A_{\xi /\xi ,\varepsilon }\) and a hyperbolic energy on the complement.
4.1 The Kovalevskian energy
4.2 The hyperbolic energy
4.3 Completion of the proof
We are now ready to prove Theorem 3.3.
Proof of Theorem 3.3
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