Propagation of polarization sets for systems of MHD type

Polarization sets were introduced by Dencker (1982) as a refinement of wavefront sets to the vector-valued case. He also clarified the propagation of polarization sets when the characteristic variety of the pseudodifferential system under study consists of two hypersurfaces intersecting tangentially (1992), or transversally (1995). In this paper, we consider the case of more than two intersecting characteristic hypersurfaces that are interesting transversally (and we give a note on the tangential case). Mainly, we consider two types of systems which we name"systems of generalized transverse type"and"systems of MHD type", and we show that we can get a result for the propagation of polarization set similar to Dencker's result for systems of transverse type. Furthermore, we give an application to the MHD equations.


Introduction
In [10], Hörmander defined the wavefront set of a distribution u, denoted by WF(u), which is a refinement of the singular support of a distribution.The wavefront sets does not only show the location of singularity, but also the direction in which the singularity occurs.Concerning the propagation of the wavefront sets, many results were given.For example, in [10] Hörmander gave the result for the propagation of the wavefront set for the solutions of partial differential equations, when considering the partial differential operator to be of real principal type, where he stated that the wavefront set is invariant under the bicharacteristic flow.Moreover, in [5], Dencker studied the propagation of singularities for pseudodifferential operator P on a smooth manifold X, having characteristics of variable multiplicity.He considered the characteristic set to be union of hypersurfaces S j , j = 1, ..., r 0 tangent at ∩ r0 j=1 S j .Under some assumptions he proved that the wavefront set of the solution of the considered pseudodifferential operator, is invariant under the union of the Hamilton flows on S j , j = 1, ..., r 0 , given that P u is smooth on X.
In [10], Hörmander defined locally the wavefront set of distributional sections u ∈ D ′ (X; E), where E → X is a vector bundle over the smooth manifold X.He defined the wavefront set of u locally as WF(u j ) where (u 1 , ..., u N ) are the components of u with respect to a local trivialization of E. However, this definition does not specify in which components u is singular, that is why Dencker defined in [2] the polarization set for vector-valued distribution u which we will denote by Pol(u).The polarization set also shows the location and the direction of the singularity as the wavefront set, but it additionally shows the most singular components of a distribution.Hence, the polarization set of a distribution is a refinement of the wavefront set, and the projection of Pol(u) \ 0 on the cotangent bundle T * X gives the wavefront set of u.Similarly, the H spolarization set is defined as a refinement of the H s -wavefront set, where H s denotes the usual Sobolev space.
In [2], Dencker defined systems of pseudodifferential operators of real principal type; note that the definition of systems of pseudodifferential operators of real principal type differs from the case of scalar pseudodifferential operators of real principal type, and he defined Hamilton orbits for systems of real principal type which are certain line bundles, and then he proved that the polarization set of a solution u of systems of real principal type P will be union of Hamilton orbits, given that P u is smooth.In [8], Gérard pointed out that the above result also holds for H s -polarization sets.
Moreover, in [6], Dencker considered pseudodifferential system having its characteristic set is union of two non-radial hypersurfaces intersecting tangentially at an involutive manifold of exactly order k 0 ≥ 1.He also assumed that the principal symbol vanishes of first order on the two-dimensional kernel at the intersection, and he assumed a Levi type of condition.Then, he defined systems satisfying these conditions to be systems of uniaxial type.Outside the intersection of the hypersurfaces the system will be of real principal type, hence the propagation result of the polarization set is already known there.In this article, Dencker has also proved a propagation result of the polarization set at the intersection.In [7], Dencker considered pseudodifferential system having its characteristic set is union of two non-radial hypersurfaces intersecting transversally at an involutive manifold of codimension 2.He also assumed that the principal symbol vanishes of first order on the two-dimensional kernel at the intersection.Systems satisfying these conditions are systems of transverse type.In this article, Dencker has proved a propagation result of the polarization set at the intersection.Outside the intersection the system is of real principal type.
We worked on extending Dencker's result stated above to pseudodifferential systems having their characteristic sets is union of several non-radial hypersurfaces intersecting transversally at an involutive manifold of codimension d 0 ≥ 2; not necessary just two hypersurfaces as in the case of systems of transverse type and systems of uniaxial type.Note that even if we assumed that the hypersurfaces are intersecting tangentially of exactly order k 0 ≥ 1 instead of intersecting transversally, we get a similar result, and for the proof we use the same weight and metric introduced by Dencker in [6] for the symbol classes S(ϑ, g) of the Weyl calculus.We have considered two cases for that: the first case is the case where we have r 0 hypersurfaces, and we assumed that the r 0 th-differential of the determinant of the principal symbol is different than zero at the intersection, and the ith-differential of the determinant of the principal symbol vanishes at the intersection for i < r 0 .Moreover, we assumed that the dimension of the kernel of the principal symbol to be r 0 at the intersection, and we assumed a condition similar to the Levi type condition considered in [6].We called systems satisfying the above conditions systems of generalized transverse type, and we proved that we have a similar propagation result of the polarization set as that for systems of transverse type.The second case, is the case where we also have r 0 hypersurfaces, and a condition similar to the Levi type condition considered in [6], but here we assumed that the (r 0 + 1)th-differential of the determinant of the principal symbol is different than zero at the intersection, and the ith-differential of the determinant of the principal symbol vanishes at the intersection for i < r 0 + 1.Moreover, we assumed that the dimension of the kernel of the principal symbol to be r 0 + 1 at the intersection.We also assumed some additional conditions that we did not assume in the case of systems of generalized transverse type.We defined systems satisfying these conditions to be systems of MHD type.We named them systems of MHD type because we have first noticed such systems when we considered the linearized ideal MHD equations.Thus we will have a section in which we study the propagation of polarization sets for the linearized ideal MHD equations.
In our work, we will assume that we have P ∈ Ψ m phg (X) an N × N system of pseudodifferential operators on a smooth manifold X of order m.Let p = σ(P ) be the principal symbol, det p the determinant of p, and Σ = (det p) −1 (0) the characteristics of P .We consider Σ to be union of several non-radial hypersurfaces intersecting transversally at an involutive manifold Σ 2 .Now, we state our main theorem in this paper regarding the propagation of polarization sets for systems of generalized transverse type, and systems of MHD type, but its proof will be postponed to Sections 3, and 4 to prove it for systems of generalized transverse type, and systems of MHD type respectively.Let be the regularity function.
Theorem 1.1.Let P ∈ Ψ m phg be an N × N system of generalized transverse type (or of MHD type) at ν 0 ∈ Σ 2 , and let A ∈ Ψ 0 phg be an N × N system such that the dimension of N A ∩ N P is equal to 1 at ν 0 , and The plan of this paper is as follows: in Section 2, we mention previous results on the propagation of polarization sets.More precisely, we will state Dencker's propagation result for systems of real principal type which was proven in [2], and Dencker's propagation result for systems of uniaxial type that was proven in [6].Also, we state Dencker's result for the propagation of polarization sets for systems of transverse type; see [7].Note that in [6] and [7], Dencker proved several results for the propagation of polarization sets under different conditions.Here we just mention the result which is similar to the result in our main theorem.In Sections 3 and 4, we define systems of generalized transverse type, and systems of MHD type, and we prove Theorem 1.1 for both types of systems.In Section 5, we give an application for the results in [9], and for the propagation of polarization sets for systems of MHD type, so we divide it into two subsections.First, we give the set of equations describing the ideal MHD, and we linearize it.In Section 5.1, we write the linearized ideal MHD equations in the form of a wave equation, and we give the characteristic variety of this wave equation under some assumptions which was calculated in [15].Then, we calculate the transport equation under these assumptions as an application to Hansen's and Röhrig's results in [9].In Section 5.2, we return to the linearized ideal MHD equations, and we calculate the eigenvalues and their multiplicities which are not constant.Then, we study the propagation of polarization sets, where we observe different cases, some in which our system is of real principal type, some in which our system is of uniaxial type, and one where our system is of MHD type.

Previous results
In this section, we state some previous results regarding the propagation of polarization sets.More precisely, we state the results for systems of real principal type, systems of uniaxial type, and systems of transverse type proven in [2], [6], and [7] respectively.First, we will state the definition of polarization sets given by Dencker in [2].
where N B = ker σ(B), and the intersection is taken over those 1 × N systems B ∈ Ψ 0 phg such that Bu ∈ C ∞ .The H s -polarization set, where H s is the usual Sobolev space is defined similarly.Definition 2.2.For u ∈ D ′ (X, C N ), the H s -polarization set is given by where N B = ker σ(B), and the intersection is taken over those 1 × N systems B ∈ Ψ 0 phg such that Bu ∈ H s .

Systems of real principal type
First, we want to state Dencker's result regarding the propagation of polarization sets for systems of real principal type.For the definition of real principal type, we will differentiate between two cases, the scalar case, and the case of system of pseudodifferential operators.
Definition 2.3.We say that P ∈ Ψ m (X) is of real principal type if the principal symbol σ(P ) = p is real and the Hamilton field H p = ∂ ξj p∂ xj − ∂ xj p∂ ξj is non-vanishing, and does not have the radial direction when p = 0. Definition 2.4 (Case of system of pseudodifferential operators).An N × N system P of pseudodifferential operators on X with principal symbol p(x, ξ) is of real principal type at (y, η) ∈ T * X \ 0 if there exists an N × N symbol p(x, ξ) such that p(x, ξ)p(x, ξ) = q(x, ξ) • Id N in a neighborhood of (y, η) where q(x, ξ) is a scalar symbol of real principal type and Id N is the identity in C N .Assume P (x, D) to be an N × N system of classical pseudodifferential operators on an n-dimensional smooth manifold X of order m.The symbol of P is an asymptotic sum of homogeneous terms: p(x, ξ) + p m−1 (x, ξ) + p m−2 (x, ξ) + ... where p is the principal symbol of P and p j is homogeneous of degree j.Assume P to be of real principal type, and let Σ = {(x, ξ) : det p(x, ξ) = 0} (2.3) be the characteristic set of P .To state the result of the propagation of polarization set given by Dencker in [2], we have to introduce first the connection he defined, and give the definition of the Hamilton orbit.Let where w is C ∞ function on T * X \ 0 with values in C N , {, } is the Poisson bracket, that is {p, p} = H pp, and p s m−1 is the subprincipal of P defined by D P is a connection on N P over Σ, that is, if w ∈ ker p at one point of a bicharacteristic of Σ, then D P w ∈ ker p along the bicharacteristic if and only if w ∈ ker p there.Hence, each parallel section (that is w such that D P w = 0) is uniquely determined by one point.D P depends on the choice of p and q, however Dencker showed that different choices of p and q only change the solution of D P w = 0 in N P by a scalar factor.This motivated him to define the following.
Definition 2.5 (Hamilton Orbit).A Hamilton orbit of a system P of real principal type is a line bundle L ⊆ N P | γ , where γ is an integral curve of the Hamilton field of Σ, and L is spanned by C ∞ section w satisfying D p w = 0.
Theorem 2.6 (Dencker's propagation result).Let P be an N × N system of pseudodifferential operators on a manifold X and let u ∈ D ′ (X, C N ).Assume that P is of real principal type at (y, η) ∈ Σ, and that (y, η) / ∈ WF(P u).Then, over a neighborhood of (y, η) in Σ, Pol(u) is a union of Hamilton orbits of P .
In [8], Gérard stated that we have similar propagation result for the H s -polarization sets for systems of real principal type.
Theorem 2.7.Let P be an N × N system of pseudodifferential operators on a manifold X of order m, and let u ∈ D ′ (X, C N ).Assume that P is of real principal type at (y, η) ∈ Σ, and that (y, η) / ∈ WF s (P u).Then over a neighborhood of (y, η) ∈ Σ, Pol s+m−1 (u) is a union of Hamilton orbits of P .

Systems of uniaxial type
Let P ∈ Ψ m phg (X) be an N × N system of classical pseudodifferential operators on a smooth manifold X.Let p be the principal symbol of P .Let Σ = (det p) −1 (0) be the characteristics of P , and let and Σ 1 = Σ \ Σ 2 .Assume that we have where S 1 and S 2 are non-radial hypersurfaces tangent at microlocally near ν 0 ∈ Σ 2 .This means that the Hamilton field of S j does not have the radial direction ξ, ∂ ξ , and it means also that the k 0 th jets of S 1 and S 2 coincide on Σ 2 , but no (k 0 + 1)th jet does.Note that we have P is of real principal type at Σ that is, p vanishes of first order on the kernel.We want to consider the limits of N P | Σ1 when we approach Σ 2 , so let T Σ2 Σ := T Σ2 S 1 = T Σ2 S 2 (note here Σ is not a manifold), and Here ∂Σ 1 is the normal bundle of Σ 2 in S 1 which is equal to the normal bundle of Σ 2 in S 2 .Let i 0 : Σ 2 → ∂Σ 1 denotes the zero section of ∂Σ 1 .By the tubular neighborhood theorem we know that there exists a diffeomorphism Φ from some neighborhood U ⊂ S j of Σ 2 to a neighborhood U 0 ∈ ∂Σ 1 of the zero section of ∂Σ 1 , and Φ identifies Σ 2 itself with the zero section.Before giving the definition of systems of uniaxial type, we need to give the definition of the limit polarizations.
Definition 2.8.For j = 1, 2, we define the limit polarizations where z k ∈ ker p(ν k ) and ∂N j P is conical in ξ and ρ, and homogeneous in the fiber, but it may have (complex) dimension > 1 at (ν, ρ).We assume that the fiber of This condition means that no element in N P | Σ2 can be the limit of polarization vectors on both characteristic surfaces, along the same direction.Dencker showed that (2.13) implies that ∂N j P is a complex line bundle over ∂Σ 1 \ (Σ 2 × 0), if we assume (2.7)-(2.10).Now, we give the definition of systems of uniaxial type.Definition 2.9.The system P is of uniaxial type at ν 0 ∈ Σ 2 , if (2.7)-(2.10)and (2.13) hold microlocally near ν 0 .
If P ∈ Ψ m phg is of uniaxial type and P u ∈ H r near ν ∈ Σ 1 , then we already know the result as Dencker showed in [2] that Pol r+m−1 (u) is a union of Hamilton orbits in N P near ν because P is of real principal type at Σ 1 .Now, we want to give Dencker's result for the propagation of the polarization set when we approach Σ 2 .Here S 1 and S 2 are tangent at Σ 2 , so their Hamilton fields are parallel on Σ 2 .Since Σ 2 is involutive, the Hamilton fields are tangent to Σ 2 .Therefore, Σ and Σ 2 are foliated by the bicharacteristics of Σ.Also, Dencker proved that ∂Σ 1 \ (Σ 2 × 0) is foliated by limit bicharactersitcs, which are liftings of bicharactersitics in Σ 2 , and that ∂N 1 P ∪ ∂N 2 P is foliated by limit Hamilton orbits, which are liftings of limits of Hamilton orbits, and are unique line bundles over limit bicharacteristics.
Theorem 2.10.Let P ∈ Ψ m phg be an N × N system of uniaxial type at ν 0 ∈ Σ 2 , and let A ∈ Ψ 0 phg be a 1 × N system such that the dimension of Moreover, in [6], Dencker showed under what assumptions we get Pol r (u) is union of limits of Hamilton orbits in N A ∩ N P near ν 0 ∈ Σ 2 .

Systems of transverse type
Finally, we want to state Dencker's result regarding the propagation of polarization sets for systems of transverse type; see [7].Let P ∈ Ψ m phg be an N × N system of classical pseudodifferential operators on a smooth manifold X, p = σ(P ) be the principal symbol, and Σ = (det p) −1 (0) be the characteristics of P .Let and Σ 1 = Σ \ Σ 2 .For systems of transverse type we have Σ is a union of two non radial hypersurfaces intersecting transversally at Σ 2 .More precisely, the systems of transverse type is defined as the following Definition 2.11.The system P is of transverse type at det p = e • q, where e = 0 and q is real valued with Hessian having rank 2 and positivity 1, (2.16) dim ker p = 2 on Σ 2 , (2.17) microlocally near ν 0 .
Similar to the case of systems of uniaxial type, if P ∈ Ψ m phg is of transverse type and P u ∈ H r near ν ∈ Σ 1 , then P is of real principal type at ν. Let N j P be as in (2.11).In [7], Dencker modified slightly the definition of limit polarizations.Definition 2.12.For j = 1, 2, the limit polarizations is defined by where z k ∈ ker p(ν k ) and ∂N j P is conical in ξ and linear in the fibers.Dencker showed that ∂N j P is a C ∞ line bundle over Σ 2 , j = 1, 2, and that Here, S 1 and S 2 are transverse at Σ 2 , so their Hamilton fields are non-parallel on Σ 2 .Since Σ 2 is involutive of codimension 2, the Hamilton fields of S j are tangent to Σ 2 and generate the two-dimensional foliation of Σ 2 .Moreover, ∂N j P is foliated by limit Hamilton orbits which are limits of Hamilton orbits in N j P , and are unique line bundles over bicharacteristics in S j at Σ 2 for j = 1, 2.
Theorem 2.13.Let P ∈ Ψ m phg be an N ×N system of transverse type at ν 0 ∈ Σ 2 , and let A ∈ Ψ 0 phg be a 1×N system such that the dimension of N A ∩ N P is equal to 1 at ν 0 , and M A = π 1 (N A ∩ N P \ 0) is a hypersurface near ν 0 .Assume that u ∈ D ′ (X, C N ) such that P u ∈ H r−m+1 and Au ∈ H r at ν 0 .Then Pol r (u) is a union of (limit) Hamilton orbits in N A ∩ N P .Here π 1 : T * X × C N → T * X is the projection along the fibers.
Note that in this case M A = S j for some j, and N A ∩ N P is a union of (limit) Hamilton orbits.

Propagation of polarization sets for systems of generalized transverse type
In this section, we generalize Dencker's result stated in Section 2.3 by considering the system to have its characteristic set is union of r 0 hypersurfaces intersecting transversally at an involutive manifold of codimension d 0 ≥ 2, with r 0 ≥ 2. Let P ∈ Ψ m phg (X) be an N × N system of classical pseudodifferential operators on a smooth manifold X.Let p = σ(P ) be the principal symbol and Σ = (det p) −1 (0) the characteristic set.Assume microlocally near (x 0 , ξ 0 ) ∈ Σ that Σ = r0 j=1 S j , r 0 ≥ 2, where S j are non-radial hypersurfaces intersecting transversally at Moreover, assume that the dimension of the fiber of N P is equal to and We want to consider the limits of N P | Σ1 when we approach Σ 2 , so let T Σ2 Σ := r0 j=1 T Σ2 S j (note that Σ is not a manifold), and ∂Σ 1 := T Σ2 Σ/T Σ 2 .Before giving the definition of systems of generalized transverse type, we need to give the definition of the limit polarizations.Definition 3.1.For j = 1, ..., r 0 , we define the limit polarizations where ∂N j P is conical in ξ and ρ, and homogeneous in the fiber.We will assume that the fiber of This condition means that no element in N P | Σ2 can be the limit of polarization vectors on all characteristic surfaces along the same direction.
Proposition 3.3.Let P ∈ Ψ 1 phg be an N × N system of generalized transverse type at ν 0 ∈ Σ 2 .Then by choosing suitable symplectic coordinates, we may assume that X = R × R n−1 , ν 0 = (0; (0, ..., 1)), and microlocally near ν 0 .Here β j are real and homogeneous of degree 1 in ξ; with where . By conjugating P with an elliptic, scalar Fourier integral operators, and multiplying with elliptic N × N systems of order 0, we may assume that microlocally near ν 0 , where E ∈ Ψ 1 phg is an elliptic (N − r 0 ) × (N − r 0 ) system and ..,β r0 as eigenvalues.Proof.We will prove it in a way similar to how Dencker proved the normal form for systems of uniaxial type.Since the result is local, we may assume X = R n .Because Σ 2 is involutive, we may choose symplectic, homogeneous coordinates (x, ξ) ∈ T * R n near ν 0 ∈ Σ 2 , so that ν 0 = (0; (0, ..., 1)) and where ξ = (ξ ′ , ξ ′′ ) ∈ R d0 × R n−d0 .We may also assume near ν 0 .Now, we rename x 1 = t, (x 2 , .., x d0 ) = x ′ , and (x d0+1 , ..., x n ) = x ′′ .Since S j is intersecting transversally with S 1 at Σ 2 , we obtain with β j real and homogeneous of degree 1 in ξ, β 1 ≡ 0, and in a conical neighborhood of ν 0 .By taking k = 1, we obtain Using that dim N P = r 0 at Σ 2 , we can find an N × N elliptic matrix b homogeneous of degree 0 in the ξ variables which maps Im p to {z ∈ C N ; z j = 0, j ≤ r 0 } over Σ 2 near ν 0 , and we can choose an N × N matrix a homogeneous of degree 0 in the ξ variables such that a −1 maps ker p onto {z ∈ C N ; z j = 0, j > r 0 } over Σ 2 near ν 0 .where its principal symbol is given by (3.16).As E is a system of order 1 which is elliptic at Σ 2 , choose J to be its microlocal parametrix of order −1.Multiply BP A from the left with Multiply also B 1 BP A from the right by Hence, we get microlocally near ν 0 , where E ∈ Ψ 1 phg is an elliptic (N − r 0 ) × (N − r 0 ) system.If f is the principal symbol for F , then conditions (3.4) and (3.7) imply , and homogeneity, we may find homogeneous system C 0 ∈ S 0 such that where det C 0 = 0 at Σ 2 .By multiplication with an elliptic system, we may assume C 0 ≡ Id r0 .Thus, det f = τ r0 i=2 (τ + β j ), which implies that k(t, x, ξ) has the eigenvalues 0, β 2 ,...,β r0 .If f 0 ∈ S 0 is the term homogeneous of degree 0 in the expansion of F , then Theorem A.4 in [6], and homogeneity give where B 0 ∈ C ∞ (R, S 0 ) is independent of τ , and B −1 ∈ S −1 .By multiplying f with an operator with symbol Id r0 −B −1 , we may assume B −1 ≡ 0. By induction over lower order terms we obtain (3.10).
We want to introduce symbol classes adapted to the functions β j defined in (3.8) for j = 2, ..., r 0 .Let where and h 2 = sup g/g σ = ξ ′ −2 .We get that g is σ temperate, and β j ∈ S(ϑ, g).Check [12, Chapter XVIII] to know more about the symbol classes S(ϑ, g) of the Weyl calculus.Moreover, using Taylor's formula we can write with a j ∈ S 0 is homogeneous of degree 0 in ξ.
It is easy to see that the condition (3.6) is satisfied.
We will introduce two spaces that we are going to use.These spaces were also given by Dencker in [7] and he showed the relation between these two spaces by a lemma that we will also state.Let H r,s be the space of u ∈ S ′ (here u depends on t and x) satisfying where u 1 ∈ H r,s and ν / ∈ WF(u 2 ).Similarly, when we have u depends only on x then the norm becomes We have S 0 ⊂ S(1, g).Note that the spaces H r,s is a particular case of the spaces B p,k introduced by Hörmander; check [11], where p = 2 and k(τ, ξ) = τ, ξ r τ, ξ ′ s .
Proposition 3.5.Assume that P is an r 0 × r 0 system of pseudodifferential operators of order 1 on R n , on the form ) and assume P u ∈ H r,s at ν 0 .Then, for every δ > 0 we can find c δ and C δ,N > 0 and v δ ∈ H r,s+1 at ν 0 , such that ) Proof.Follow the proof of Proposition A.1 in [7].
Let H r,s * be the Banach space of u ∈ S ′ , satisfying If we have u depends only on x then the norm becomes Hence, u * r,s = u r,s when u depends only on x.If u ∈ H * r,s then we get u| t=ρ ∈ H r,s for almost all ρ, by Fubini's theorem.If u ∈ S ′ satisfies (3.37), then where s ± = max(±s, 0).Hence, we lose only O(δ) derivatives when taking restriction to {t = r}, for almost all r.
Note that H r = H r,0 is the usual Sobolev space.Changing the notation, let and ξ are weights for the metric g defined by (3.45) Let Ψ r,s = Op S r,s be the corresponding pseudodifferential operators, which maps H r,s into L 2 .Returning to the old notation where using t instead of x 1 , and assume that P ∈ C ∞ (R, Op(S(ϑ, g))) be as in (3.27), we get P ∈ Ψ 0,1 .In order to prove the main theorem, we need to study the regularity of a Cauchy problem that we will state.Consider the following N × N system Here q is a scalar operator with symbol where β j ∈ S(ϑ, g) is homogeneous and satisfies (3.8).We will assume with A i k ′ ∈ Op S(ϑ k ′ , g).We are going to study the following Cauchy Problem: (3.49) We are going to assume that the ξ = 0 in WF(u).Hence, the restrictions are well defined.
Proof.We are going to prove it in a similar way as the proof of Lemma 5.2 in [4].As k is diagonalizable in S(1, g), we get that k = r0 j=1 β j π j , with β 1 = 0, where π j (t, x, ξ) ∈ S(1, g) is the projection on the eigenvectors corresponding to the eigenvalues β j along the others when ξ ′ = 0. k is symmetrizable with symmetrizer M = π * j π j , that is M > 0 amd M k is symmetric.Note that k is symmetrizable means there exists symmetric N × N system M (t, x, ξ) ∈ S(1, g) such that 0 < c ≤ M and M k − (M k) * ∈ S(1, g).If we put V (t) to be the L 2 norm in the x-variables, depending on t, and we put (3.57)By Grönwall's inequality we get, for bounded t, so (3.56) and integration gives the result.
The above proposition is similar to Proposition 7.2 in [6].To prove Proposition 7.2 in [6], Dencker used the parametrix constructed in [5] for P = D t Id N +K, where K ∈ Op S(ϑ, g) has principal symbol k satisfying the conditions in Proposition 7.2, with ϑ, and g are given as (3.8), and (3.9) in [6] respectively, and he used the microlocal uniqueness; see [5].In our case; case of generalized transverse type, we are using different weight and metric, but we can still construct a parametrix for the N 0 × N 0 matrix, where K ∈ Op S(ϑ, g) has principal symbol k satisfying the conditions in Proposition 3.10, and we can prove microlocal uniqueness as in [5].The steps are similar to that in [5], except some details are changed.Thus, we will not write about the construction of the parametrix and the microlocal uniqueness and we invite the readers to check [5].
Proof of Proposition 3.10.The argument is all the same as in the proof of Proposition 7.2 in [6], except that to get U ∈ C ∞ (R, D ′ (R n−1 , C N )) ∩ L 2 , we need to prove (5.6) in [4] for our case, which we proved in Lemma 3.9.For the parametrix and the microlocal uniqueness part we have mentioned above that it can be proven in our case in a similar way.
Note that here S j are transverse at Σ 2 , so their Hamiton fields are non-parallel on Σ 2 .That is why we considered in the main theorem, Theorem 1.1, that M A is a hypersurface near ν 0 , so one can consider M A = S j for some j.Now, we will prove Theorem 1.1 for the case of systems of generalized transverse type.

Propagation of polarization sets for systems of MHD type
In this section, we define systems of MHD type which are also systems having their characteristic sets are union of r 0 hypersufaces intersecting transversally at an involutive manifold of codimension d 0 ≥ 2 and r 0 ≥ 2. However, they satisfy some assumptions different than that in case of systems of generalized transverse type.We named them systems of MHD type because we first noticed these types of systems when we considered studying the propagation of polarization sets of the linearized ideal MHD equations.Let P ∈ Ψ m phg (X) be an N × N system of classical pseudodifferential operators of order m on a smooth manifold X.Let p = σ(P ) be the principal symbol of P , det p the determinant of p and Σ = (det p) −1 (0) the characteristic set of P .Assume microlocally near ν 0 = (x 0 , ξ 0 ) ∈ Σ, that Σ =  We are interested in finding the propagation of polarization set at Σ 2 , as in our application; see Section 5, we know the result on Σ \ Σ 2 .We assume that Σ 2 is an involutive manifold of codimension d 0 ≥ 2, (4.2) d j (det p) = 0 for j ≤ r 0 and d r0+1 (det p) = 0 at Σ 2 .Moreover, assume that t P co the adjugate matrix of P can be written as with R being a scalar pseudodifferential operator of order m, with σ(R) vanishing on S i0 to the first order.L 1 , and L 2 are N × N system of pseudodifferential operators of order m(N − 2), and m(N − 3) respectively.Assume also that with Σ = {f = 0}, and df = 0 at S i \ Σ 2 for i = {1, ..., r 0 }.We are using same notation as the previous section.
We want to write systems of MHD in a normal form.
We use the same weight and metric introduced in the previous section; see (3.24), and (3.25).Thus, we have β j ∈ S(ϑ, g).

Application
Magnetohydrodynamics, or MHD couples Maxwell's equations with hydrodynamics to describe the behavior of electrically conducting fluids under the influence of electromagnetic fields.In this section, we want to consider the simplest form of MHD, which is the Ideal MHD to study the propagation of polarization set of the solution of the linearized ideal MHD equations.Ideal MHD, assumes that the fluid has so little resistivity that it can be treated as a perfect conductor.See [15] to know more about MHD equations.
We will show, under some assumptions, that the linearized ideal MHD equations is of real principal type.As we mentioned before, systems of real principal type were defined by Dencker; see [2], who studied the propagation of their solutions, and showed that the propagation of polarization sets is governed by a certain connection on sections of the kernel subbundle, ker p 2 where p 2 is the principal symbol of the system.In [9], Hansen and Röhrig merged the theory of real principal type systems with the calculus of Fourier integral operators and constructed a Fourier integral solution for system of real principal type, and derived a transport equation for the principal symbol of this solution (note that disregarding half densities this transport equation is the connection introduced by Dencker).
We will divide this section into two subsections: first, we introduce the ideal MHD equations and its linearization.In Section 5.1, we write the linearized ideal MHD equations in the form of a wave equation P β = 0 where β is the displacement vector and P is a second order 3 × 3 system; see [15, Lecture 20], and we show that under some assumptions, the characteristic variety of P is disjoint union of the Shear Alfvén wave, the slow magnetosonic wave and the fast magnetosonic wave; see [15,Lecture 24].Moreover, we show that, under the considered assumptions, P is of real principal type and we calculate the transport equation on Char P , as an application to the result in [9].In Section 5.2, we return to the linearized ideal MHD equations, and we study the propagation of polarization sets.It turns out that we can consider different cases, some in which we have our system is of real principal type, some in which our system is of uniaxial type, and we have a case where our system is of MHD type.
The set of equations describing the ideal MHD are where ρ, p ∈ R denotes the density and the pressure respectively.u ∈ R 3 is the fluid velocity, H ∈ R 3 is the magnetic field, and γ is the adiabatic index, see [15,Lecture 20].Assuming a stationary equilibrium the linearized equations of (5.1) about (ρ, H, p) is: ) where (ρ, H, p) are the values in the equilibrium state (that is the solutions of the Ideal MHD equations when ∂/∂t = 0, and as we assumed stationary equilibrium we have u = 0).Note that we used that which we get from the stationary equilibrium assumption.

The ideal MHD wave equation and the transport equation
We can write the linearized ideal MHD equations (5.2) in the form of a wave equation; see [15,Lecture 20].Consider the displacement β to be defined by u = ∂β ∂t .
(5.4) Substituting (5.4) in (5.2a), (5.2c), and (5.2d) respectively, and integrating with respect to t we get Replace now (5.5b), (5.5c), and (5.4) in (5.2b) to get Equation (5.6) is the ideal MHD wave equation.Consider from now on P where (5.7) Now, we want to give the Characteristic variety of P under some assumptions; for this part we refer to [15, Lectures 23 and 24].
Proof.We have with p 2 is the principal symbol of P .Considering v = H/ √ ρ, equation (5.9) can be written as Without loss of generality, let v = |H|/ √ ρ êz , ξ = ξ ⊥ êx + ξ êz with ξ 2 = ξ 2 ⊥ + ξ 2 , and β = β x êx + β y êy + β z êz with êx , êy , and êz are unit vectors that points in the direction of the x-axis, y-axis, and z-axis respectively.Substituting this in equation (5.10), we find x-component: y-component: z-component: (5.11c) Notice that the y-component decouples from the x-and z-components.This immediately gives This is the shear Alfvén wave.The characteristic equation for the coupled x-and z-component is Hence, we get (5.14) Still we want to prove that {q 1 = 0}, {q 2 = 0}, and {q 3 = 0} are disjoint.Dividing (5.13) by ρ 2 , it can be written as . Suppose that the conditions of Lemma 5.1 are satisfied.Now, we are interested in calculating the transport equation as in [9].The full symbol of P is p 2 + p 1 + p 0 , with p 2 is the principal symbol of P homogeneous of degree 2, and p 1 and p 0 are homogeneous terms of degree 1 and 0 respectively.One can check that the principal symbol of P is (5.17) and Using (5.3), we get (5.18) One can check (5.17), (5.18), and the calculations given below by using "mathematica" for example.Let Γ 1 , Γ 2 , and Γ 3 be disjoint conic neighborhoods of {q 1 = 0}, {q 2 = 0}, and {q 3 = 0} respectively.Set q = q 1 in Γ 1 , q = q 2 in Γ 2 , and q = q 3 in Γ 3 .Proposition 5.2.P is of real principal type with respect to the Hamilton field H q of Char P = {q = 0}.
Proof.We have q 1 , q 2 , and q 3 are scalar real principal type.Let In Γ 1 we take to get p2 p 2 = q 1 Id 3 .In Γ 2 we take to get p2 p 2 = q 2 Id 3 .In Γ 3 we take to get p2 p 2 = q 3 Id 3 .
In what follows, let X = R × R 3 , Λ ⊂ T * X \ 0 be a closed Lagrangian submanifold of the characteristic set of P , and let Ω 1/2 Λ denote the half-density bundle of Λ. S µ+1 (Λ, (Ω 1/2 Λ ) 3 ) is the space of symbols of the space of Lagrangian distributions I µ (X, Λ; (Ω 1/2 X ) 3 ); see [13,Section 25.1] to read more about Lagrangian distributions.From [9], we know that there is a first order differential operator T P,Hq on Λ, uniquely determined by P and H q which maps a a 3-vector of half densities with p 2 a = 0 to 3-vector of half densities where T P,Hq a = L Hq a + 1 2 {p 2 , p 2 }a + ip 2 p s a. (5.30) Here is the subprincipal symbol of P , L Hq is the Lie derivative with respect to H q , and {.} is the Poisson bracket.
Proof.Differentiating p 2 we get Therefore, the subprincipal is given as follows: We have 2ip 2 p s π = 2iπp s π on Char P.
(5.32) Using that 2ip s is a 3 × 3 skew-symmetric matrix with zero entries on the diagonal, and π is a symmetric matrix we get that 2iπp s π is a 3 × 3 skew-symmetric matrix.Therefore to prove that it vanishes, it suffices to show that its rank is < 2. Since π is projection we have rank π =trace π.Calculating the trace of π we get that rank π=1 and hence we proved the lemma.
The proof of this lemma is very similar to the explanation given in [14, Appendix A] except here we have the additional eigenvalue λ 4 = 0. Also, here we will not state all the eigenspaces as in [14].
Note: As an application for systems of generalized transverse type, one can consider the linearized isentropic MHD equations, which is 7 × 7 matrix; check [14,Appendic A] where the first order term of the linearized isentropic MHD equations and its eigenvalues are given, and then we can easily check the type of the system as we did in this section for linearized ideal MHD equations.

S 2
j , r 0 ≥ 2, where S j are non-radial hypersurfaces intersecting transversally at Σ