Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

In this paper, we study the Cauchy problem for the Landau Hamiltonian wave equation, with time-dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional-type solutions under conditions when such solutions also exist.


Introduction
The purpose of this paper is to establish the well-posedness results for the wave equation for the Landau Hamiltonian with irregular electromagnetic field and similarly irregular velocity. We are especially interested in distributional irregularities appearing, for example, when modelling electric shocks by δ-function type behaviour. While this leads to fundamental mathematical difficulties for the usual distributional interpretation of the equation (and of the Cauchy problem) due to impossibility of multiplication of distributions (see Schwartz [38]) we are able to establish the wellposedness using a notion of very weak solutions introduced in [15] in the context of space-invariant hyperbolic problems. This notion also allows us to recapture the classical/distributional solution to the Cauchy problem for the Landau Hamiltonian under conditions when it does exist.
Thus, we consider a non-relativistic particle with mass m and electric charge e moving in a given electromagnetic field. We concentrate on the 2D version and then indicate in Sect. 8 the changes for the multidimensional case in R 2d . To describe the electromagnetic field in the plane, one usually uses the electromagnetic scalar and vector potentials q, A. In a brief description of the physical model below, we follow [2] which we can also refer for some more details. The dynamics of a particle with mass m and charge e on the Euclidean x y-plane interacting with a perpendicular homogeneous electromagnetic field is determined by the Hamiltonian (see [22]) where h denotes Planck's constant, c is the speed of light and i the imaginary unit (see also Sect. 8). We denote by 2B > 0 the strength of the magnetic field, and can choose the symmetric gauge given by A = (−By, Bx).
As usual for mathematical models, we set m = e = c = h = 1 in (1.1), which leads to the Landau Hamiltonian acting on the Hilbert space L 2 (R 2 ). It is well known (see [13,21]) that the spectrum of the operator H consists of infinite number of eigenvalues with infinite multiplicity of the form λ n = (2n + 1)B, n = 0, 1, 2, . . . . (1.3) These eigenvalues are called the Euclidean Landau levels. We denote the eigenspace of H corresponding to the eigenvalue λ n in (1. 3
In this paper, we are interested in the wave equation for the Landau Hamiltonian with time-dependent irregular electric potential and varying in time electromagnetic field. More precisely, for a distributional propagation speed function a = a(t) ≥ 0 and for the distributional electromagnetic scalar potential q = q(t), we consider the Cauchy problem for the Landau Hamiltonian H in the form (1.7) A special feature of our analysis is that we want to allow a and q to be distributions. For instance, if the electric potential produces shocks, these can be modelled with δ-distributions; for example, by taking q = δ 1 , the δ-distribution at time t = 1. Moreover, if the velocity a(t) also contained δ-type terms, as an example of such an equation we could consider (1.8) Moreover, we could also look at discontinuous speeds given by, e.g. the Heaviside function h(t) such that h(t) = 1 for t < 1 and h(t) = 2 for t ≥ 1, and singular electric fields, e.g. q(t) = δ 1 + h(t), in which case the Cauchy problem (1.7) would take the form (1.9) The physical problem that we are interested in is as follows: How to understand the Cauchy problems (1.7)-(1.9) and their well-posedness? There are several difficulties already at the fundamental level for such problems: first of all, in general, impossibility of multiplying distributions due to the famous Schwartz' impossibility result [38]. Second, even if we could somehow make sense of the product aq being a distribution by, e.g. imposing wave front conditions, we would still have to multiply it with u(t, x) which, a priori, may also have singularities in t, thus leading to another multiplication problem. Moreover, another difficulty [for the global in space analysis of (1.7)] is that the coefficients of H increase in space thus leading to potential problems at infinity if we treat the problem only locally.
In our analysis, we assume that a is a positive distribution so that the Cauchy problem (1.7) is of hyperbolic type, at least when a and q are regular. More precisely, we will assume that there exists a constant a 0 > 0 such that where a ≥ a 0 means that a − a 0 ≥ 0, i.e. a − a 0 , ψ ≥ 0 for all ψ ∈ C ∞ 0 (R), ψ ≥ 0. Incidentally, the structure theory of distributions implies that a is a Radon measure but this does not remove the multiplication problems or problem with understanding the meaning of the well-posedness of the Cauchy problem (1.7).
Nevertheless, we are able to study the well-posedness of (1.7) using an adaptation of the notion of very weak solutions introduced in [15] in the context of hyperbolic problems with distributional coefficients in R n .
As noted, the Eq. (1.7) cannot be, in general, understood distributionally, so we are forced to weaken the notion of solutions. However, we want to do it in a way so that we can recapture classical solutions should they exist. Thus, in this paper we will show the following facts: • The Cauchy problem (1.7) admits a very weak solution even for distributional-type Cauchy data u 0 and u 1 . The very weak solution is unique in an appropriate sense. • If the coefficients a and q are regular so that the Cauchy problem (1.7) has a 'classical' solution, the very weak solution recaptures this classical solution in the limit of the regularising parameter. This shows that the introduced notion of a very weak solution is consistent with classical solutions should the latter exist.
• When the classical solution does not exist, the very weak solution comes with an explicit numerical scheme modelling the limiting behaviour of regularised solutions.
We also note that at the same time our analysis will yield results for the modified problem for distributions a, q with a ≥ a 0 > 0 for some constant a 0 .
For second-order operators H independent of x, the Cauchy problems of this type have been intensively studied; however, for more regular (starting from Hölder) coefficients, see for example [5][6][7][8]10] and references therein. For the setting of distributional coefficients, see [15].
The analysis of this paper is different from the one in [15] that was adapted to constant coefficients in R n . At the same time, the techniques of the present paper may be extended to treat more general operators; however, since such analysis is more abstract and requires more background material, it will appear elsewhere.
The description of appearing function spaces is carried out in the spirit of [11] using the general development of nonharmonic type analysis carried out by the authors in [33] which is, however, 'harmonic' in the present setting. The treatment of the global wellposedness in the appearing function spaces is an extension of the method developed in [14] in the context of compact Lie groups.
In Sect. 7, we show an extension of the construction to also consider the inhomogeneous wave equation The structure of the paper is as follows. In Sect. 2, we formulate our main results. In Sect. 3, we discuss elements of the global Fourier analysis associated to the Landau Hamiltonian as a special case of abstract constructions that have been developed in [33]. In Sect. 4, we prove Theorem 2.1 and in Sect. 5 we prove Theorem 2.4. In Sect. 6, we establish the uniqueness of very weak solutions and their consistence with 'classical' solutions when they exist. In Sect. 7, we give an extension of our constructions to the inhomogeneous wave equation. In Sect. 8, we discuss an extension to higher dimensions, namely to the Landau Hamiltonian in R 2d .

The main results
In our results below, concerning the Cauchy problem (1.7), as the preliminary step we first carry out the analysis in the strictly hyperbolic regular case with the norm f H s The same result is true also for the Cauchy problem (1.10).
Anticipating the material of the next section, using Plancherel's identity (3.5), in our case we can express the Sobolev norm as with e j ξ as in (1.6). In Theorem 2.1, the assumption of q being real valued is actually enough to assure the well-posedness; however, we assume that q ≥ 0 to facilitate the proofs of the distributional results later.
We now describe the notion of very weak solutions and formulate the corresponding results for distributions a, q ∈ D ([0, T ]). The first main idea is to start from the distributional coefficient a to regularise it by convolution with a suitable mollifier ψ obtaining families of smooth functions (a ε ) ε , namely and ω(ε) is a positive function converging to 0 as ε → 0 to be chosen later. Here, ψ is a Friedrichs-mollifier, i.e. ψ ∈ C ∞ 0 (R), ψ ≥ 0 and ψ = 1. It turns out that the net (a ε ) ε is C ∞ -moderate, in the sense that its C ∞ -seminorms can be estimated by a negative power of ε. More precisely, we will make use of the following notions of moderateness.
In the sequel, the notation K R means that K is a compact set in R.

Definition 2.2 (i) A net of functions
We note that the conditions of moderateness are natural in the sense that regularisations of distributions are moderate, namely we can regard by the structure theorems for distributions. Thus, while a solution to the Cauchy problems may not exist in the space of distributions on the left-hand side of (2.4), it may still exist (in a certain appropriate sense) in the space on its right-hand side. The moderateness assumption will be crucial allowing to recapture the solution as in (2.1) should it exist. However, we note that regularisation with standard Friedrichs mollifiers will not be sufficient, hence the introduction of a family ω(ε) in the above regularisations.
We can now introduce a notion of a 'very weak solution' for the Cauchy problem (1.7).
is a very weak solution of order s of the Cauchy problem (1.7) if there exist C ∞ -moderate regularisations a ε and q ε of the coefficients a and q, such that (u ε ) ε solves the regularised problem for all ε ∈ (0, 1], and is C ∞ ([0, T ]; H s H )-moderate. We note that according to Theorem 2.1 the regularised Cauchy problem (2.3) has a unique solution satisfying estimate (2.1).
In [15], the authors studied weakly hyperbolic second-order equations with timedependent irregular coefficients, assuming that the coefficients are distributions. For such equations, the authors of [15] introduced the notion of a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. We now apply a modification of this notion to the Cauchy problems (1.7) and (1.10).
In the following theorem, we assume that a is a strictly positive distribution, which means that there exists a constant a 0 > 0 such that a − a 0 is a positive distribution. In other words, , and q is positive if q, ψ ≥ 0 whenever ψ ≥ 0. The main results of this paper can be summarised as the following solvability statement complemented by the uniqueness and consistency in Theorems 6.2 and 2.5. Since s is allowed to be negative, the Cauchy data are allowed to be H-distributions (i.e. elements of H s H with negative s). In Theorem 6.2, we show that the very weak solution is unique in an appropriate sense.
But now let us formulate the theorem saying that very weak solutions recapture the classical solutions; in the case, the latter exist. This happens, for example, under conditions of Theorem 2.  Here, the very weak solution is understood according to Definition 2.3. We now proceed with preparation for proving theorems in this section. R 2 in our setting is unbounded, the following constructions carry over without any significant changes. Moreover, there is a significant simplification since the appearing Fourier analysis is self-adjoint. A more general version of these constructions under weaker conditions can be found in [34]. For application of the general non-self-adjoint analysis to the spectral analysis, we refer to [12].
where Dom(H k ) is the domain of the operator H k , in turn defined as The Fréchet topology of C ∞ H (R 2 ) is given by the family of norms is the space of all linear continuous functionals on C ∞ H (R 2 ). For w ∈ D H (R 2 ) and ϕ ∈ C ∞ H (R 2 ), we shall write For any ψ ∈ C ∞ H (R 2 ), the functional . Taking into account the fact that the eigenfunctions of the Landau Hamiltonian in (1.5) come in pairs, it will be convenient to group them together in the way suggested by the notation (1.6). This leads to the following definitions. Let S(N 2 0 ) denote the space of rapidly decaying functions ϕ : That is, ϕ ∈ S(N 2 0 ) if for any M < ∞ there exists a constant C ϕ,M such that The topology on S(N 2 0 ) is given by the seminorms p k , where k ∈ N 0 and We now define the H-Fourier transform on C ∞ H (R 2 ) as the mapping by the formula is given by so that the Fourier inversion formula becomes The Plancherel identity takes the form which we can take as the definition of the norm on the Hilbert space 2 (N 2 0 ), and where f (ξ ) 2 HS = Tr( f (ξ ) f (ξ )) is the Hilbert-Schmidt norm of the matrix f (ξ ).
One can readily check that test functions and distributions on R 2 can be characterised in terms of their Fourier coefficients. Thus, we have In addition, for distributions, we have In general, given a linear continuous operator L : where Le ξ means that we apply L to the matrix components of e ξ (x), provided that e ξ (x) is invertible in a suitable sense. In this case, we may prove that (3.6) The correspondence between operators and symbols is one-to-one. The quantization (3.6) has been extensively studied in [35,36] in the setting of compact Lie groups, and in [33] in the setting of (non-self-adjoint) boundary value problems, to which we may refer for its properties and for the corresponding symbolic calculus. However, the situation with the Landau Hamiltonian is now much simpler since this operator can be treated as an 'invariant' operator in the corresponding global calculus. The operator H acts as a Fourier multiplier in its own Fourier calculus; therefore, its symbol σ H (ξ ) is independent of x, and since H is formally self-adjoint and positive we can always write it in the form for some ν j (ξ ) ≥ 0. Indeed, we have ν 2 j (ξ ) = B(1 + 2ξ 2 ) for j = 1, 2. Consequently, we can also define Sobolev spaces H s H associated to H. Thus, for any s ∈ R, we set justifying the expression (2.2).

Proof of Theorem 2.1
We will prove the result for the Cauchy problem (1.7) since equation (1.10) can be treated by the same argument with minor modification.
The operator H has the symbol (3.7), which we can write in matrix components as with δ mk standing for the Kronecker's delta. Taking the H-Fourier transform of (1.7), we obtain the collection of Cauchy problems for matrix-valued Fourier coefficients: where I is the identity 2 × 2 matrix. Writing this in the matrix form, we see that this is equivalent to the system Rewriting (4.1) in terms of matrix coefficients u(t, ξ) = ( u(t, ξ) mk ) 1≤m,k≤2 , we get the equations The main point of our further analysis is that we can make an individual treatment of the equations in (4.2) and then collect the estimates together using the H-Plancherel theorem.
We then study the Cauchy problem with ξ, m being parameters, and want to derive estimates for v(t, ξ). Combined with the characterisation (3.9) of Sobolev spaces this will yield the well-posedness results for the original Cauchy problem (1.7). In the sequel, for fixed m, we set Hence, the equation in (4.3) can be written as We now proceed with a standard reduction to a first-order system of this equation and define the corresponding energy. The energy estimates will be given in terms of t and ν(ξ ) and we then go back to t, ξ and m using (4.4).
We can now do the natural energy construction for (4.5). We use the transformation It follows that the Eq. (4.5) can be written as the first-order system where V is the column vector with entries V 1 and V 2 and Note that the matrix A has eigenvalues ± a(t) 1 + q(t) ν 2 (ξ ) and its symmetriser is given by i.e. we have It is immediate to prove that (4.8) where (·, ·) and | · | denote the inner product and the norm in C, respectively.
Since a(t) > a 0 ≥ 0, q(t) ≥ 0, and a, q ∈ C([0, T ]), it is clear that there exist constants a 1 > 0 and a 2 > 0 such that Hence, (4.8) implies that with c 1 , c 2 > 0. We then define the energy We get, from (4.9), that Since a(t) 1 + q(t) ν 2 (ξ ) is bounded on [0, T ] and for all ξ , we obtain for some constant c > 0. A part of the subsequent application of the Gronwall's lemma is standard (see [25]) but we give it for completeness and clarity. By applying Gronwall's lemma to inequality (4.10), we conclude that for all T > 0 there exists c > 0 such that Hence, inequalities (4.9) yield for constants independent of t ∈ [0, T ] and ξ . This allows us to write the following statement: there exists a constant C 1 > 0 such that for all t ∈ [0, T ] and ξ . Hence, Recalling the notation v(t, ξ) = u(t, ξ) mk and ν 2 (ξ ) = (B + 2Bξ 2 ), this means for all t ∈ [0, T ], ξ ∈ N 2 0 and 1 ≤ m, k ≤ 2, with the constant C 1 independent of ξ , m, k. Now we recall that by Plancherel's equality, we have Hence, the estimate (4.12) implies that where the constant C > 0 does not depend on t ∈ [0, T ]. More generally, multiplying (4.12) by powers of (B + 2Bξ 2 ), for any s, we get (4.14) Taking the sum over ξ , m and k as above, this yields the estimate (2.1).

Proof of Theorem 2.4
Again, in this section we deal with the Cauchy problem (1.7) and the proof for equation (1.10) can be done by minor modifications. We now assume that the equation coefficients are distributions with compact support contained in [0, T ]. Since the formulation of (1.7) in this case might be impossible in the distributional sense due to issues related to the product of distributions, we replace (1.7) with a regularised equation. In other words, we regularise a, q by convolution with a mollifier in C ∞ 0 (R) and get nets of smooth functions as coefficients. More precisely, let ψ ∈ C ∞ 0 (R), ψ ≥ 0 with ψ = 1, and let ω(ε) be a positive function converging to 0 as ε → 0, with the rate of convergence to be specified later. Define , Since a is a positive distribution with compact support (hence a Radon measure) and ψ ∈ C ∞ 0 (R), supp ψ ⊂ K, ψ ≥ 0, identifying the measure a with its density, we can write with a positive constantã 0 > 0 independent of ε.
By the structure theorem for compactly supported distributions, we have that there exist L 1 , L 2 ∈ N and c 1 , c 2 > 0 such that for all k ∈ N 0 and t ∈ [0, T ]. We note that the numbers L 1 and L 2 may be related to the distributional orders of a and q but we will not be needing such a relation in our proof. Hence, a ε , q ε are C ∞ -moderate regularisations of the coefficients a, q. Now, fix ε ∈ (0, 1], and consider the regularised problem with S ε corresponding to (4.7), and Gronwall's lemma, we get the estimate where the coefficients L 1 and L 2 are from (5.1).
Put ω(ε) ∼ log(ε) −1 . Then the estimate (5.3) transforms to with possibly new constants L 1 , L 2 . To simplify the notation, we continue denoting them by the same letters. Now, let us show that there exist N ∈ N 0 , c > 0 and, for all k ∈ N 0 there exist N k > 0 and c k > 0 such that for all t ∈ [0, T ], and ε ∈ (0, 1]. Applying (4.9) and (4.10) to the problem with a ε and q ε , and by taking account the properties of a ε and q ε , we get for all t ∈ [0, T ], ξ ∈ N 2 0 and 1 ≤ m, k ≤ 2, with the constant C independent of ξ , m, k. Thus, we obtain Acting by the iterations of ∂ t and by H on the equality and taking it in L 2 -norms, we conclude that u ε is C ∞ ([0, T ]; H s H )-moderate. This shows that the Cauchy problem (1.7) has a very weak solution.

Consistency with the classical well-posedness
In this section, we show that when the coefficients are regular enough then the very weak solution coincides with the classical one: this is the content of Theorem 2.5 which we will prove here.
Moreover, we show that the very weak solution provided by Theorem 2.4 is unique in an appropriate sense. For formulating the uniqueness statement, it will be convenient to use the language of Colombeau algebras. Definition 6. 1 We say that (u ε ) ε is C ∞ -negligible if for all K R, for all α ∈ N and for all ∈ N there exists a constant c > 0 such that We now introduce the Colombeau algebra as the quotient For the general analysis of G(R), we refer to Oberguggenberger [27].
We can now consider the Cauchy problem H 1+s