Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field
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Abstract
In this paper, we study the Cauchy problem for the Landau Hamiltonian wave equation, with timedependent 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 Friedrichstype 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 distributionaltype solutions under conditions when such solutions also exist.
Keywords
Wave equation Wellposedness Electromagnetic field Cauchy problem Landau HamiltonianMathematics Subject Classification
35D99 35L81 35Q99 42C10 58J451 Introduction
The purpose of this paper is to establish the wellposedness 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 \(\delta \)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 spaceinvariant 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.
We can also mention papers [19, 26], where the authors investigated properties of eigenfunctions of perturbed Hamiltonians, and in [20, 23, 24, 30, 31, 32, 37] asymptotics of the eigenvalues for perturbed Landau Hamiltonians were described.
How to understand the Cauchy problems (1.7)–(1.9) and their wellposedness?
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 \({\mathcal H}\) increase in space thus leading to potential problems at infinity if we treat the problem only locally.
Nevertheless, we are able to study the wellposedness 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 \(\mathbb R^{n}\).

The Cauchy problem (1.7) admits a very weak solution even for distributionaltype 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.
For secondorder operators \({\mathcal H}\) 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 \({{\mathbb R}^n}\). 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.
2 The main results
Theorem 2.1
In Theorem 2.1, the assumption of q being real valued is actually enough to assure the wellposedness; however, we assume that \(q\ge 0\) to facilitate the proofs of the distributional results later.
In the sequel, the notation \(K\Subset \mathbb R\) means that K is a compact set in \(\mathbb R\).
Definition 2.2
 (i)A net of functions \((f_\varepsilon )_\varepsilon \in C^\infty (\mathbb R)^{(0,1]}\) (i.e. \((f_\varepsilon )_{\varepsilon \in (0,1]}\subset C^\infty (\mathbb R)\)) is said to be \(C^\infty \)moderate if for all \(K\Subset \mathbb R\) and for all \(\alpha \in \mathbb N_{0}\) there exist \(N=N_{\alpha }\in \mathbb N_{0}\) and \(c=c_{\alpha }>0\) such thatfor all \(\varepsilon \in (0,1]\).$$\begin{aligned} \sup _{t\in K}\partial ^\alpha f_\varepsilon (t)\le c\varepsilon ^{N\alpha }, \end{aligned}$$
 (ii)A net of functions \((u_\varepsilon )_\varepsilon \in C^\infty ([0,T];{H}^{s}_{\mathcal H})^{(0,1]}\) is said to be \(C^\infty ([0,T];{H}^{s}_{\mathcal H})\)moderate if there exist \(N\in \mathbb N_{0}\) and \(c_k>0\) for all \(k\in \mathbb N_{0}\) such thatfor all \(t\in [0,T]\) and \(\varepsilon \in (0,1]\).$$\begin{aligned} \Vert \partial _t^k u_\varepsilon (t,\cdot )\Vert _{H^{s}_{\mathcal H}}\le c_k\varepsilon ^{Nk}, \end{aligned}$$
Thus, while a solution to the Cauchy problems may not exist in the space of distributions on the lefthand side of (2.4), it may still exist (in a certain appropriate sense) in the space on its righthand 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 \(\omega (\varepsilon )\) in the above regularisations.
We can now introduce a notion of a ‘very weak solution’ for the Cauchy problem (1.7).
Definition 2.3
Let \(s\in \mathbb R\) and \(u_{0},u_{1}\in {H}^{s}_{\mathcal H}\). The net \((u_\varepsilon )_\varepsilon \in C^\infty ([0,T];{H}^{s}_{\mathcal H})\) is a very weak solution of order s of the Cauchy problem (1.7) if there exist
\(C^\infty \)moderate regularisations \(a_{\varepsilon }\) and \(q_{\varepsilon }\) of the coefficients a and q,
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 secondorder 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).
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.
Theorem 2.4
(Existence) Let the coefficients a and q of the Cauchy problem (1.7) be positive distributions with compact support included in [0, T], such that \(a\ge a_{0}\) for some constant \(a_{0}>0\). Let \(s\in \mathbb R\) and let the Cauchy data \(u_0, u_1\) be in \({H}^{s+1}_{\mathcal H}\). Then the Cauchy problem (1.7) has a very weak solution of order s.
The same result is true also for the Cauchy problem (1.10).
Since s is allowed to be negative, the Cauchy data are allowed to be \({\mathcal H}\)distributions (i.e. elements of \({H}^{s}_{\mathcal H}\) 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 2.1. So, we can compare the solution given by Theorem 2.1 with the very weak solution in Theorem 2.4 under assumptions when Theorem 2.1 holds.
Theorem 2.5
Here, the very weak solution is understood according to Definition 2.3. We now proceed with preparation for proving theorems in this section.
3 Fourier analysis for the Landau Hamiltonian
In this section, we recall the necessary elements of the global Fourier analysis that has been developed in [33] and applied to the present setting. Although the domain \(\mathbb R^{2}\) 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 selfadjoint. A more general version of these constructions under weaker conditions can be found in [34]. For application of the general nonselfadjoint analysis to the spectral analysis, we refer to [12].
4 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.
5 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.
This shows that the Cauchy problem (1.7) has a very weak solution.
6 Consistency with the classical wellposedness
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
Theorem 6.2
The same statement holds also for the Cauchy problem (1.10).
Here, \(\mathcal G([0,T]; H^s_{{\mathcal H}})\) stands for the space of families which are in \(\mathcal G([0,T])\) with respect to t and in \(H^s_{{\mathcal H}}\) with respect to x.
Proof
The last case is when \(V_{\varepsilon }(t,\xi )\le c \, \omega (\varepsilon )^{\alpha }\) for some constant c and \(\alpha >0\). Indeed, it completes the proof of Theorem 6.2. \(\square \)
Proof of Theorem 2.5
7 Inhomogeneous equation case
Theorem 7.1
Let us formulate definition of the very weak solution for the inhomogeneous wave equation (7.1).
Definition 7.2

\(C^\infty \)moderate regularisations \(a_{\varepsilon }\) and \(q_{\varepsilon }\) of the coefficients a and q,

\(C^\infty ([0,T];{H}^{s}_{\mathcal H})\)moderate regularisation \(f_{\varepsilon }\) of the source term f,
Without significant changes in the proofs of Theorems 2.4, 2.5 and 6.2, we conclude the following modified results for the Cauchy problem (7.1) for the inhomogeneous wave equation.
Theorem 7.3
(Existence) Let the coefficients a and q of the Cauchy problem (7.1) be positive distributions with compact support included in [0, T], such that \(a\ge a_{0}\) for some constant \(a_{0}>0\), and let the source term \(f(\cdot , x)\) be a distribution with compact support included in [0, T]. Let \(s\in \mathbb R\) and let the Cauchy data \((u_0, u_1)\) be in \({H}^{s+1}_{\mathcal H}\times {H}^{s}_{\mathcal H}\) and the source term \(f(t, \cdot )\) be in \({H}^{s}_{\mathcal H}\). Then the Cauchy problem (7.1) has a very weak solution of order s.
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 7.1. So, we can compare the solution given by Theorem 7.1 with the very weak solution in Theorem 7.3 under assumptions when Theorem 7.1 holds.
Theorem 7.4
Theorem 7.5
8 Landau Hamiltonian in \(\mathbb R^{2d}\)
Consequently, all the statements of Theorems 2.1, 2.5 and 6.2 continue to hold for the Cauchy problems (8.3) and (8.4). Namely, we have
Theorem 8.1
We also have the corresponding very weak solutions’ result.
Theorem 8.2

(Existence) The Cauchy problem (8.3) has a very weak solution of order s.

(Uniqueness) There exists an embedding of the coefficients a and q into \(\mathcal G([0,T])\) such that the Cauchy problem (8.3) has a unique solution \(u\in \mathcal G([0,T]; H^s_{{\mathcal H}})\).

(Consistency) Let u be a very weak solution of (8.3). If \(a, q\in L_{1}^{\infty }([0,T])\) are such that \(a(t)\ge a_0>0\) and \(q(t)\ge 0\), then for any regularising families \(a_{\varepsilon }, q_{\varepsilon }\), any representative \((u_\varepsilon )_\varepsilon \) of u converges in \(C([0,T],{H}^{1+s}_{\mathcal H}) \cap C^1([0,T],{H}^{s}_{\mathcal H})\) as \(\varepsilon \rightarrow 0\) to the unique classical solution in \(C([0,T],{H}^{1+s}_{\mathcal H}) \cap C^1([0,T],{H}^{s}_{\mathcal H})\) of the Cauchy problem (8.3) given by Theorem 8.1.
These theorems follow by an easy adaptation of the corresponding 2D proofs, so we omit them.
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