On the existence and uniqueness of a generalized solution of the Protter problem for ( 3 + 1 ) $(3+1)$ -D Keldysh-type equations

Popivanov, Nedyu; Hristov, Tsvetan; Nikolov, Aleksey; Schneider, Manfred

doi:10.1186/s13661-017-0757-1

Cited by 9 publications

(11 citation statements)

References 24 publications

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“…Let us mention here that, in problem PK, unlike Tricomi case, a data on the degenerate boundary is not prescribed (similar to the elliptic case) and derivative can have singularity on it, but up to the prescribed level. On the other hand, the results in [1] and the results of the present paper show some similarities between problem PK and problems 1, 2: the infinite-dimensional cokernel of the problem and the existence of generalized solutions with isolated singularities.…”

Section: History Of the Problem And Motivationmentioning

confidence: 76%

“…In our recent paper [1], we proved the following results on the existence and uniqueness of a generalized solution of problem PK.…”

Section: Lemmamentioning

confidence: 99%

“…Theorem 3 (see [1]). If ∈ (0, 4/3), then there exists at most one generalized solution of problem PK in Ω .…”

Section: Lemmamentioning

confidence: 99%

“…(ii) What kind of singularity may have the generalized solution in this case? The a priori estimate, obtained in [1], shows that when the function ( , ) is a "harmonic polynomial" the generalized solution may have at most a polynomial growth. Are there exist singular solutions with an exponential growth, as it is in the case of the Protter problems for the usual wave equation?…”

Section: Open Problemsmentioning

confidence: 99%

“…Definition 2 (see [1]). We call a function ( , ) a generalized solution of problem PK in Ω , 0 < < 4/3, for (1), if…”

Section: Lemmamentioning

confidence: 99%

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Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations

Popivanov

Hristov

Nikolov

et al. 2017

Advances in Mathematical Physics

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We study a boundary value problem for (3 + 1)-D weakly hyperbolic equations of Keldysh type (problem PK). The Keldysh-type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces. Problem PK is not well-posed since it has infinite-dimensional cokernel. Actually, this problem is analogous to a similar one proposed by M. Protter in 1952, but for Tricomi-type equations which, in part, are closely connected with transonic fluid dynamics. We consider a properly defined, in a special function space, generalized solution to problem PK for which existence and uniqueness theorems hold. It is known that it may have a strong power-type singularity at one boundary point even for very smooth right-hand sides of the equation. In the present paper we study the asymptotic behavior of the generalized solutions of problem PK at the singular point. There are given orthogonality conditions on the right-hand side of the equation, which are necessary and sufficient for the existence of a generalized solution with fixed order of singularity.

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Section: History Of the Problem And Motivationmentioning

confidence: 76%

“…In our recent paper [1], we proved the following results on the existence and uniqueness of a generalized solution of problem PK.…”

Section: Lemmamentioning

confidence: 99%

“…Theorem 3 (see [1]). If ∈ (0, 4/3), then there exists at most one generalized solution of problem PK in Ω .…”

Section: Lemmamentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

“…Definition 2 (see [1]). We call a function ( , ) a generalized solution of problem PK in Ω , 0 < < 4/3, for (1), if…”

Section: Lemmamentioning

confidence: 99%

See 3 more Smart Citations

Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations

Popivanov

Hristov

Nikolov

et al. 2017

Advances in Mathematical Physics

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Asymptotics to all orders of the Euler–Darboux equation in a triangle

Mauersberger

2019

Journal of Mathematical Analysis and Applications

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In Einstein's theory of relativity, the interaction of two collinearly polarized plane gravitational waves can be described by a Goursat problem for the Euler-Darboux equation in a triangular domain. In this paper, using a representation of the solution in terms of Abel integrals, we give a full asymptotic expansion of the solution near the diagonal of the triangle. The expansion is related to the formation of a curvature singularity of the spacetime. In particular, our framework allows for boundary data with derivatives which are singular at the corners. This level of generality is crucial for the application to gravitational waves.PDEs in the case of colliding electromagnetic plane waves [3,7]. In the linear limit, both of these coupled equations reduce to the Euler-Darboux equation (1.1).Of particular interest is the behavior of the solution of (1.1) near the triangle's diagonal edge x+y = 1. Indeed, this behavior is related to the formation of a curvature singularity of the spacetime by the mutual focusing of the colliding waves. In [26], it was observed that the solution must behave like ln(1 − x − y) as x + y → 1. In this paper, using an Abel integral representation of the solution, we derive an asymptotic expansion to all orders as x + y → 1.Our first result (Theorem 1) establishes a mathematically precise version of the classical Abel integral representation of the solution of the Goursat problem for (1.1) with given boundary data. This type of representation is well known (cf. e.g. [11,14,15]) and the main purpose of Theorem 1 is to provide a formulation that is suitable for our needs. We discuss regularity, the behavior at the boundary of D, and uniqueness of the solution under reasonable assumptions on the boundary data. In this context, reasonable means that our results can be applied to the common examples of collinear solutions (cf. e.g. [9,19,26]). In particular, we allow for boundary data with singular derivatives at the corners of D. As a consequence, our representation of the solution of (1.1) contains singular integrands, which is the main challenge in the analysis of the Goursat problem.Our second result (Theorem 2) describes the asymptotic behavior of the solution near the diagonal of D. We show that V (x, 1 − x − ) admits an asymptotic expansion to all orders of the formwhere the coefficients f j , g j are given explicitly in terms of the boundary data V 0 (x) = V (x, 0) and V 1 (y) = V (0, y), and where the error term is uniform on compact subsets

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