Very weak solutions to hypoelliptic wave equations

In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent Hölder (or more regular) non-negative propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$ p -evolution equations for higher order operators on ${{\mathbb{R}}}^{n}$ R n or on groups, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In particular, we describe an interesting local loss of regularity phenomenon depending on the step of the group (for stratified groups) and on the order of the considered operator.

show abstract

“…We remark that in the meantime, also models with more singular coefficients have been treated in the context of very weak solutions, see [44].…”

Section: Introductionmentioning

confidence: 99%

Time-Dependent Wave Equations on Graded Groups

Ruzhansky

Taranto

2021

Acta Appl Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, we take inspiration from the study of weakly hyperbolic homogeneous equations done by Spagnolo and Taglialatela in [ST07] in the 1-space dimensional case. Very weak solutions have been recently employed to study different classes of PDEs with singularities in [RT,RY,ARST20,ARST20b,ARST20c]. A numerical analysis of very weak solutions has also been initiated in [ART19,MRT19] for the wave equation with irregular time-dependent coefficients leading to the discovery of new interesting echo-effects (see [MRT19,SW]).…”

Section: Claudia Garettomentioning

confidence: 99%

On the wave equation with multiplicities and space-dependent irregular coefficients

Garetto¹

2021

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

In this paper we study the well-posedness of the Cauchy problem for a wave equation with multiplicities and space-dependent irregular coefficients. As in Garetto and Ruzhansky [Arch. Ration. Mech. Anal. 217 (2015), pp. 113–154], in order to give a meaningful notion of solution, we employ the notion of very weak solution, which construction is based on a parameter dependent regularisation of the coefficients via mollifiers. We prove that, even with distributional coefficients, a very weak solution exists for our Cauchy problem and it converges to the classical one when the coefficients are smooth. The dependence on the mollifiers of very weak solutions is investigated at the end of the paper in some instructive examples.

show abstract

“…Later, this analysis was applied to other hyperbolic type equations with singular coefficients [3][4][5][6]. The wave type equations with time-dependent coefficients on graded Lie groups were analysed in [7] for Hölder coefficients, and in [8] for distributional time-dependent coefficients, using the notion of very weak solutions. All these works deal with the time-dependent equations and in the recent papers [9][10][11][12], the authors start to develop the notion of very weak solutions for equations with (irregular) space-depending coefficients.…”

Section: Introductionmentioning

confidence: 99%

Fractional Klein-Gordon equation with singular mass. II: hypoelliptic case

Chatzakou

Ruzhansky

Tokmagambetov

2021

Complex Variables and Elliptic Equations

Self Cite

View full text Add to dashboard Cite

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part [Altybay et al. Fractional Klein-Gordon equation with singular mass. Chaos Solitons Fractals. 2021;143:Article ID 110579] which was devoted to the classical Euclidean Klein-Gordon equation.

show abstract

Very weak solutions to hypoelliptic wave equations

Cited by 28 publications

References 12 publications

Time-Dependent Wave Equations on Graded Groups

Time-Dependent Wave Equations on Graded Groups

On the wave equation with multiplicities and space-dependent irregular coefficients

Fractional Klein-Gordon equation with singular mass. II: hypoelliptic case

Contact Info

Product

Resources

About