This is a concept for treating strong singularities in equations, such as delta distribution.
… various PDEs: hyperbolic systems, Heat, Schrödinger, Wave equations: Acoustic waves, Landau Hamiltonian, Water wave equations describing tsunamis
… with irregular coefficients in time and space
… and on groups and manifolds
The notion of very weak well-posedness was introduced for the first time in [GR:2015]
for hyperbolic equations with time-dependent singular (distributional) coefficients.
The fundamental idea of the very weak solution concept is to model irregular objects in the (system of) equations by approximating nets of regular functions, converging or not, but with moderate asymptotics.
Papers that followed prove this concept to be useful and “easy to use” in applications.
“The approach of very weak solutions opens up a whole new research area where one can deal with problems with singularities in a way that is consistent with stronger notions of solutions should they exist. “
The singular coefficients often appear in physical problems in the presence of singular backgrounds and geometries. With the very weak solutions as a tool, such equations can be analysed rigorously even if the weak or distributional interpretation of equations does not make sense since operations like multiplication of distributions or division of functions with singularities are not allowed in the classical theory of distributions.
“this is a very promising far-reaching research with further mathematical developments and many expected applications in other sciences “
to apply this approach to a variety of problems in PDEs with singular coefficients but also to follow theoretical directions
Analysis of nets: regularity theory and microlocal analysis
Microlocal and harmonic analysis allowing different types of singularities
Pseudo-differential operators with irregular coefficients
Spectral problems for singular operators or in singular domains
!!!NEW!!! [ARST1] A. Altybay, M. Ruzhansky, M. E. Sebih, N. Tokmagambetov. Fractional Klein-Gordon equation with strongly singular mass term. Preprint, arXiv:2004.10145 (2020).
!!!NEW!!! [ARST2] A. Altybay, M. Ruzhansky, M. E. Sebih, N. Tokmagambetov. Fractional Schrödinger Equations with potentials of higher-order singularities. Preprint, arXiv:2004.10182 (2020).
!!!NEW!!! [ARST3] A. Altybay, M. Ruzhansky, M. E. Sebih, N. Tokmagambetov. The heat equation with singular potentials. Preprint, arXiv:2004.11255 (2020).
!!!NEW!!! [ARST4]A. Altybay, M. Ruzhansky, M. E. Sebih, N. Tokmagambetov. Tsunami propagation for singular topographies Preprint, arXiv:2005.11931 (2020).
!!!NEW!!! [ART19] Altybay A., Ruzhansky M., Tokmagambetov N., Wave equation with distributional propagation speed and mass term: numerical simulations. Appl. Math. E-Notes, 19, 552-562, 2019.
!!!NEW!!! [G] C. Garetto. On the wave equation with multiplicities and space-dependent irregular coefficients. Preprint, arXiv:2004.09657 (2020).
[GR:2015] C. Garetto, M. Ruzhansky. Hyperbolic second order equations with non-regular time de- pendent coefficients. Arch. Rational Mech. Anal., 217, no. 1, 113–154, 2015.
[MRT19J. C. Munoz, M. Ruzhansky and N. Tokmagambetov. Wave propagation with irregular dissipation and applications to acoustic problems and shallow waters, J. Math. Pures Appl., 9 (123), 127–147, 2019.
[RT17] M. Ruzhansky, N. Tokmagambetov. Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field. Lett. Math. Phys., 107:591-618, 2017.
[RT17a] Ruzhansky M. and Tokmagambetov N. Wave equation for operators with discrete spectrum and irregular propagation speed. Arch. Ration. Mech. Anal., 226: 1161–1207, 2017.
[RT19] M. Ruzhansky, N. Tokmagambetov. Wave propagation with irregular dissipation and ap- plications to acoustic problems and shallow waters, J. Math. Pures Appl., 123, 127147, 2019.
[RY20] Ruzhansky, Michael, Yessirkegenov, Nurgissa Very weak solutions to hypoelliptic wave equations. J. Differential Equations, 268 (2020), no. 5, 2063–2088.
[SW] M.E. Sebih, J. Wirth. On a wave equation with singular dissipation. Preprint, Arxiv:2002.00825 (2020).
For links to the papers see, for example, here.