Very weak solutions

 

This is newly (in [GR2015]) introduced concept for solving PDEs  for treating strong singularities in PDE’s. Following papers [RT, RT17a, …, and most recent [ARST 1-4] prove 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. “

 

Fundamental idea in very weak solution concept is to model irregular objects in the (system of) equations by approximating nets of smooth functions, converging or not, but with moderate asymptotics.

 

“this is a very promising far-reaching research with further mathematical developments and many expected applications in other sciences “

 

References:

[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.

[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.

[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.

[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.

[ARST1] A. Altybay, M. Ruzhansky, M. E. Sebih, N. Tokmagambetov. Fractional Klein-Gordon equation with strongly singular mass term. Preprint, arXiv:2004.10145 (2020).

[ARST2] A. Altybay, M. Ruzhansky, M. E. Sebih, N. Tokmagambetov. Fractional Schrödinger Equa- tions with potentials of higher-order singularities. Preprint, arXiv:2004.10182 (2020).

[ARST3] A. Altybay, M. Ruzhansky, M. E. Sebih, N. Tokmagambetov. The heat equation with singular potentials. Preprint, arXiv:2004.11255 (2020).

[ARST4]A. Altybay, M. Ruzhansky, M. E. Sebih, N. Tokmagambetov. Tsunami propagation for singular topographies Preprint, arXiv:2005.11931 (2020).

[G] C. Garetto. On the wave equation with multiplicities and space-dependent irregular coeffi- cients. Preprint, arXiv:2004.09657 (2020).

[SW] M.E. Sebih, J. Wirth. On a wave equation with singular dissipation. Preprint, Arxiv:2002.00825 (2020).

[RY20] Ruzhansky, Michael,  Yessirkegenov, Nurgissa  Very weak solutions to hypoelliptic wave equations. J. Differential Equations 268 (2020), no. 5, 2063–2088.