UK Network on Hyperbolic Equations and Related Topics, 2017-2021

Supported by the LMS (Scheme 3 Grant Ref 31703) and by the Edinburgh Mathematical Society

Departments

School of Mathematics, The University of Edinburgh, Maxwell Institute for Mathematical Sciences

School of Mathematical Sciences, Queen Mary University of London

Department of Mathematical Sciences, Loughborough University

Department of Mathematics, Imperial College London (2017-2018)

Organisers

Pieter Blue (Edinburgh)
Claudia Garetto (Loughborough)
Michael Ruzhansky (Imperial College London –> Queen Mary University of London)

Linear and nonlinear hyperbolic partial differential equations (PDEs) arise in basically all sciences (physics, chemistry, medicine, engineering, astronomy, etc.). In physics, they model several important phenomena, from propagation of waves in a medium (for instance propagation of seismic waves during an earthquake) to refraction in crystals and gas-dynamics. The purpose of this UK network on hyperbolic equations and related topics is to bring together the expertise on hyperbolic equations of three different mathematics department (Edinburgh, Imperial, Loughborough), to strengthen the existing research collaborations and to create new ones. Three 1-day workshops per year are planned focused on different approaches to hyperbolic equations and related topics (inverse problems, kinetic theory, imaging, microlocal analysis, general relativity, etc.).

NEW: Meeting 9

9 December 2021 (Thursday), Online (ZOOM)

https://us02web.zoom.us/j/88645590199?pwd=NzhGZlkyTlJCSk5oZWJOWDh0TmxNdz09

Meeting ID: 886 4559 0199 Passcode: 569414

Website of the Network and of the Meeting

Planned program (UK time)

10:00-11:00 Juan Valiente Kroon (QMUL) Degenerate symmetric hyperbolic systems, conformal structure and totally characteristic hypersurfaces

11:00-12:00 Marica Minucci (QMUL) A conformal approach to the stability of Einstein spaces with spatial sections of negative scalar curvature and beyond

12:00-13:00 Giovanni Taglialatela (Bari) The Cauchy problem for properly hyperbolic equations in one space variable

break

14:00-15:00 Marianna Chatzakou (Ghent) Very weak solutions of hypoelliptic equations

15:00-16:00 Priscila Leal-Da-Silva (Loughborough) Existence and uniqueness of solutions for the rotation-Camassa-Holm equation

Abstracts and additional information 

All are welcome to attend.

The Ghent Analysis & PDE Center is launching a new junior seminar:

Methusalem Junior Analysis & PDE Seminar is run by PhD students and postdocs at Ghent Analysis & PDE Center. It provides an opportunity to young researchers in various areas of analysis and PDEs to share their ideas and to learn new exciting things related to the topics of Analysis & PDEs as well as broader mathematical subjects.

All are welcome to attend!

The seminar currently takes place on ZOOM.

Website of the seminar

Vishvesh Kumar has just given a nice talk at the Virtual Math Fest 2020 (Symposium on Analysis and PDE)

The talk was about Lp-Lq boundedness of Fourier multipliers associated with the anharmonic oscillator,

on the joint work with Marianna Chatzakou from Imperial College London.

Well done, Vishvesh!

• Chatzakou M., Kumar V., Lp-Lq boundedness of Fourier multipliers associated with the anharmonic oscillator. arxiv
• Chatzakou M., Delgado J., Ruzhansky M., On a class of anharmonic oscillators. arxiv
• Cardona D., Kumar V., Ruzhansky M., Tokmagambetov N., L^p–L^q boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations. arxiv

Watch the talk on YouTube

Congratulations to Linda Botchway, for receiving the grant of USD 15,500 from the Carnegie Corporation of New York (CCNY), for her PhD at the University of Ghana, Legon, under our supervision.

 Well done, Linda!

We are looking forward to Linda’s visits to our Analysis & PDE Center at Ghent University, and to her enrolment to UGent PhD program for a joint PhD.

Research of Linda on pseudo-differential operators on started while she was a Master student at AIMS Ghana. This resulted in a substantial (41 pages) publication, a great achievement for a Master Student:

Botchway L., Kibiti G., Ruzhansky M., Difference equations and pseudo-differential operators on Zⁿ, J. Funct. Anal., 278 (2020), no. 11, 108473, 41pp. link (open access), arxiv

In the above paper one makes a consistent development of the calculus of pseudo-differential operators which can be called pseudo-difference operators in the context of the lattice  Some interesting questions to answer:

• • how to define Hörmander type or other symbol classes on ?
  • what is the relation to the analysis on the torus ?
  • can it be used to derive new properties of operators on ?
  • how to apply it to solving difference equations and to finding properties of their solutions?

Another of our former postdocs (still at Imperial College London), and an associated collaborator of our Ghent Analysis and PDE Centre, Dr Durvudkhan Suragan, featured on Kazakh TV, in a special program about him and his achievements.

Congratulations, Durvudkhan!

It is very rewarding to see current and former students and postdocs being recognised for their achievements!

Watch it on Youtube:

Also, with the appearance of Dr Berikbol Torebek:

Recently: another of our associated member, Dr Erkinjon Karimov, on Uzbek State TV, read here.

Our books with Durvudkhan Suragan:

1. Ashyralyev A., Kalmenov T., Ruzhansky M., Sadybekov M., Suragan D. (Eds.) Functional Analysis in Interdisciplinary Applications II, Springer Proceedings in Mathematics & Statistics, Springer, 2020, to appear
2. Ruzhansky M., Suragan D., Hardy inequalities on homogeneous groups: 100 years of Hardy inequalities, Progress in Mathematics, Vol. 327, Birkhauser, 2019. xvi+588pp. link, free download (open access book), poster, the winner of  Ferran Sunyer I Balaguer Prize 2018
3. Ruzhansky M., Sadybekov M., Suragan D., Spectral geometry of partial differential operators, Monographs and Research Notes in Mathematics, Chapman and Hall/CRC Press, 2020. 378pp. link, free download (open access book), poster

Our papers with Durvudkhan Suragan:

1. Restrepo J., Ruzhansky M., Suragan D., Explicit representations of solutions for linear fractional differential equations with variable coefficients. arxiv
2. Kassymov A., Ruzhansky M., Suragan D., Reverse Stein-Weiss, Hardy-Littlewood-Sobolev, Hardy, Sobolev and Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups, arxiv
3. Kassymov A., Ruzhansky M., Suragan D., Reverse integral Hardy inequalities on metric measure spaces. arxiv
4. Ruzhansky M., Sabitbek B., Suragan D., Geometric Hardy inequalities on starshaped sets. arxiv
5. Ruzhansky M., Sabitbek B., Suragan D., Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups. Bull. Math. Sci., to appear. arxiv
6. Kassymov A., Ruzhansky M., Suragan D., Anisotropic fractional Gagliardo-Nirenberg, weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type inequalities, and applications to Riesz potentials and p-sub-Laplacian systems. arxiv
7. Ruzhansky M., Sabitbek B., Suragan D., Subelliptic geometric Hardy type inequalities on half-spaces and convex domains. Ann. Funct. Anal., to appear. arxiv
8. Ruzhansky M., Suragan D., Yessirkegenov N., Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups. Semigroup Forum, to appear. link (open access), arxiv
9. Ruzhansky M., Suragan D., Green’s identities, comparison principle and uniqueness of positive solutions for nonlinear p-sub-Laplacian equations on stratified Lie groups, Potential Analysis, to appear. arxiv, link
10. Ruzhansky M., Sabitbek B., Suragan D., Hardy and Rellich inequalities for anisotropic p-sub-Laplacians, Banach J. Math. Anal., 14 (2020), 380-398. link, arxiv
11. Kassymov A., Ruzhansky M., Suragan D., Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups. NoDEA Nonlinear Differential Equations Appl., 27 (2020), no. 1, Paper No. 7. link
12. Ruzhansky M., Suragan D., Critical Hardy inequalities, Ann. Acad. Sci. Fenn. Math., 44 (2019), 1159-1174. link (download), arxiv
13. Kassymov A., Ruzhansky M., Suragan D., Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups. Integral Transforms and Special Functions, 30 (2019), 643-655. link, arxiv
14. Ruzhansky M., Sabitbek B., Suragan D., Weighted anisotropic Hardy and Rellich type inequalities for general vector fields, NoDEA Nonlinear Differential Equations Appl., 26 (2019), no. 2, 26:13. link, arxiv
15. Kalmenov T. Sh., Ruzhansky M., Suragan D., On spectral and boundary properties of the volume potential for the Helmholtz equation. Math. Model. Nat. Phenom., 14 (2019), no. 5, Art. 502, 11 pp. link
16. Ozawa T., Ruzhansky M., Suragan D., L^p-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups, Quart. J. Math., 70 (2019), 305-318. arxiv, link, offprint (open access)
17. Ruzhansky M., Sabitbek B., Suragan D., Weighted L^p-Hardy and L^p-Rellich inequalities with boundary terms on stratified Lie groups, Rev. Mat. Complutense, 32 (2019), 19-35. enhanced pdf (open access), arxiv, link
18. Ruzhansky M., Suragan D., A note on stability of Hardy inequalities, Ann. Funct. Anal., 9 (2018), 451-462. arxiv, link
19. Ruzhansky M., Suragan D., A comparison principle for nonlinear heat Rockland operators on graded groups, Bull. Lond. Math. Soc., 50 (2018), 753-758. enhanced pdf (open access), arxiv, link
20. Ruzhansky M., Suragan D., Elements of potential theory on Carnot groups, Funct. Anal. Appl., 52 (2018), 158-161. link(eng), link (rus)
21. Ruzhansky M., Suragan D., Yessirkegenov N., Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces, Fract. Calc. Appl. Anal., 21 (2018), 577-612. arxiv, link
22. Ruzhansky M., Suragan D., Yessirkegenov N., Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability and superweights for Lp-weighted Hardy inequalities, Trans. Amer. Math. Soc. Ser. B, 5 (2018), 32-62. offprint (open access), link, arxiv
23. Ruzhansky M., Suragan D., Yessirkegenov N., Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups, Integral Equations Operator Theory, 90 (2018), no. 1, 90:10. enhanced pdf (open access), arxiv, link
24. Ruzhansky M., Suragan D., Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups, Adv. Math., 317 (2017), 799-822. offprint (open access), arxiv, link
25. Ruzhansky M., Suragan D., Anisotropic L²-weighted Hardy and L²-Caffarelli-Kohn-Nirenberg inequalities, Commun. Contemp. Math., 19 (2017), no. 6, 1750014, 12pp. offprint (open access), arxiv, link
26. Ruzhansky M., Suragan D., Yessirkegenov N., Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups, NoDEA Nonlinear Differential Equations Appl, 24 (2017), no. 5, 24:56. offprint (open access), arxiv, link
27. Ruzhansky M., Suragan D., Geometric maximizers of Schatten norms of some convolution type integral operators, J. Math. Anal. Appl., 456 (2017), 444-456. arXiv, link
28. Ruzhansky M., Suragan D., Yessirkegenov N., Extended Caffarelli-Kohn-Nirenberg inequalities and superweights for L^p-weighted Hardy inequalities, C. R. Acad. Sci. Paris, 355 (2017), 694-698. offprint (open access), link
29. Ruzhansky M., Suragan D., Uncertainty relations on nilpotent Lie groups, Proc. R. Soc. A, 473 (2017), no. 2201, 20170082, 12pp. offprint (open access), arxiv, link
30. Ruzhansky M., Suragan D., Isoperimetric inequalities for some integral operators arising in potential theory, in Functional Analysis in Interdisciplinary Applications. FAIA 2017, 320-329, Springer Proceedings in Mathematics & Statistics, vol 216, Springer, 2017. arxiv, link
31. Ruzhansky M., Suragan D., Local Hardy and Rellich inequalities for sums of squares of vector fields, Adv. Diff. Equations, 22 (2017), 505-540. arxiv, link
32. Garofalo N., Ruzhansky M., Suragan D., On Green functions for Dirichlet sub-Laplacians on H-type groups, J. Math. Anal. Appl., 452 (2017), 896-905. offprint (open access), link, arxiv
33. Ruzhansky M., Suragan D., Layer potentials, Green formulae, Kac problem, and refined Hardy inequality on homogeneous Carnot groups, Adv. Math., 308 (2017), 483-528. offprint (open access), arxiv, link
34. Ruzhansky M., Suragan D., On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and p-sub-Laplacian inequalities on stratified groups, J. Differential Equations, 262 (2017), 1799-1821. offprint (open access), arxiv, link
35. Ruzhansky M., Suragan D., On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries, Bull. Math. Sci., 6 (2016), 325-334. offprint (open access), arxiv, link
36. Rozenblum G., Ruzhansky M., Suragan D., Isoperimetric inequalities for Schatten norms of Riesz potentials, J. Funct. Anal., 271 (2016), 224-239. offprint (open access), arxiv, link
37. Ruzhansky M., Suragan D., On Schatten norms of convolution-type integral operators, Russ. Math. Surv., 71 (2016),157-158. link
38. Ruzhansky M., Suragan D., On Kac principle of not feeling the boundary for the Kohn Laplacian on the Heisenberg group, Proc. Amer. Math. Soc., 144 (2016), 709-721. arxiv, link
39. Ruzhansky M., Suragan D., Isoperimetric inequalities for the logarithmic potential operator, J. Math. Anal. Appl.,434 (2016), 1676-1689. offprint (open access), arxiv, link

Further success by Dr Erkinjon Karimov, receiving the Doctor of Science degree by the Supreme Attestation Commission of the Republic of Uzbekistan:

Congratulations, Erkinjon!

Dr Erkinjon Karimov on State Uzbekistan TV

Watch the YouTube Video about Dr Erkinjon Karimov (in Russian):

Dr Erkinjon Karimov, a long-term collaborator and an associated member of our Ghent Analysis & PDE Center was on the Uzbekistan State TV, celebrated for his scientific achievements. A special program was filmed about him.

Congratulations, Erkinjon!

Erkinjon has been our long-term visitor in 2019, and is planning to join us as a researcher from Autumn 2020, in the framework of our Odysseus project.

Erkinjon has just co-organised the Fractional Calculus Workshop with us.

Watch the YouTube Video (in Russian):

Finally the presentation of our Ghent Analysis & PDE (GAP) Center is ready. Many thanks to Ljubica Oparnica for the design and for putting it together!

Click on it to explore:

The movie is coming up as well.