Research Interests

My research interests are in the analysis of partial differential equations (PDEs), in particular evolution equations. I am particularly interested in questions related to the well-posedness or ill-posedness of Cauchy problems. This includes studying questions such as the existence and uniqueness of solutions in particular function spaces. I am also interested in the continuity of the data-to-solution map. This is closely related to asymptotic and orbital stability of solutions to these PDEs.

I am interested in finding general techniques and theorems which can explain a wide class of phenomenon and results for classes of equations and/or large classes of solutions.

Graduate Students, Visitors, and Postdocs Supervised:


Michael Harmon: Optimal portfolio allocation problems and mathematical simulations, MS thesis. Graduated 2022. Currently a graduate student at Imperial College London.


Rajan Puri: Visiting Assistant Professor at Wake Forest University.

Junpei Li: The Heath--Jarrow--Morton model for interest rates, affine yield models, and their robustness, MS thesis. Graduated 2021. Currently a bond analyst in Hong Kong

Publications:

  1. J. Holmes, B. Keyfitz and F. Tiglay, Weak diffeomorphisms and solutions to conservation laws, accepted for publication in the inaugural issue of La Matematica, the Official Journal of the Association for Women in Mathematics.

  2. J. Holmes, R. Thompson and F. Tiglay, Nonuniform dependence of the r-b-family system in Besov spaces, ZAMM - Journal of Applied Mathematics and Mechanics, 2021, e202000329.

  3. J. Holmes, R. Thompson and F. Tiglay, Continuity of the data-to-solution map for the FORQ equation in Besov spaces, Differential and Integral Equations, {\bf 34}, 2021, 295-314.

  4. J. Holmes and R. Puri, Non-uniqueness for the ab-family of equations, Journal of Mathematical Analysis and Applications, {\bf 493}, 2021, e124563.

  5. J. Holmes and F. Tiglay, Non-uniform dependence of the data-to-solution map for the Hunter-Saxton equation in Besov spaces, Journal of Evolution Equations {\bf18}, 2018, 1173-1187.

  6. J. Holmes and F. Tiglay, Continuity properties of the solution map for the Euler-Poisson equation, Journal of Mathematical Fluid Mechanics {\bf 20}, 2018, 757-769.

  7. J. Holmes, B. Keyfitz and F. Tiglay, Nonuniform dependence on initial data for compressible gas dynamics: The Cauchy problem on $\mathbb{R}^2$, SIAM Journal on Mathematical Analysis {\bf 50}, 2018, 1237-1254.

  8. J. Holmes and R. Thompson, Well-posedness and continuity properties of the Fornberg-Whitham equation in Besov spaces, Journal of Differential Equations, {\bf 263}, 2017, 4355-4381.

  9. J. Holmes, Well--posedness and regularity of the generalized Burgers equation in periodic Gevrey spaces, Journal of Mathematical Analysis and Applications, {\bf 454}, 2017, 18-40.

  10. J. Holmes and R. Thompson, Classical solutions of the generalized Camassa-Holm equation, Advances in Differential Equations, {\bf 22}, 2016, 339-362.

  11. J. Holmes, Well--posedness of the Fornberg-Whitham equation on the circle, Journal of Differential Equations, {\bf 260}, 2016, 8530-8549.

  12. J. Holmes, The Cauchy problem for several nonlinear evolution equations. The University of Notre Dame Ph.D. Dissertation.

  13. J. Holmes, Continuity properties of the data-to-solution map for the generalized Camassa-Holm equation, Journal of Mathematical Analysis and Applications. {\bf 417}, 2014, 635-642.

  14. A. Himonas and J. Holmes, Holder continuity of the solution map for the Novikov equation, Journal of Mathematical Physics {\bf 54}, 2013, 1-11.

  15. M. Axtell, J. Stickles and W. Trampbachls, Zero-divisor ideals and realizable zero-divisor graphs, Involve {\bf 2} 2009, 17-27.

  16. J. Holmes and A. Shull, Properties of ideal divisors, Pi Mu Epsilon Journal, {\bf 13} 2009, 33-36.

Conferences Organized:


AMS special session on Analysis of Nonlinear Evolution Equations at the AMS Spring Central Sectional Meeting Spring 2022.


AMS special session on Nonlinear Evolution Equations at the AMS Spring Central Sectional Meeting Spring 2018.