Math 290: PDE Learning Seminar (Spring 2024)


The seminar has three parts. The first part is a continuation of last semester's learning seminar on critical wave equations. In particular, several papers will be discussed which adapt the technique of profile decomposition (discussed last semester for energy-critical wave equations) to new problems. The second part of the semester is focused on classical and modern ways of understanding singularity formation for the compressible Euler equations, and more generally quasilinear hyperbolic systems. The plan for this part is inspired by the survey article [8]. The first five topics are relatively self-contained, with the drawback that they do not provide a very detailed picture outside of radial or planar symmetry. These results will help clarify what is meant by ``describing a shock,'' and will hopefully provide a benchmark for the last talk which is focused on work done after the publication of Christodoulous's pioneering monograph [14], with the goal of understanding recent developments in the subject. The third part of the seminar will be on the topic of dispersive shocks. As a comment, the reference list is incomplete and a bit arbitrary.


The seminar meets on Fridays at 2:10 PM to 4:00 PM in Evans 740. If you would like to join the mailing list, please click here. Notes for each talk will be posted to the following Google Drive folder.


Date Topic Ref. Speaker
2/9 No talk
2/16 Profile Decomposition for energy-supercritical wave equations [1] Xiaoyu
2/23 Profile decomposition for the Navier-Stokes equations + quantitative approach [2,3] Jason
3/1 Introduction to BV theory for hyperbolic systems. [5,17] Amy
3/8 Breakdown of smooth solutions to hyperbolic systems in 1 spatial dimension. [4] Tom
3/15 No talk
3/22 Intro to breakdown of solutions to hyperbolic systems in d=3. [4,6,7,8] Ovidiu
3/29 No talk (Spring break)
4/5 Shock continuation in d=3. [9,10,11] Ryan
4/12 Shock description in 3d under the assumption of radial symmetry. [12,13] Ely
4/19 No talk
4/26 Introduction to the geometric method to describing shock formation. [8,14,15] Yuchen
4/30 C^0-inextendibility of the Schwartzchild spacetime (note: HADES timeslot, room 732) [16] Ning
5/2 Introduction to dispersive shocks (note: Thursday, 3pm, room 748). [20,21,22,23] Zhongkai
5/3 More about dispersive shocks. [20,21,22,23] Federico
5/10 Construction of global solutions to 3D compressible Euler with zero vorticity [18,19] Dongxiao


[1] R. Killip and M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions
[2] I. Gallagher, G. S. Koch and F. Planchon, A profile decomposition approach to the L_t^{\infty}(L_x^3) Navier-Stokes regularity criterion
[3] T. Tao, Quantitative bounds for critically bounded solutions to the Navier-Stokes equations
[4] L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations
[5] C. Dafermos Hyperbolic conservation laws in continuum physics
[6] T. Sideris, Formation of singularities in three-dimensional compressible fluids
[7] F. John, Blow-up of radial solutions of u_{tt}=c^2(u_t)\Delta u in three space dimensions.
[8] G. Holzegel, S. Klainerman, J. Speck, W. Wong, Small-data shock formation in solutions to 3D quasilinear wave equations: an overview.
[9] A. Majda, The existence and stability of multi-dimensional shock fronts
[10] A. Majda, The existence of multi-dimensional shock fronts
[11] A. Majda, The stability of multi-dimensional shock fronts
[12] Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with spherical initial data
[13] D. Christodoulou, A. Lisibach, Shock Development in Spherical Symmetry
[14] D. Christodoulou, The formation of shocks in 3-Dimensional Fluids
[15] J. Speck, Shock formation in small-data solutions to 3D quasilinear wave equations
[16] J. Sbierksi, The C^0-inextendibility of the Schwartzchild spacetime and the spacelike diameter in Lorentizan geometry.
[17] A. Bressan, Lecture notes on Hyperbolic Conservation Laws.
[18] J. Luk, J. Speck, The hidden null structure of the compressible Euler equations and a prelude to applications.
[19] D. Yu, Nontrivial global solutions to some quasilinear wave equations in three space dimensions.
[20] P. D. Lax, C. D. Levermore, The small dispersion limit of the Korteweg-de Vries equation. I
[21] P. D. Lax, C. D. Levermore, The small dispersion limit of the Korteweg-de Vries equation. II
[22] P. D. Lax, C. D. Levermore, The small dispersion limit of the Korteweg-de Vries equation. III
[23] T. Claeys, T. Grava, Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach