Math 290: PDE Learning Seminar (Spring 2024)
Shocks!
Abstract
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 energycritical 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 selfcontained, 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.
Coordinates
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 energysupercritical wave equations

[1]

Xiaoyu

2/23

Profile decomposition for the NavierStokes 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^0inextendibility 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

References
[1] 
R. Killip and M. Visan, The defocusing energysupercritical 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) NavierStokes regularity criterion 
[3] 
T. Tao,
Quantitative bounds for critically bounded solutions to the NavierStokes equations

[4] 
L. HÃ¶rmander,
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 threedimensional compressible fluids

[7] 
F. John,
Blowup 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,
Smalldata shock formation in solutions to 3D quasilinear wave equations: an overview.

[9] 
A. Majda,
The existence and stability of multidimensional shock fronts

[10] 
A. Majda,
The existence of multidimensional shock fronts

[11] 
A. Majda,
The stability of multidimensional shock fronts

[12] 
Yin,
Formation and construction of a shock wave for 3D 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 3Dimensional Fluids

[15] 
J. Speck,
Shock formation in smalldata solutions to 3D quasilinear wave equations

[16] 
J. Sbierksi,
The C^0inextendibility 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 Kortewegde Vries equation. I

[21] 
P. D. Lax, C. D. Levermore,
The small dispersion limit of the Kortewegde Vries equation. II

[22] 
P. D. Lax, C. D. Levermore,
The small dispersion limit of the Kortewegde Vries equation. III

[23] 
T. Claeys, T. Grava,
Universality of the Breakup Profile for the KdV Equation in the Small Dispersion Limit Using the RiemannHilbert Approach
