Osaka City University Advanced Mathematical Institute (OCAMI)MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics

OCAMI Reports Vol. 8 (2021)
doi: 10.24544/ocu.20210908-001

Quasi-linear PDEs in fluids

Organized byKen AbeKyudong Choi
Kazuo Yamazaki

February 21–22, 2021

Abstract. This workshop held by online to conduct international researchexchanges on quasi-linear partial differential equations related with fluid mechanics.

2020 Mathematics Subject Classification.
35Q35, 35K90.

Key words and Phrases.
vortex rings, Euler equations, singularity formation, quasi-geostrophic equation,Burgers equation, ill-posedness, martingale solutions© 2021 OCAMI.
OCAMI. Quasi-linear PDEs in fluids. OCAMI Reports. Vol. 8, Osaka City University Advanced
Mathematical Institute. 2021, 11pp. doi: 10.24544/ocu.20210908-001

2OCAMI Reports Vol. 8 (2021)

PrefaceThis volume of OCAMI Reports is proceedings of the online workshop “Quasilinear PDEs in fluid” held from February 20th to 21st in 2021. The workshop aimed atexchanging researches on quasi-linear PDEs related to fluid mechanics among young
researchers. It was realized by completely online due to COVID-19 though originallyplaned as a face-to-face meeting at Osaka in the summer of 2020.
The 7 speakers gave 40 mins lectures by zoom. In the 1st day, blow-up and illposedness results on the Euler equations are reported by Profs. Elgindi and Jeong. Profs.
Choi and Aiki reported stability and large time behavior of vortex rings. In the 2ndday, Prof. Inahama reported local wellposedness of the stochastic QG equation. Prof.
Iwabuchi reported large time behavior of the Burgers equation. Prof. Rongchan Zhu
reported non-uniqueness of Markov solutions to the stochastic Euler equations. Theprogram is available at https://sites.google.com/view/qpdes.
The workshop had about 80 participants from Japan, South Korea, India, UK,
Canada and US, and conducted international and interdisciplinary exchanges on quasilinear PDEs.

August 2021
Ken AbeQPDEs in fluids

3Organizers
Ken AbeDepartment of Mathematics, Graduate School of Science, Osaka City University, 3-3-138,
Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
E-mail address: kabe@osaka-cu.ac.jp
Kyudong Choi
Department of Mathematical Sciences, Ulsan National Institute of Science and Technology,
UNIST-gil 50, Ulsan, 44919, Republic of Korea
E-mail address: kchoi@unist.ac.kr
Kazuo Yamazaki
Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 794091042, US
E-mail address: kyamazak@ttu.edu

4

OCAMI Reports Vol. 8 (2021)

Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Organizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23Masashi AikiOn the Head-on Collision of Coaxial Vortex Rings

5. . . . . . . . . . . . . .

Kyudong Choi
Stability of some exact solutions of incompressible Euler equations. . . . .

6

. . . . . . . . . . . . . . . .

7

Yuzuru Inahama
Paracontrolled quasi-geostrophic equation with space-time white noise . . . .

8Tsukasa Iwabuchi
Analyticity and large time behavior for the Burgers equation with the criticaldissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9In-jee Jeong
Ill-posedness for incompressible fluid models at critical Sobolev regularity . .

10Rongchan Zhu
On ill- and well-posedness of dissipative martingale solutions to stochastic 3D
Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11Tarek Elgindi
Singularity formation in the 3D Euler equation

QPDEs in fluids

5

On the Head-on Collision of Coaxial Vortex Rings
Masashi Aiki
The study of the interaction of coaxial vortex rings dates back to the pioneering
paper by Helmholtz (1858). Helmholtz considered vortex motion in a incompressible
and inviscid fluid based on the Euler equations. His study includes the motion of
circular vortex filaments, and he observed that motion patterns such as head-on collision may occur. Since then, many researches have been done on head-on collision of
coaxial vortex rings, and interaction of coaxial vortex rings in general. Most of theseresearch are either experiments conducted in a laboratory or numerical simulationsof the Navier–Stokes equations, and the rigorous mathematical treatment of head-on
collision of vortex rings are very scarce. In light of this, we consider the head-on collision of two coaxial vortex rings, which have circulations of opposite sign, describedas the motion of two coaxial circular vortex filaments under the localized induction
approximation. A vortex filament is a space curve on which the vorticity of the fluid
is concentrated. In our present work, we approximated thin vortex structures, such asvortex rings, by vortex filaments, and described the motion as the motion of a curvein the three-dimensional Euclidean space. In this formulation, a vortex ring is a spacecurve in the shape of a circle. We prove the existence of solutions to a system of nonlinear partial differential equations modelling the interaction of two vortex filaments
proposed by the speaker which exhibit head–on collision. We also give a necessary
and sufficient condition for the initial configuration and parameters of the filaments forhead-on collision to occur. Our results suggest that there exists a critical value γ∗ > 1for the ratio γ of the magnitude of the circulations satisfying the following. Whenγ ∈ [1, γ∗ ], two approaching rings will collide, and when γ ∈ (γ∗ , ∞), the ring with thelarger circulation passes through the other and then separate indefinitely. As far as
the speaker knows, the existence of such threshold γ∗ is only indirectly suggested via
numerical investigations of the head-on collision of coaxial vortex rings, for example byInoue, Hattori, and Sasaki (2000). Hence, our result is the first to obtain the thresholdin a way that is possible to numerically calculate γ∗ , as well as prove that the thresholdexists in a framework of a mathematical model.

6OCAMI Reports Vol. 8 (2021)

Stability of some exact solutions of incompressible
Euler equations
Kyudong Choi
We visit results on stability of exact vortex solutions of the incompressible Euler
equations including circular vortex patches, Lamb’s dipoles, Hill’s spherical vortices.
They are solutions minimizing energy under natural constraints so that their stability
comes from calculus of variations. This talk is based on joint works with K. Abe (OsakaCity Univ.) and with D. Lim (UNIST).

QPDEs in fluids

7Singularity formation in the 3D Euler equation
Tarek Elgindi
I discuss various issues related to the blow-up problem for the 3D Euler equation.

8OCAMI Reports Vol. 8 (2021)

Paracontrolled quasi-geostrophic equation withspace-time white noise
Yuzuru Inahama
We study the stochastic dissipative quasi-geostrophic equation with space-time
white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is toformulate and solve this equation locally in time in the framework of paracontrolledcalculus when the differential order of the main term, the fractional Laplacian, is larger
than 7/4. No renormalization has to be done for this model. (This is a joint work withYoshihiro Sawano)QPDEs in fluids

9Analyticity and large time behavior for the Burgers
equation with the critical dissipation
Tsukasa Iwabuchi
We consider the Burgers equation with the critical dissipation.
∂t u + (−∂x2 )1/2 + u∂x u = 0, t > 0, x ∈ Ru(0, x) = u0 (x)We impose that the initial data is smooth and integrable, and it is known that thesolution exists globally in time. In this talk, it will be shown that every solution
behaves like the Poisson kernel as time tends to infinity. Analyticity in spacetime willbe also discussed.

10OCAMI Reports Vol. 8 (2021)

Ill-posedness for incompressible fluid models at
critical Sobolev regularity
In-jee Jeong
We consider the incompressible fluid equations including the Euler and SQG equations in critical Sobolev spaces, which are Sobolev spaces with the same scaling as theLipschitz norm of the velocity. We show that initial value problem for the equations
are ill-posed at critical regularity. This is based on joint works with Tarek Elgindi,Tsuyoshi Yoneda, and Junha Kim.

QPDEs in fluids

11

On ill- and well-posedness of dissipative martingalesolutions to stochastic 3D Euler equations
Rongchan Zhu
We are concerned with the question of well-posedness of stochastic three dimensional incompressible Euler equations. In particular, we introduce a novel class ofdissipative solutions and show that (i) existence; (ii) weak–strong uniqueness; (iii)
non-uniqueness in law; (iv) existence of a strong Markov solution; (v) non-uniquenessof strong Markov solutions; all hold true within this class. Moreover, as a byproduct of(iii) we obtain existence and non-uniqueness of probabilistically strong and analyticallyweak solutions defined up to a stopping time and satisfying an energy inequality. This
talk is based on joint work with Martina Hofmanova and Xiangchan Zhu.