In this post, I discuss a rather classical roadmap khổng lồ obtain the non-uniqueness of
weak solutions lớn the classical incompressible Euler equations; namely, focusing on the two-dimensional case, which reads in the vorticity formulation for vorticity function
:
on
, with initial vorticity in
(hence, vorticity remains in
for all times). It’s known, going far back to Yudovich ’63, that weak solutions with bounded vorticity are unique, leaving mở cửa the question of uniqueness of solutions whose vorticity is only in
for
. This blog post is khổng lồ discuss the possible quick roadmap to lớn proving nonuniqueness arising from the instability nature of fluid models, focusing on the Euler equations (1).
Consider the classical incompressible Euler equations in two dimension; namely written in the vorticity formulation, the transport equation for the unknown scalar vorticity
,
posed on a spatial domain name
, where the velocity field
is obtained through the Biot-Savart law
. By construction, the velocity field
is incompressible:
. When dealing with domains with a boundary,
is defined together with a Dirichlet boundary condition that corresponds khổng lồ the no-penetration condition of fluids
on the boundary.
The global well-posedness theory of 2 chiều Euler is classical: (1) smooth initial data give rise khổng lồ solutions that remain smooth for all times (e.g. The Beale-Kato-Majda criterium holds, as vorticity is uniformly bounded for all times; in fact, being transported along the volume-preserving flow, all
norms of vorticity are conserved), và (2) weak solutions with bounded vorticity are quality (see Yudovich ’63).
The major open problem is lớn understand the large time behavior of solutions lớn the 2 chiều Euler equations. This is notoriously difficult, being time-reversible Hamiltonian system and having conserved energy & many invariant Casimir’s
that are conserved for all times, for any reasonable function
. A great reference that discusses in depth this topics and other questions in fluid dynamics is this lecture notes by V. Sverak. Continue Reading »
The Cauchy-Kovalevskaya theorem is a classical convenient tool khổng lồ construct analytic solutions to partial differential equations, which allows one khổng lồ view và treat them as if they are ordinary differential equations:
for unknown functions
in
và
(or some spatial domain). Roughly, if
is locally analytic near a point
then the PDE has a chất lượng solution
which is analytic near
, as established by Cauchy (1842) và generalized by Kovalevskaya as part of her dissertation (1875). The theorem has found many applications such as in fluid dynamics và kinetic theory where it is used to lớn provide existence of analytic solutions including those that are obtained at certain asymptotic limits (e.g., inviscid limit, Boltzmann-to-fluid limit, Landau damping, inviscid damping,…). There are modern formulations of the abstract theorem: see, for instance, Asano, Baouendi & Goulaouic, Caflisch, Nirenberg, và Safonov. In this blog post, I present generator functions, as an alternative approach khổng lồ the use of the Cauchy-Kovalevskaya theorem, recently introduced in my joint work with E. Grenier (ENS Lyon), and discuss the versatility & simplicity of their use lớn applications. In addition lớn provide existence of analytic solutions, I will also mention another use of generator functions to capture some physics that would be otherwise missed for analytic data (namely, an analyticity framework khổng lồ capture physical phenomena that are not seen for analytic data!). Continue Reading »
Sanchit Chaturvedi (Stanford), Jonathan Luk (Stanford), & I just submitted the paper “The Vlasov–Poisson–Landau system in the weakly collisional regime”, where we prove Landau damping & extra dissipation for plasmas modeled by the physical Vlasov-Poisson-Landau system in the weakly collisional regime
, where
is the collisional parameter. The results are obtained for Sobolev data that are
-close khổng lồ global Maxwellians on the torus
. While Landau damping is a classical subject in plasma physics that predicts mixing và relaxation without dissipation of the electric field in a plasma, extra dissipation arises due to the interplay between phase mixing & entropic relaxation, or between transport and diffusion, which enhances decay to lớn a faster rate than the usual diffusion rate. The result is similar to lớn that obtained by Bedrossian ’17 for the Vlasov–Poisson–Fokker–Planck equation & by Masmoudi-Zhao ’19 for Navier-Stokes equations near Couette. However, unlike these works, the linearized operator cannot be inverted explicitly due to the complexity of the Landau collision operator. A framework follows, which is purely energy method, combining Guo’s weighted energy method with the hypocoercive energy method & the Klainerman’s vector field method. In this blog post, I give a flavor of our proof.
For the full post, follow this link hosted by Penn State University.
Dafermos và Rodnianski introduced an
-weighted vector field method lớn obtain boundedness và decay of solutions to lớn wave equations on a Lorentzian background, avoiding to use global vector field multipliers và commutators with weights in
và thus proving lớn be more robust in dealing with đen holes such as in Schwarzschild và Kerr background metrics. The approach has found numerous applications; see, for instance, Dafermos-Rodnianski, Moschidis, or Keir for many insightful discussions. In this blog post, I will give some basic details of this approach, based on lecture notes of S. Klainerman, focusing mostly on the flat Minkowski spacetime.
