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Local Existence and Finite-Time Singularity Formation in the Vlasov-Poisson-Isotropic Landau System

Jin Woo Jang, Junsung Kim

Abstract

The isotropic Landau (Coulomb) operator was introduced in kinetic theory by Krieger and Strain (Comm. Partial Differential Equations, 2012). In this work, we study the spatially inhomogeneous Vlasov--Poisson--isotropic Landau system. We first establish a local--in--time existence theory for the Cauchy problem: for initial data satisfying a suitable smallness condition in an appropriate norm, there exists a non--negative solution on a time interval $[0,T]$, where the lifespan $T$ depends on the size of the initial data. Beyond the local theory, we investigate a mechanism that may lead to the breakdown of global existence. We show that finite--time singularity formation can occur in the gravitationally attractive case, provided that the weak solution satisfies certain a priori regularity and decay assumptions, the initial gravitational field energy exceeds the kinetic energy, and the resulting energy gap dominates the diffusive effect of the collision operator. As a consequence, if the solution is further assumed to belong to a suitable measure space up to the maximal existence time, it collapses to a single point in physical space at that time. The proof of finite--time singularity formation is based on deriving an upper bound for the second spatial moment, which becomes negative in finite time.

Local Existence and Finite-Time Singularity Formation in the Vlasov-Poisson-Isotropic Landau System

Abstract

The isotropic Landau (Coulomb) operator was introduced in kinetic theory by Krieger and Strain (Comm. Partial Differential Equations, 2012). In this work, we study the spatially inhomogeneous Vlasov--Poisson--isotropic Landau system. We first establish a local--in--time existence theory for the Cauchy problem: for initial data satisfying a suitable smallness condition in an appropriate norm, there exists a non--negative solution on a time interval , where the lifespan depends on the size of the initial data. Beyond the local theory, we investigate a mechanism that may lead to the breakdown of global existence. We show that finite--time singularity formation can occur in the gravitationally attractive case, provided that the weak solution satisfies certain a priori regularity and decay assumptions, the initial gravitational field energy exceeds the kinetic energy, and the resulting energy gap dominates the diffusive effect of the collision operator. As a consequence, if the solution is further assumed to belong to a suitable measure space up to the maximal existence time, it collapses to a single point in physical space at that time. The proof of finite--time singularity formation is based on deriving an upper bound for the second spatial moment, which becomes negative in finite time.
Paper Structure (20 sections, 25 theorems, 271 equations, 1 table)

This paper contains 20 sections, 25 theorems, 271 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background on Landau-type Collisional Kinetic Equations
  3. The Vlasov–Poisson–Isotropic Landau System and Main Results
  4. Relationship between Landau and Isotropic Landau Operator and these Basic Properties
  5. Singularity Formation in the Attractive Case
  6. Main Theorems and Strategies
  7. Main Strategies
  8. Construction of the Approximate Sequence
  9. To Study the Finite-Time Singularity Formation
  10. Related Results in the Literature
  11. On the isotropic Landau equation.
  12. On the Weighted Elliptic Regularity Theory.
  13. On Singularity Formation in Various Kinetic Equations
  14. Notations
  15. Outline of the Paper
  16. ...and 5 more sections

Key Result

Theorem 2.3

We simultaneously consider the plasma and gravitational cases $E_f = -(\nabla K\star_x\rho_f)$, for which $K(x)=\pm\frac{1}{4\pi|x|}$. Let $m>3$, $1<p_0<\frac{3}{2}$ and $3<q_0<\frac{11}{3}$, where $p_0,q_0$ denote the integrability exponent associated with the (weighted) Sobolev space in $X_T^m$. T for some $C \in [0,C_0]$, then there exists a time $T = T\!\left(\|f^\textup{in}\|_{\mathcal{C}^2_{

Theorems & Definitions (60)

  • Remark 1.1: Formal conservation and monotonicity properties for VPiL
  • Definition 2.1
  • Definition 2.2: Weak solution
  • Theorem 2.3: Local existence with polynomially decaying initial data
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: Finite-time singularity formation for the gravitationally attractive case
  • Definition 2.8
  • Corollary 2.9: Collapse of mass
  • ...and 50 more