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Blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$

Tristan Buckmaster, Jiajie Chen

Abstract

In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$. This work builds on the earlier results of Shao, Wei, and Zhang [SWZ2024a,SWZ2024b], reducing the order of the nonlinearity from $29$ to $7$ in $\mathbb{R}^{4+1}$. As in [SWZ2024a,SWZ2024b], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations. As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result in [SWZ2024a] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism.

Blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$

Abstract

In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in . This work builds on the earlier results of Shao, Wei, and Zhang [SWZ2024a,SWZ2024b], reducing the order of the nonlinearity from to in . As in [SWZ2024a,SWZ2024b], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations. As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result in [SWZ2024a] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism.

Paper Structure

This paper contains 79 sections, 32 theorems, 450 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Self-similar ansatz
  3. Main results
  4. Comparison with the methods in existing works
  5. Related results
  6. Smooth Implosions
  7. Shock formation
  8. Computer-assisted proofs in fluids
  9. Autonomous ODE for the self-similar profile
  10. ODE for the profile
  11. Constraints on the parameters
  12. Double roots and the sonic point
  13. Notations and parameters
  14. Renormalization
  15. Change of coordinates
  16. ...and 64 more sections

Key Result

Theorem 1.1

Let $d = 4, p = 7, \ell = \frac{4}{p-1}+1$, and $n$ be an odd number and large enough. There exists $\gamma_n$ accumulating at $\ell^{-1/2}$ with $\gamma_n >\ell^{-1/2} , b_n = \frac{ d-1 }{ \ell (\gamma_n + 1)} - 1>0$ such that the ODE eq:ODE admits a smooth solution $V^ { (\gamma_n) } \in C^{\in for any $Z > 0$, and $V(Z) = Z V(Z^2)$ for some function $V \in C^{\infty}[0, \infty)$. Moreover,

Figures (2)

  • Figure 1: Illustrations of phase portrait of the $(Z, V)$-ODE \ref{['eq:ODE']}. The solution curve is in black and $Z_{\pm}(V), Z_{V}(V)$ defined in \ref{['eq:root_mono_ZV0']} are roots of $\Delta_Z, \Delta_V$.
  • Figure 2: Illustrations of phase portrait of the $(Y, U)$-ODE \ref{['eq:ODE']}. The black curve represents the $C^{\infty}$ solution curve, $(Y_F, U_F)$ (orange) defined in \ref{['eq:QO_glue']} is the solution curve near $Q_O= (1/d, \infty)$, $B_l^f$ (purple) and $B_u^f$ (green) are barrier functions defined in \ref{['eq:bar_far_l']}, \ref{['eq:bar_far_u']}, $U_{\Delta_Y}$ (red) and $U_{\Delta_U}$ (blue) defined in \ref{['eq:root_UY']} are roots of $\Delta_Y, \Delta_U$. The domain $\Omega_B^f$ is defined in \ref{['eq:bar_region']} and $\Omega_{ {tri},i}$ in \ref{['eq:dom_tri']}.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2: Theorem 1.1 shao2024blow
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1: Lemma 2.1 shao2024self
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 45 more