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Formation and construction of a shock wave for one dimensional $n\times n$ strictly hyperbolic conservation laws with small smooth initial data

Min Ding, Huicheng Yin

TL;DR

The paper addresses shock formation in 1-D $n\times n$ strictly hyperbolic conservation laws with small smooth data by developing a geometric blowup framework. It introduces a blowup system via a coordinate transform and a careful variable redefinition, establishes solvability and detailed bounds near the blowup point, and proves the existence of a weak entropy $i$-shock emanating from the unique blowup point with a cusp envelope. The results yield precise local asymptotics for the blowup rate and cusp structure, and construct approximate shock solutions that converge to a genuine shock satisfying Rankine–Hugoniot and Lax entropy conditions; the approach extends to several large-scale hyperbolic systems. These findings provide a rigorous mechanism for shock formation and construction in multidimensional contexts, with applications to 2-D Euler, 1-D MHD, and elastic-wave models. The framework offers a pathway to analyze and simulate shock development in complex hyperbolic systems under small smooth perturbations.

Abstract

Under the genuinely nonlinear assumption for 1-D $n\times n$ strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic nondegenerate condition. At first, near the unique blowup point we give a precise description on the space-time blowup rate of the smooth solution and meanwhile derive the cusp singularity structure of characteristic envelope. These results are established through extending the smooth solution of the completely nonlinear blowup system across the blowup time. Subsequently, by utilizing a new form on the resulting 1-D strictly hyperbolic system with $(n-1)$ good components and one bad component, together with the choice of an efficient iterative scheme and some involved analyses, a weak entropy shock wave starting from the blowup point is constructed. As a byproduct, our result can be applied to the shock formation and construction for the 2-D supersonic steady compressible full Euler equations ($4\times 4$ system), 1-D MHD equations ($5\times 5$ system), 1-D elastic wave equations ($6\times 6$ system) and 1-D full ideal compressible MHD equations ($7\times 7$ system).

Formation and construction of a shock wave for one dimensional $n\times n$ strictly hyperbolic conservation laws with small smooth initial data

TL;DR

The paper addresses shock formation in 1-D strictly hyperbolic conservation laws with small smooth data by developing a geometric blowup framework. It introduces a blowup system via a coordinate transform and a careful variable redefinition, establishes solvability and detailed bounds near the blowup point, and proves the existence of a weak entropy -shock emanating from the unique blowup point with a cusp envelope. The results yield precise local asymptotics for the blowup rate and cusp structure, and construct approximate shock solutions that converge to a genuine shock satisfying Rankine–Hugoniot and Lax entropy conditions; the approach extends to several large-scale hyperbolic systems. These findings provide a rigorous mechanism for shock formation and construction in multidimensional contexts, with applications to 2-D Euler, 1-D MHD, and elastic-wave models. The framework offers a pathway to analyze and simulate shock development in complex hyperbolic systems under small smooth perturbations.

Abstract

Under the genuinely nonlinear assumption for 1-D strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic nondegenerate condition. At first, near the unique blowup point we give a precise description on the space-time blowup rate of the smooth solution and meanwhile derive the cusp singularity structure of characteristic envelope. These results are established through extending the smooth solution of the completely nonlinear blowup system across the blowup time. Subsequently, by utilizing a new form on the resulting 1-D strictly hyperbolic system with good components and one bad component, together with the choice of an efficient iterative scheme and some involved analyses, a weak entropy shock wave starting from the blowup point is constructed. As a byproduct, our result can be applied to the shock formation and construction for the 2-D supersonic steady compressible full Euler equations ( system), 1-D MHD equations ( system), 1-D elastic wave equations ( system) and 1-D full ideal compressible MHD equations ( system).
Paper Structure (17 sections, 18 theorems, 458 equations, 4 figures)

This paper contains 17 sections, 18 theorems, 458 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Geometric blowup of the hyperbolic system
  3. Simplification of \ref{['Y-1']} and the resulting blowup system
  4. Reformulation of blowup system \ref{['eq:2.1']}
  5. Boundary values of blowup system \ref{['eq:2.1']} on the $i$-th characteristics
  6. Solvability of the blowup system \ref{['eq:2.1']} and proof of Theorem \ref{['T:2.1']}
  7. Precise descriptions on the $i-$shock formation
  8. Analysis on the pre-shock wave near the blowup point
  9. The behavior of $y^{\epsilon}_{\pm}(x, t)$ in cusp domain
  10. Estimates on the pre-shock wave
  11. Approximate shock solutions
  12. Reformulated problem
  13. Iteration schemes for resulting initial-value and initial-boundary value problems
  14. The proof of Lemma $4.1$
  15. Convergence of the approximate shock solutions and proofs of Theorem \ref{['T:2.14']} and Theorem \ref{['DY-1']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Provided that the generic nondegenerate condition Eq:2.1 holds and $eq:1.1_1$ is genuinely nonlinear with respect to the $i$-th eigenvalue $\lambda_i(u)$ ($1\leq i\leq n$), there exists a unique solution $u(x,t)\in C(\mathbb{R}\times[0, T_{\epsilon}])\cap C^1(\mathbb{R}\times [0, T_{\epsilon}))$ to

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 31 more