Formation of singularity in the Euler-Possion equations

we consider the solution of the system

\[U_t+A(U) U_x =(E,0)^{T} \]

where \(U=(u,\rho)^T,E=-V_x, V_{xx}=-s \rho\)

• Theorem 1
  If either of \((u(x,0) \pm \gamma^{1/2} (2/(\gamma-1)) \rho^{(\gamma-1)/2})_x\) is sufficiently negative at some point , then the solution \(u,\rho\) leaves \(C^1\) in finite time.

• Proof
  the eigenvalue of \(A(U)\) are \(\mu_{\pm}=u \pm \gamma^{1/2} \rho^{(\gamma-1)/2}\)
  the corresponding right eigenvector are \(r_{\pm}\). The Riemann invariant \(w_{\pm}\) satisfy that \(D_{(u,\rho)} w_{\pm} \cdot r_{\pm}=0\)

multiplying the system by \(D w_{\pm}\) , we find that

\[(w_j)_t +\mu_j (w_j)_x =D w_j \cdot (E,0)^{T}=f'_{\pm} E \]

Integral along the characteristics, we find that $$|w_j(t)-w_j(0)| < |f^{'}j E|$$

where \(x_j\). satisfy

\[\frac{d}{dt}. x_j =\mu_j(x(t),t) \]

Let \(p_j=(w_j )_x\), \(L_j:=\partial_t +\mu_j \partial_x\)
we have. the Ricatii ODE for \(p_j\)

\[L_j p_j +1/4( 3-\gamma)p_i p_j =s\rho -1/4(1+\gamma) p_j^2 \]