Quasilinear Systems in One Space Dimension

• Quasilinear Systems in One Space Dimension
  We will consider here quasilinear \(N \times N\) systems of the form

\[U_t +A(U) U_x=0 \]

For $2 \times 2 $ systems, we can use the Riemann invariants to make the system diagonal and show a decoupling of the two modes in finite time, thus reducing the problem to the well known scalar case. This allows us to prove finite time blowup for systems with at least one nonlinearly degenerate eigenvalue.

For general systems, only the case of small data is well understood; approximate decoupling occurs then before blowup, and the behavior of the solution can be partially described up to the time \(\tilde{T}\) (the unknown lifespan). The proof of this striking result makes an essential use of certain boundedness properties of first order derivatives of the solution in \(L_1\)norm . It is then rather easy to obtain upper bounds and lower bounds for the lifespan, by considering the system as an ODE along characteristics and using elementary results on blowup or existence for ODEs.

• $ N=1$
  we assume that \(A\) is genuinely nonlinear in the sense that ** \(A_x(0) \neq 0\)**
  If the smooth initial value \(u_0\) is compactly supported, the lifespan is then given by

\[\frac{1}{T} =\max{\| -A'(u_0)(u_0)_x \|} \]

On each characteristic line $$ x=x_0+ A(u_0(x_0)) t $$

The derivative $q(t)=\partial_x u $ satisfy

\[\boxed{ q'+A'(u_0(x_0)) q^2=0 } \]

\(q\) blow up at time \(\tilde{T}\) like $$ \frac{C}{\tilde{T} -t}$$

• \(L^1\)-boundedness
  For \(t< \tilde{T}\) ,the solution is given by blow up system

\[\begin{align} \partial_T \phi &=A(v) \\ \partial_T v &=0\\ \phi(x,0) &=x\\ v(x,0) &=u_0(x) \end{align} \]

thorough the change of variables

\[x=\phi, t=T \]

then $$ \int | u_x | dx =\int |u'_0 | dX$$
In other words, \(u_x\) does not blow up in the \(L^1\) norm.

• Riemann invariants, simple waves, \(L^1\) boundedness
  For a general system, we denote by \(\lambda_1 < \lambda _2...<\lambda_N\) the real and distinct eigenvalues of \(A(u)\) , with corresponding left and right eigenvectors \(\ell_j ,r_j\).

Integral curves of the field \(L_j=\partial_t +\lambda_j \partial_x\) are called \(j-\) characteristics.
make the change of variables \(u=\Phi(U)\),the system becomes

\[U_t+(\Phi')^{-1} A \Phi' U_x =0 \]

• Riemann invariants

\[r_j \partial_u R^j_k=0,k=1,...N-1 \]

In the case \(N=2\), we set \(R^1_1=w_2,R^2_1=w_1\)
we have $$ (w_j)_t +\lambda_j (w_j)_x =0$$

we will then rewrite the system as

\[w_t+\Lambda(w) w_x =0,\Lambda(w)=diag(\lambda_1,\lambda_2) \]

• Simple wave
  The waves are solutions of the form

\[u=v(\zeta) \]

\[\zeta_t +\lambda_j(v) \zeta_x=0 \]

we will call such a solution a \(j-\) simple wave.

\[u_x=r_j \zeta_x \]

• \(L^1\) boundeness
  Consider a \(C^2\) solution of \(U_t+A(U)U_x=0\) and

\[u_x=\sum w_j r_j \]

Then the \(w_j\) satisfy a system

\[L_i w_i =\sum \gamma_{ijk} w_j w_k,i=1,...,N \]

where \(L_i=\partial_t +\lambda_i \partial_x\)

then

\[\int_\gamma |w_i d\zeta| \leq \int |w_i| dx+\int_D |\sum \Gamma_{ijk} w_j w_k | dxdt \]