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The onset of instability in first-order systems

(this post was also posted here on my new blog address:http://blog.toannguyen.org/ )

Nicolas Lerner, Ben Texier and I have just submitted to arxiv our long paper on “The onset of instability in first-order systems”, in which we prove the Hadamard’s instability for first-order quasilinear systems that lose its hyperbolicity in positive times. Precisely, we consider the Cauchy problem for the following first-order systems of partial differential equations:

\displaystyle \left\{ \begin{aligned} u_t + \sum_{1 \leq j \leq d} A_j(t,x,u)u_{x_j} &= F(t,x,u), \\ u_{\vert_{t=0}} & = u_0 \end{aligned}\right. \ \ \ \ \ (1)

where ${t \geq 0,}$ ${x \in {\mathbb R}^d,}$ ${u \in {\mathbb R}^N,}$ the maps ${A_j}$ are smooth from ${{\mathbb R}_+ \times {\mathbb R}_x^d \times {\mathbb R}_u^N}$ to the space of ${N\times N}$ real matrices and ${F}$ is smooth from ${{\mathbb R}_+ \times {\mathbb R}_x^d \times {\mathbb R}^N_u}$ into ${{\mathbb R}^N.}$ Many models in physics can be written in the above form: for instance, the p-system in gas dynamics, compressible Euler equations, or general conservation laws.

The wellposedness (i.e., existence, uniqueness, and continuity of the solution flow, say in ${H^s}$ Sobolev spaces) for (1) has been studied extensively by many authors; for instance, Friedrichs, G\r{a}rding, Hörmander, Lax, Majda, among many others. A nice reference on the subject is this book by Métivier. Essentially, it asserts that the (strict) hyperbolicity is sufficient for the local-in-time well-posedness. Here, by hyperbolicity, we mean all the eigenvalues of the principal symbol matrix

$\displaystyle \mathcal{A}(x,u,\xi):=\sum_{1 \leq j \leq d} A_j(0,x,u) \xi_i \ \ \ \ \ (2)$

are real, for all ${(x,u,\xi)\in \mathbb{R}^d \times \mathbb{R}^N\times \mathbb{R}^d}$. The hyperbolicity is strict, when in addition the eigenvalues are simple. The strict hyperbolicity for the local wellposedness can be relaxed to the symmetrizability or even microlocal symmetrizability of the system (in fact, this is exactly how the classical proof goes, using the symmetry to obtain a priori Friedrichs type of energy estimates).

In this paper, we establish a result in the reverse direction, or often refer to as a Lax-Mizohata’s theorem: necessity of the hyperbolicity for well-posedness. There are instabilities or ill-posedness in the case of the loss of hyperbolicity. For instance, consider a simple toy model problem:

$\displaystyle u_t + i u_x = 0,\ \ \ \ \ (3)$

with the principal symbol being ${i\xi}$, which is not real. Here, ${u}$ is a scalar complex function. We can solve it by the Fourier transform: ${\hat u(t,\xi) = e^{\xi t} \hat u_0(\xi)}$, in which ${\hat u}$ denotes the Fourier transform of ${u}$ with respect to variable ${x}$. Hence, we can see that any ${C^0}$ solution, with initial data ${\hat u_0}$ has support contained in the positive real line, must have the analytic initial data ${u_0}$ or in the other words, there is no ${C^0}$ solution for non-analytic initial data. This type of backward regularization is well-known for elliptic operators. Such a strong analytic instability is however limited to scalar complex equations. As for systems, Lax, and then Mizohata, proved the necessity of hyperbolicity for well-posedness in a ${C^\infty}$ framework in the case of systems of linear equations.

In term of the regularity of the solution map, we consider, for instance, initial data of the form ${u_0(x) = \epsilon^{N}e^{ix/\epsilon}}$, the unique solution is ${u(x,t) =\epsilon^N e^{(t + ix)/\epsilon}}$. The ${H^m}$ norm of the initial data is arbitrarily small of order ${\epsilon^{N-m}}$, whereas the ${L^2}$ norm of the solution is growing at an exponential rate: ${\epsilon^Ne^{t/\epsilon}}$, which is arbitrarily large within a very short time: ${t_\epsilon = 2N\epsilon \log 1/\epsilon}$. The ${L^2}$ norm of the solution at ${t=t_\epsilon}$ is of order ${\epsilon^{-N}}$. Following the same calculations, we can in fact show that for all ${m}$, for all ${\alpha \in (0,1]}$, there are ${x_\epsilon \rightarrow 0, t_\epsilon \rightarrow 0}$ and solutions ${u_\epsilon}$ of (3) so that

$\displaystyle \lim_{\epsilon \rightarrow 0}\frac{\| u_\epsilon(t_\epsilon)\|_{L^2 ([-x_\epsilon, x_\epsilon])}}{\| u_\epsilon (0)\|^\alpha_{H^m([-1,1])}} = +\infty. \ \ \ \ \ (4)$

Instability occurs in a very weak norm (indeed, one can replace ${L^2}$ by any negative-index Sobolev norm: ${H^{-s}}$ in the above estimate), within a vanishing time ${t_\epsilon \rightarrow 0}$ and in a vanishing spatial region. Metivier obtained such an instability result for general quasi-linear systems of equations where the hyperbolicity is violated at initial time.

In this paper, we are interested in the transition from hyperbolicity to non-hyperbolicity, that is the limiting case where hyperbolicity holds at initial time, but is violated at positive times. For example, the scalar complex equation

$\displaystyle u_t + 2i t u_x = 0$

has a similar instability as in (4), despite the fact that the equation is hyperbolic at initial time ${t=0}$ (there is still an exponential amplify: ${e^{t^2/\epsilon}}$ in any positive time). Another example is the following complex Burgers equation:

$\displaystyle u_t + uu_x = i, \qquad u_{\vert_{t=0}} = \omega(x).$

Assume that ${\omega}$ is real-valued. The equation is thus hyperbolic at ${t=0}$, however it is non-hyperbolic in any positive time since ${u}$ instantaneously becomes complex due to the ${i}$ on the right-hand side.

Our results also apply to the Van der Waals systems in gas dynamics, which read in Lagrangian coordinates

\displaystyle \left\{ \begin{aligned} \partial_t u_1 + \partial_x u_2 & = 0, \\ \partial_t u_2 + \partial_x p(u_1) & = 0. \end{aligned}\right. \ \ \ \ \ (5)

The system is hyperbolic when ${p'(u_1) >0}$ and elliptic when ${p'(u_1) <0}$. Precisely, we prove that if ${(\phi_1,\phi_2)}$ is a smooth solution such that, for some ${x_0 \in {\mathbb R},}$ there holds

$\displaystyle {\rm (i)}\, p'(\phi_1(x_0,0)) < 0, \quad{ \mbox{or} } \quad {\rm (ii)}\,p'(\phi_1(0,x_0)) = 0, \,\,p''(\phi_1(0,x_0)) \partial_x \phi_2(0,x_0) > 0,$

for some point ${x_0}$, then the initial-value problem for (5) is ill-posed in any neighborhood of ${\phi.}$ Condition (i) is an ellipticity assumption and condition (ii) is an open condition on the boundary of the domain of hyperbolicity, which we shall explain our derivation below.

To formulate our assumptions and results, we introduce the linearized principal symbol at a smooth background solution ${\phi}$:

$\displaystyle A(t,x, \xi) := \sum_{1 \leq j \leq d} \xi_jA_j(t,x,\phi(t,x)) \ \ \ \ \ (6)$

and the characteristic polynomial ${P}$ of the principal symbol:

$\displaystyle P(t,x,\xi,\lambda) := \det\big(\lambda I - A(t,x,\xi)\big).\ \ \ \ \ (7)$

Initial ellipticity condition means that there is a zero ${(0,x_0,\xi_0, \lambda_0)}$ of the characteristic polynomial ${P}$ so that ${\lambda_0}$ is not real. The delicate situation is when the symbol is initially hyperbolic, but instantaneously loses its hyperbolicity. In this case, we assume that the real roots occur at most double:

$\displaystyle P(0,x_0,\xi_0, \lambda_0) = \partial_\lambda P(0,x_0,\xi_0, \lambda_0) = 0, \quad \partial_\lambda^2 P(0,x_0,\xi_0, \lambda_0) \neq 0. \ \ \ \ \ (8)$

Would there be an instability? Note that since ${\partial_\lambda P = 0}$ and ${\partial_\lambda^2 P \not =0}$ at ${(0,x_0,\xi_0, \lambda_0)}$, there is some ${\nu = \nu(t,x,\xi)}$ near ${\lambda_0}$ so that we can write

$\displaystyle P(t,x,\xi, \lambda) = P(t,x,\xi, \nu) + (\lambda - \nu)^2 e_0(t,x,\xi,\lambda).$

If we assume in addition that ${\partial_t P \not =0}$, then we can further write ${P(t,x,\xi, \nu) = e_1(t,x,\xi) (t - \theta(x,\xi))}$, and so

$\displaystyle P(t,x,\xi, \lambda) = e_1(t,x,\xi) (t - \theta(x,\xi)) + (\lambda - \nu)^2 e_0(t,x,\xi,\lambda),$

for some smooth functions ${\theta, \nu, e_0, e_1}$ in their arguments. That is, a zero of ${P(t,x,\xi,\lambda)}$ with ${\lambda}$ becomes non-real, if ${e_0 e_1>0}$ and if ${t>\theta(x,\xi)}$. Note that at ${(0,x_0,\xi_0, \lambda_0)}$, ${e_0 = \partial_\lambda^2 P}$ and ${e_1 = \partial_t P}$. This motivates us to impose the following sign condition:

$\displaystyle (\partial^2_\lambda P \partial_t P)(0,x_0,\xi_0, \lambda_0) > 0, \ \ \ \ \ (9)$

leading to instability. We note that our assumption can be calculated in term of given initial data (using the evolution equation (1)). One of our main theorems, which applies to Van der Waals and Klein-Gordon-Zakharov systems, then read:

Theorem 1 Assume that either the initial ellipticity condition or conditions (8)(9) hold for some ${(0,x_0,\xi_0, \lambda_0)}$, and that the other eigenvalues of ${A(0,x_0,\xi_0)}$ are simple. Then the Cauchy problem for system (1) is ill-posed in the vicinity of the reference solution ${\phi}$.

In our paper, we in fact establish the instability result, under a much more general assumption posed on the symbolic flow of the principal symbol. One of the main difficulties in obtaining such an instability result, even in the elliptic case, is the unboundedness of the unstable spectrum, which results in the serious lack of solution representation (or some type of the Duhamel’s principle, which is needed for any available instability analysis in treating nonlinearity). Our analysis in this paper relies on an approximation Duhamel’s presentation for pseudo-differential flows. See our paper for extensive discussions on our results, and their detailed proof in the paper.