Blowup of small data solutions for a quasilinear wave equation in two space dimensions

For the quasilinear wave equation \partial_t^2u - \Delta u = u_t u_{tt}, we analyze the long-time behavior of classical solutions with small (not rotationally invariant) data. We give a complete asymptotic expansion of the lifespan and describe the solution close to the blowup point. It turns out that this solution is a ``blowup solution of cusp type,'' according to the terminology of the author.Comment: 31 pages, published versio

Semilinear geometric optics with boundary amplification

We study weakly stable semilinear hyperbolic boundary value problems with highly oscillatory data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency $\beta$ in the hyperbolic region. As a consequence of this degeneracy there is an amplification phenomenon: outgoing waves of amplitude O(\eps^2) and wavelength \eps give rise to reflected waves of amplitude O(\eps), so the overall solution has amplitude O(\eps). Moreover, the reflecting waves emanate from a radiating wave that propagates in the boundary along a characteristic of the Lopatinskii determinant. An approximate solution that displays the qualitative behavior just described is constructed by solving suitable profile equations that exhibit a loss of derivatives, so we solve the profile equations by a Nash-Moser iteration. The exact solution is constructed by solving an associated singular problem involving singular derivatives of the form \partial_{x'}+\beta\frac{\partial_{\theta_0}}{\eps}, $x'$ being the tangential variables with respect to the boundary. Tame estimates for the linearization of that problem are proved using a first-order calculus of singular pseudodifferential operators constructed in the companion article \cite{CGW2}. These estimates exhibit a loss of one singular derivative and force us to construct the exact solution by a separate Nash-Moser iteration. The same estimates are used in the error analysis, which shows that the exact and approximate solutions are close in $L^\infty$ on a fixed time interval independent of the (small) wavelength \eps. The approach using singular systems allows us to avoid constructing high order expansions and making small divisor assumptions

Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions

We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.Comment: 43 page

Space Propagation of Instabilities in Zakharov Equations

In this paper we study an initial boundary value problem for Zakharov's equations, describing the space propagation of a laser beam entering in a plasma. We prove a strong instability result and prove that the mathematical problem is ill-posed in Sobolev spaces. We also show that it is well posed in spaces of analytic functions. Several consequences for the physical consistency of the model are discussed.Comment: 39

Smooth type II blow up solutions to the four dimensional energy critical wave equation

International audienceWe exhibit $\mathcal C^{\infty}$ type II blow up solutions to the focusing energy critical wave equation in dimension $N=4$. These solutions admit near blow up time a decomposiiton u(t,x)=\frac{1}{\lambda^{\frac{N-2}{2}}(t)}(Q+\e(t))(\frac{x}{\lambda(t)}) \ \ \mbox{with} \ \ \|\e(t),\pa_t\e(t)\|_{\dot{H}^1\times L^2}\ll1 where $Q$ is the extremizing profile of the Sobolev embedding $\dot{H}^1\to L^{2^*}$, and a blow up speed $$\lambda(t)=(T-t)e^{-\sqrt{|\log (T-t)|}(1+o(1))} \ \ \mbox{as} \ \ t\to T.$

Generalized harmonic spatial coordinates and hyperbolic shift conditions

We propose a generalization of the condition for harmonic spatial coordinates analogous to the generalization of the harmonic time slices introduced by Bona et al., and closely related to dynamic shift conditions recently proposed by Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic spatial coordinates imply a condition for the shift vector that has the form of an evolution equation for the shift components. We find that in order to decouple the slicing condition from the evolution equation for the shift it is necessary to use a rescaled shift vector. The initial form of the generalized harmonic shift condition is not spatially covariant, but we propose a simple way to make it fully covariant so that it can be used in coordinate systems other than Cartesian. We also analyze the effect of the shift condition proposed here on the hyperbolicity of the evolution equations of general relativity in 1+1 dimensions and 3+1 spherical symmetry, and study the possible development of blow-ups. Finally, we perform a series of numerical experiments to illustrate the behavior of this shift condition.Comment: 18 pages and 12 figures, extensively revised version explaining in the new Section IV how the shift condition can be made 3-covarian

Global Solutions for Incompressible Viscoelastic Fluids

We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near equilibrium initial data. The results hold in both two and three dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals and mixture problems.Comment: We prove the existence of global smooth solutions to the Cauchy problem for the incompressible viscoelastic system of Oldroyd-B type in the case of near equilibrium initial dat

Asymptotically simple solutions of the vacuum Einstein equations in even dimensions

We show that a set of conformally invariant equations derived from the Fefferman-Graham tensor can be used to construct global solutions of the vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich's result on the future hyperboloidal stability of Minkowski space-time, and extends its validity to even dimensions.Comment: 25p