When given an explicit solution to an evolutionary partial differential equation, it is natural to ask whether the solution is stable, and if yes, what is the mechanism for stability and whether this mechanism survives under perturbations of the equation itself. Many familiar linear equations enjoy some notion of stability for the zero solution: solutions of the heat equation dissipate and decay uniformly and exponentially to zero, solutions of the Schrödinger equations disperse at a polynomial rate in time depending on spatial dimension, while solutions of the wave equation enjoy radiative decay (in the presence of at least two spatial dimensions) also at polynomial rates.
For this set of short course sessions, we will focus on the wave equation and its nonlinear perturbations. As mentioned above, the stability mechanism for the linear wave equation is that of radiative decay. Radiative decay depends on the number of spatial dimensions, and hence so does the stability of the zero solution for nonlinear wave equations. By the mid-1980s it was well understood that the stability mechanism survives generally (for “smooth nonlinearities”) when the spatial dimension is at least four, but for lower dimensions (two and three specifically; in dimension one there is no linear stability mechanism to start with) obstructions can arise when the nonlinearities are “stronger” than can be controlled by radiative decay. This led to the discovery of the null condition as a structural condition on the nonlinearities preventing the aforementioned obstructions. But what happens when the null condition is violated? This development spanning a quarter of a century, from F. John’s qualitative analysis of the spherically symmetric case, though S. Alinhac’s sharp control of the asymptotic lifespan, and culminating in D. Christodoulou’s full description of the null geometry, is the subject of this short course.
(1) We will start by reviewing the radiative decay mechanism for wave equations, and indicate the nonlinear stability results for high spatial dimensions. We then turn our attention to the case of three spatial dimensions: after a quick discussion of the null condition for quasilinear wave equations, we sketch, at the semilinear level, what happens when the null condition fails (in particular the asymptotic approximation of the solution by a Riccati equation).
(2) The semilinear picture is built up using a version of the method of characteristics associated with the standard wave operator. Turning to the quasilinear problem we will hence need to understand the characteristic geometry for a variable coefficient wave operator. This leads us to introduce the optical/acoustical function and its associated null structure equations.
(3) From this modern geometric perspective we next discuss, in some detail, the blow-up results obtained in the mid-1980s by F. John for quasilinear wave equations assuming radial symmetry.
(4) Finally, we indicate the main difficulties in extending the analysis to the non-radially-symmetric case, and how they can be resolved à la the recent tour de force of D. Christodoulou. While some knowledge of Lorentzian geometry and dynamics of wave equations will be helpful, this short course should be accessible to also graduate students with training in partial differential equations.
Imperial College London, United Kingdom E-mail address: email@example.com
École Polytechnique Fédérale de Lausanne, Switzerland E-mail address: firstname.lastname@example.org
- Partial Differential Equations Seminar