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Register a new userOn the existence and stability of blowup for wave maps into a negatively curved target
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We consider wave maps on $(1+d)$-dimensional Minkowski space. For each dimension $dgeq 8$ we construct a negatively curved, $d$-dimensional target manifold that allows for the existence of a self-similar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem.
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We consider the energy-critical half-wave maps equation $$partial_t mathbf{u} + mathbf{u} wedge | abla| mathbf{u} = 0$$ for $mathbf{u} : [0,T) times mathbb{R} to mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $mathbb{S}^2$. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Mobius group on the target $mathbb{S}^2$ applied to half-harmonic maps from $mathbb{R}$ to $mathbb{S}^2$. Complementing our classification result, we carry out a detailed analysis of the linearized operator $L$ around half-harmonic maps $mathbf{Q}$ with arbitrary degree $m geq 1$. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $mathbf{Q}$. Moreover, we give a full description of the spectrum of $L$ by finding all its $L^2$-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for $L$ and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $mathbb{R}$ to the unit circle $mathbb{S}$. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.
We consider the heat flow of corotational harmonic maps from $mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators.
We consider co-rotational wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere for $dgeq 3$ odd. This is an energy-supercritical model which is known to exhibit finite-time blowup via self-similar solutions. Based on a method developed by the second author and Schorkhuber, we prove the asymptotic nonlinear stability of the ground-state self-similar solution.
We establish Strichartz estimates for the radial energy-critical wave equation in 5 dimensions in similarity coordinates. Using these, we prove the nonlinear asymptotic stability of the ODE blowup in the energy space.
We consider co-rotational wave maps from (1+3)-dimensional Minkowski space into the three-sphere. This model exhibits an explicit blowup solution and we prove the asymptotic nonlinear stability of this solution in the whole space under small perturbations of the initial data. The key ingredient is the introduction of a novel coordinate system that allows one to track the evolution past the blowup time and almost up to the Cauchy horizon of the singularity. As a consequence, we also obtain a result on continuation beyond blowup.
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