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Hans Lindblad

Publications and source records attributed to Hans Lindblad.

At least 37 records · Page 2Linked to original sources
arXiv2016

A sharp counterexample to local existence of low regularity solutions to Einstein's equations in wave coordinates

We are concerned with how regular initial data have to be to ensure local existence for Einstein's equations in wave coordinates. Klainerman-Rodnianski and Smith-Tataru showed that there in general is local existence for data in Sobolev spaces H^s with regularity s>2. We give an example of data in Sobolev spaces with regularity s=2 for which there is no local solution in this space.

math.AP↗
arXiv2014

Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities

We study the 1D Klein-Gordon equation with variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case where the worst of this resonant behavior is absent, we prove L-Infinity scattering as well as a certain kind of strong smoothness for the solution at time-like infinity with the help of several new normal-forms transformations. Some explicit examples are also given which suggest qualitatively different behavior in the case where the strongest cubic resonances are present.

math.AP↗
arXiv2013

The Strauss conjecture on Kerr black hole backgrounds

We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the Strauss conjecture on the Schwarzschild and Kerr, with small angular momentum, black hole backgrounds. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and a localized energy estimate on the black hole background, which handles the behavior in the remaining compact set.

math.AP↗
arXiv2011

Asymptotic decay for a one-dimensional nonlinear wave equation

We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation $-u_{tt} + u_{xx} = |u|^{p-1} u$, where $p > 1$. Standard energy methods guarantee global existence, but do not directly say much about the behaviour of $u(t)$ as $t \to \infty$. Note that in contrast to higher-dimensional settings, solutions to the linear equation $-u_{tt} + u_{xx} = 0$ do not exhibit decay, thus apparently ruling out perturbative methods for understanding such solutions. Nevertheless, we will show that solutions for the nonlinear equation behave differently from the linear equation, and more specifically that we have the average $L^\infty$ decay $\lim_{T \to +\infty} \frac{1}{T} \int_0^T \|u(t)\|_{L^\infty_x(\R)}\ dt = 0$, in sharp contrast to the linear case. An unusual ingredient in our arguments is the classical Radamacher differentiation theorem that asserts that Lipschitz functions are almost everywhere differentiable.

math.AP↗
arXiv2010

The global stability of the Minkowski space-time in harmonic gauge

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach shows that the Einstein-vacuum and the Einstein-scalar field equations with general asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time. The proof utilizes the classical harmonic (also known as de Donder or wave coordinate) gauge.

math.AP↗
arXiv2009

A priori estimates for the free-boundary 3-D compressible Euler equations in physical vacuum

We prove a priori estimates for the three-dimensional compressible Euler equations with moving {\it physical} vacuum boundary, with an equation of state given by $p(ρ) = C_γρ^γ$ for $γ>1$. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic {\it free-boundary} system to which standard methods of symmetrizable hyperbolic equations cannot be applied.

math.AP↗
arXiv2005

Global Stability for Charged Scalar Fields on Minkowski Space

We prove that the charge-scalar field (also known as the massless Maxwell-Klein-Gordon) equations are globally stable on (3+1) dimensional Minkowski space for small initial data in certain gauge covariant weighted Sobolev spaces. These spaces can be chosen as to be almost scale invariant with respect to the homogeneity of the equations. This result is valid for initial data with non-zero charge that is also non-stationary at space-like infinity. The method of proof is a tensor-geometric approach which is based on a certain family of weighted bilinear L2 space-time estimates.

math.AP↗
arXiv2005

A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation

We give a short proof of asymptotic completeness and global existence for the cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing with the long range behavior of the asymptotic solution is by reducing it, in hyperbolic coordinates to the study of an ODE. Similar arguments extend to higher dimensions and other long range type nonlinear problems.

math.AP↗
arXiv2005

Well posedness for the motion of a compressible liquid with free surface boundary

We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a "physical condition", related to the fact that the pressure of a fluid has to be positive.

math.AP↗
arXiv2004

Global existence for the Einstein vacuum equations in wave coordinates

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with Schwartzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are "unstable in the large" and provides an alternative approach to the stability problem originally solved (for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman. Using the wave coordinate gauge we recast the Einstein equations as a system of quasilinear wave equations and, in absence of the classical null condition, establish a small data global existence result. In our previous work we introduced the notion of a eak nul condition and showed that the Einstein equations in wave coordinates satisfy this condition. The result of this paper relies on this observation and combines it with the vector field method based on the symmetries of the standard Minkowski space. In a forthcoming paper we will address the question of stability of Minkowski space for the Einstein vacuum equations in wave coordinates for all "small" asymptotically flat data and the case of the Einstein equations coupled to a scalar field.

math.AP↗
arXiv2004

Well-posedness for the motion of an incompressible liquid with free surface boundary

We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a "physical condition", related to the fact that the pressure of a fluid has to be positive.

math.AP↗