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Sebastian Herr

Publications and source records attributed to Sebastian Herr.

At least 37 records · Page 2Linked to original sources
arXiv2020

The Zakharov-Kuznetsov equation in high dimensions: Small initial data of critical regularity

The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces and it is proved that solutions scatter to free solutions as $t \to \pm \infty$. The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension $d=4$.

math.AP↗
arXiv2020

Biharmonic wave maps: Local wellposedness in high regularity

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.

math.AP↗
arXiv2019

Biharmonic wave maps into spheres

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation.

math.AP↗
arXiv2019

Critical well-posedness and scattering results for fractional Hartree-type equations

Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the unboundedness of a third order derivative of the flow map in the super-critical range.

math.AP↗
arXiv2018

Unconditional Uniqueness Results for the Nonlinear Schrödinger Equation

We study the problem of unconditional uniqueness of solutions to the cubic nonlinear Schrödinger equation. We introduce a new strategy to approach this problem on bounded domains, in particular on rectangular tori. It is a known fact that solutions to the cubic NLS give rise to solutions of the Gross-Pitaevskii hierarchy, which is an infinite-dimensional system of linear equations. By using the uniqueness analysis of the Gross-Pitaevskii hierarchy, we obtain new unconditional uniqueness results for the cubic NLS on rectangular tori, which cover the full scaling-subcritical regime in high dimensions. In fact, we prove a more general result which is conditional on the domain. In addition, we observe that well-posedness of the cubic NLS in Fourier-Lebesgue spaces implies unconditional uniqueness.

math.AP↗
arXiv2018

Scattering for stochastic nonlinear Schrödinger equations

We study the scattering behavior of global solutions to stochastic nonlinear Schrödinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is defocusing,we prove that the solutions scatter at infinity in the pseudo-conformal space and in the energy space respectively, including the energy-critical case. Moreover, in the case where the noise is large, non-conservative and has infinite quadratic variation, we show that the solutions scatter at infinity with high probability for all energy-subcritical exponents.

math.PR↗
arXiv2018

I-BEAT: New ultrasonic method for single bunch measurement of ion energy distribution

The shape of a wave carries all information about the spatial and temporal structure of its source, given that the medium and its properties are known. Most modern imaging methods seek to utilize this nature of waves originating from Huygens' principle. We discuss the retrieval of the complete kinetic energy distribution from the acoustic trace that is recorded when a short ion bunch deposits its energy in water. This novel method, which we refer to as Ion-Bunch Energy Acoustic Tracing (I-BEAT), is a generalization of the ionoacoustic approach. Featuring compactness, simple operation, indestructibility and high dynamic ranges in energy and intensity, I-BEAT is a promising approach to meet the needs of petawatt-class laser-based ion accelerators. With its capability of completely monitoring a single, focused proton bunch with prompt readout it, is expected to have particular impact for experiments and applications using ultrashort ion bunches in high flux regimes. We demonstrate its functionality using it with two laser-driven ion sources for quantitative determination of the kinetic energy distribution of single, focused proton bunches.

physics.plasm-ph↗
arXiv2018

On the Division Problem for the Wave Maps Equation

We consider Wave Maps into the sphere and give a new proof of small data global well-posedness and scattering in the critical Besov space, in any space dimension $n \geq 2$. We use an adapted version of the atomic space $U^2$ as the single building block for the iteration space. Our approach to the so-called division problem is modular as it systematically uses two ingredients: atomic bilinear (adjoint) Fourier restriction estimates and an algebra property of the iteration space, both of which can be adapted to other phase functions.

math.AP↗
arXiv2017

On the Majorana condition for nonlinear Dirac systems

For arbitrarily large initial data in an open set defined by an approximate Majorana condition, global existence and scattering results for solutions to the Dirac equation with Soler-type nonlinearity and the Dirac-Klein-Gordon system in critical spaces in spatial dimension three are established.

math.AP↗
arXiv2017

Conditional large data scattering results for the Dirac-Klein-Gordon system

We obtain conditional results on the global existence and scattering for large solutions of the Dirac-Klein-Gordon system in critical spaces in dimension $1+3$. In particular, for bounded solutions we identify a space-time Lebesgue norm controlling the global behaviour. The proof relies on refined nonlinear estimates involving the controlling norm.

math.AP↗
arXiv2017

Transference of Bilinear Restriction Estimates to Quadratic Variation Norms and the Dirac-Klein-Gordon System

Firstly, bilinear Fourier Restriction estimates --which are well-known for free waves-- are extended to adapted spaces of functions of bounded quadratic variation, under quantitative assumptions on the phase functions. This has applications to nonlinear dispersive equations, in particular in the presence of resonances. Secondly, critical global well-posedness and scattering results for massive Dirac-Klein-Gordon systems in dimension three are obtained, in resonant as well as in non-resonant regimes. The results apply to small initial data in scale-invariant Sobolev spaces exhibiting a small amount of angular regularity.

math.AP↗
arXiv2016

The cubic Dirac equation: Small initial data in $H^{\frac12}(\mathbb{R}^2)$

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space $H^{\frac12}(\mathbb{R}^2)$ is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein-Gordon equation in dimension $n=2$, which is captured by constructing an adapted systems of coordinate frames.

math.AP↗
arXiv2016

Trigonometric time integrators for the Zakharov system

The main challenge in the analysis of numerical schemes for the Zakharov system originates from the presence of derivatives in the nonlinearity. In this paper a new trigonometric time-integration scheme for the Zakharov system is constructed and convergence is proved. The time-step restriction is independent from a spatial discretization. Numerical experiments confirm the findings.

math.NA↗
arXiv2015

Small data scattering for semi-relativistic equations with Hartree type nonlinearity

We prove that the initial value problem for the equation \[ - i\partial_t u + \sqrt{m^2-Δ} \, u= (\frac{e^{-μ_0 |x|}}{|x|} \ast |u|^2)u \ \text{in} \ \mathbb R^{1+3}, \quad m\ge 0, \ μ_0 >0\] is globally well-posed and the solution scatters to free waves asymptotically as $t \to \pm \infty$ if we start with initial data which is small in $H^s(\mathbb R^{3})$ for $s>\frac12$, and if $m>0$. Moreover, if the initial data is radially symmetric we can improve the above result to $m\ge 0$ and $s>0$, which is almost optimal, in the sense that $L^2(\mathbb R^{3})$ is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, $L^2(\mathbb R^{1+3})$ bilinear estimates for free waves and the application of the $U^p$ and $V^p$ function spaces.

math.AP↗
arXiv2015

Well-posedness and scattering for the Zakharov system in four dimensions

The Cauchy problem for the Zakharov system in four dimensions is considered. Some new well-posedness results are obtained. For small initial data, global well-posedness and scattering results are proved, including the case of initial data in the energy space. None of these results is restricted to radially symmetric data.

math.AP↗