arXiv subjects

Tobias Schmid

Publications and source records attributed to Tobias Schmid.

12 recordsLinked to original sources

arXiv2026

On soliton clusters and collision blow up for the $L^2$-critical Hartree equation

We consider the $L^2$-critical nonlinear Hartree equation in $\mathbb{R}^{1+4}$ and multisoliton solutions for which the trajectories are approximated to leading order by an $m$-body law. We obtain soliton clusters asymptotically following hyperbolic-parabolic trajectories of the corresponding $m$-body problem. By pseudo-conformal invariance, we then conclude finite-time collision blow-up with any number of clusters, each consisting of an arbitrary number of solitons, colliding simultaneously at distinct prescribed points.

math.AP↗

arXiv2026

Multisoliton solutions and blow up for the $L^2$-critical Hartree equation

We construct multisoliton solutions for the $L^2$-critical Hartree equation with trajectories asymptotically obeying a many-body law for an inverse square potential. Precisely, we consider the $m$-body hyperbolic and parabolic non-trapped dynamics. The pseudo-conformal symmetry then implies finite-time collision blow up in the latter case and a solution blowing up at $m$ distinct points in the former case. The approach we take is based on the ideas of [Krieger-Martel-Raphaël, 2009] and the third author's recent extension [Wu, 2026]. The approximation scheme requires new aspects in order to deal with a certain degeneracy for generalized root space elements.

math.AP↗

arXiv2025

Blow up dynamics for the 3D energy-critical Nonlinear Schrödinger equation

We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $ d = 3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $ u(t,x) = e^{i α(t)}λ(t)^{\frac{1}{2}}W(λ(t) x) + η(t, x )$, $ t \in[0, T)$, $ x \in \mathbb{R}^3$, where $ W( x) = \big( 1 + \frac{|x|^2}{3}\big)^{-\frac{1}{2}}$ is the ground state solution, $λ(t) = (T-t)^{- \frac12 - ν} $ for suitable $ ν> 0 $, $ α(t) = α_0 \log(T - t)$ and $ T= T(ν, α_0) > 0 $. Further $ \|η(t) - η_T\|_{\dot{H}^1 \cap \dot{H}^2} = o(1)$ as $ t \to T^-$ for some $ η_T \in \dot{H}^{1} \cap~ \dot{H}^2$.

math.AP↗

arXiv2024

Finite time blow up for the energy critical Zakharov system I: approximate solutions

We construct approximate solutions $ (ψ_*, n_*)$ of the critical 4D Zakharov system which collapse in finite time to a singular renormalization of the solitary bulk solutions $ (λe^{i θ}W, λ^2 W^2)$ . To be precise for $ N \in \mathbb{Z}_+,\;N \gg1 $ we obtain a magnetic envelope/ion density pair of the form $$ ψ_*(t, x)= e^{iα(t)}λ(t) W(λ(t)x) + η(t, x), \;n_*(t,x) = λ^2(t) W^2(λ(t) x) + χ(t,x), $$ where $ W(x) = (1 + \frac{|x|^2}{8})^{-1}$, $α(t) = α_0 \log(t)$, $λ(t)= t^{-\frac{1}{2}-ν}$ with large $ν> 1 $ and further $$ i \partial_t ψ_* + Δψ_* + n_* ψ_* = \mathcal{O}(t^N),\; \Box n_* - Δ(|ψ_*|^2) = \mathcal{O}(t^N),\;\;η(t) \to η_0, χ(t) \to χ_0, $$ as $ t \to 0^+$ in a suitable sense. The method of construction is inspired by matched asymptotic regions and approximation procedures in the context of blow up solutions introduced by the first author jointly with W. Schlag and D. Tataru, as well as the subsequently developed methods in the Schrödinger context by G. Perelman et al.

math.AP↗

arXiv2024

Finite time blow up for the energy critical Zakharov system II: exact solutions

Based on our companion paper [Krieger-Schmid, 2024], we show that the 4D energy critical Zakharov system admits finite time type II blow up solutions. The main new difficulty this work deals with is the appearance of a term in the linearization around the approximate solution, which is non-local with respect to both space and time. In particular this cannot be handled by straightforward adaptation of the methods developed in [Krieger-Schlag-Tataru, 2008/09]. The key new ingredients we use are a type of approximate modulation theory, taking advantage of frequency localisations, and the exploitation of an inhomogeneous wave equation with both a non-local, as well as a local potential term. These terms arise for the main non-perturbative component of the ion density $n$ and can be solved via inversion of a certain Fredholm type operator, as well as by using distorted Fourier methods. Our result relies on a number of numerical non-degeneracy assumptions.

math.AP↗

arXiv2022

Energy bounds for a fourth-order equation in low dimensions related to wave maps

For compact, isometrically embedded Riemannian manifolds $ N \hookrightarrow \mathbb{R}^L$, we introduce a fourth-order version of the wave map equation. By energy estimates, we prove an $\textit{a priori}$ estimate for smooth local solutions in the energy subcritical dimension $ n = 1,2$. The estimate excludes blow-up of a Sobolev norm in finite existence times. In particular, combining this with recent work of local well-posedness of the Cauchy problem, it follows that for smooth initial data with compact support, there exists a (smooth) unique global solution in dimension $n = 1,2$. We also give a proof of the uniqueness of solutions that are bounded in these Sobolev norms.

math.AP↗

arXiv2021

Global results for a Cauchy problem related to biharmonic wave maps

We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space $\dot{B}^{2,1}_{\frac{d}{2}}(\mathbb{R}^d) \times \dot{B}^{2,1}_{\frac{d}{2}-2}(\mathbb{R}^d)$ for $ d \geq 3 $. Since the solution persists higher regularity of the initial data, we obtain a small data global regularity result for the biharmonic wave maps equation for a certain class of target manifolds including the sphere.

math.AP↗

arXiv2021

Formal Verification of a Fail-Operational Automotive Driving System

A fail-operational system for highly automated driving must complete the driving task even in the presence of a failure. This requires redundant architectures and a mechanism to reconfigure the system in case of a failure. Therefore, an arbitration logic is used. For functional safety, the switch-over to a fall-back level must be conducted in the presence of any electric and electronic failure. To provide evidence for a safety argumentation in compliance with ISO 26262, verification of the arbitration logic is necessary. The verification process provides confirmation of the correct failure reactions and that no unintended system states are attainable. Conventional safety analyses, such as the failure mode and effect analysis, have its limits in this regard. We present an analytical approach based on formal verification, in particular model checking, to verify the fail-operational behaviour of a driving system. For that reason, we model the system behaviour and the relevant architecture and formally specify the safety requirements. The scope of the analysis is defined according to the requirements of ISO 26262. We verify a fail-operational arbitration logic for highly automated driving in compliance with the industry standard. Our results show that formal methods for safety evaluation in automotive fail-operational driving systems can be successfully applied. We were able to detect failures, which would have been overlooked by other analyses and thus contribute to the development of safety critical functions.

cs.SE↗

arXiv2020

Experiences from Large-Scale Model Checking: Verification of a Vehicle Control System

In the age of autonomously driving vehicles, functionality and complexity of embedded systems are increasing tremendously. Safety aspects become more important and require such systems to operate with the highest possible level of fault tolerance. Simulation and systematic testing techniques have reached their limits in this regard. Here, formal verification as a long established technique can be an appropriate complement. However, the necessary preparatory work like adequately modeling a system and specifying properties in temporal logic are anything but trivial. In this paper, we report on our experiences applying model checking to verify the arbitration logic of a Vehicle Control System. We balance pros and cons of different model checking techniques and tools, and reason about our choice of the symbolic model checker NuSMV. We describe the process of modeling the architecture, resulting in ~1500 LOC, 69 state variables and 38 LTL constraints. To handle this large-scale model, we automate and optimize the model checking procedure for use on multi-core CPUs and employ Bounded Model Checking to avoid the state explosion problem. We share our lessons learned and provide valuable insights for architects, developers, and test engineers involved in this highly present topic.

cs.SE↗

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↗

arXiv2010

Environment-assisted quantum transport and trapping in dimers

We study the dynamics and trapping of excitations for a dimer with an energy off-set $Δ$ coupled to an external environment. Using a Lindblad quantum master equation approach, we calculate the survival probability $Π(t)$ of the excitation and define different lifetimes $τ_s$ of the excitation, corresponding to the duration of the decay of $Π(t)$ in between two predefined values. We show that it is not possible to always enhance the overall decay to the trap. However, it is possible, even for not too small environmental couplings and for values of $Δ$ of the order ${\cal O}(1)$, to decrease certain lifetimes $τ_s$, leading to faster decay of $Π(t)$ in these time intervals: There is an optimal environmental coupling, leading to a maximal decay for fixed $Δ$.

quant-ph↗

arXiv2009

Dissipative Dynamics with Trapping in Dimers

The trapping of excitations in systems coupled to an environment allows to study the quantum to classical crossover by different means. We show how to combine the phenomenological description by a non-hermitian Liouville-von Neumann Equation (LvNE) approach with the numerically exact path integral Monte-Carlo (PIMC) method, and exemplify our results for a system of two coupled two-level systems. By varying the strength of the coupling to the environment we are able to estimate the parameter range in which the LvNE approach yields satisfactory results. Moreover, by matching the PIMC results with the LvNE calculations we have a powerful tool to extrapolate the numerically exact PIMC method to long times.

quant-ph↗