arXiv subjects

Max Sauerbrey

Publications and source records attributed to Max Sauerbrey.

11 recordsLinked to original sources

arXiv2026

Parabolic-hyperbolic splitting in support propagation for stochastic porous media equations

We develop a randomly-localized energy method to derive qualitative results on the support propagation of stochastic porous media equations with linear conservative noise. Unlike in previous works, where energies are localized by weighting them with a spatial bump function, we weight them with profiles which are solutions to stochastic transport equations. This modulates out the support propagation due to the conservative noise term, and energy arguments---which otherwise fail in this setting---are again applicable. As a result, finite speed of propagation along with sufficient and necessary conditions on the existence of waiting time phenomena (modulo stochastic transport) are proven. These methods and results demonstrate, on short time scales, that the support propagation may be disintegrated into two independent parts, one due to the evolution of the porous media equation and one due to stochastic transport.

math.AP↗

arXiv2026

The Schr\"odinger equation with fluctuating nonlinearity in the energy space

We study nonlinear Schr\"odinger equations with nonlinear Stratonovich noise \begin{equation*} \mathrm{d} u\,=\, i\bigl[ \Delta u \,+\, \lambda|u|^{p-1}u\bigr] \, \mathrm{d} t \,+\,i|u|^{(q-1)/2}u\circ \mathrm{d} {W}, \end{equation*} in their energy space $H^1(\mathbb R^d;\mathbb C)$. By combining the stochastic Strichartz estimates derived in [Potential Anal. 41 (2014), pp.\ 269--315] with the approach from [Ann.\ Inst.\ H.\ Poincar\'e Phys.\ Th\'eor.\ 46 (1987), pp.\ 113--129] we obtain local well-posedness for all energy-subcritical nonlinearities $p,q\in [1, 1+4/(d-2)_+)$ together with a corresponding blow-up alternative. For a linear multiplicative noise $q=1$, a real-valued noise $W$ and a defocusing nonlinearity $\lambda\le 0$, we check this blow-up condition using a bound on the energy, resulting in the global well-posedness of the equation. If both nonlinearities are mass-subcritical, i.e., $p,q\in [1, 1+4/d)$, we provide an improved blow-up criterion involving the $L^2(\mathbb R^d;\mathbb C)$-norm. Using the conservation of mass for real-valued $W$, we obtain global well-posedness also in this case. Compared to previous results on stochastic nonlinear Schr\"odinger equations, we thereby improve the range of exponents $p$ and $q$ and the spatial regularity assumption on the noise.

math.AP↗

arXiv2026

Finite speed of propagation and waiting time phenomena for stochastic porous media equations with nonlinear conservative noise

Starting from localized energy estimates, we prove finite speed of propagation for kinetic solutions to stochastic porous media equations with nonlinear conservative noise, the existence and uniqueness of which has recently been established. In particular, we propose a novel iteration technique which allows us to obtain a Stampacchia-type inequality involving one single integral quantity, despite the possibly different scaling behaviors of the porous media and the noise term. This allows us to apply stochastic filtering arguments developed for the case of linear source-type noise. Using related ideas, we identify flatness conditions on initial data which guarantee locally the occurrence of a waiting time phenomenon, i.e., the onset of forward propagation of the solution's support is locally delayed. The condition for the latter matches the one for the deterministic porous media equation up to a logarithmic correction in the case of critical nonlinearity in the noise, but it requires more and more flatness of the initial data as the nonlinearity tends towards linear conservative noise. This is in line with the expected behavior: In the case of linear conservative noise, instantaneous forward motion is possible due to the effects of stochastic transport, no matter how flat the initial profile is.

math.AP↗

arXiv2026

The stochastic Keller--Segel system in critical spaces

We study stochastic, parabolic-parabolic Keller--Segel equations on the $d$-dimensional torus in scaling critical Besov spaces, for $d \geq 3$. Using stochastic maximal regularity estimates, we prove local well-posedness of the equation, i.e., that there exists a unique solution up to a maximal time of existence. We show that the time of existence can be made arbitrarily large with arbitrarily high probability, provided that the initial data is sufficiently small in these critical spaces. Contribution to the Oberwolfach Report for the Seminar 'Stochastic Partial Differential Equations in Critical Spaces' organized by Antonio Agresti and Mark Veraar.

math.PR↗

arXiv2026

The Incompressible Navier--Stokes--Fourier System with Thermal Noise

We establish a solution theory for the incompressible Navier--Stokes--Fourier system with thermal noise, posed on the three-dimensional torus. While in the incompressible deterministic setting the equation for the velocity can be solved independently of the temperature, the inclusion of the effects of thermal fluctuations by means of the GENERIC framework leads to a nonlinear gradient noise term, which couples the dynamics of both variables. Therefore, the analysis poses new challenges, which are absent in the deterministic incompressible Navier--Stokes--Fourier equations. In particular, the a priori estimates used in the deterministic setting are not readily generalizable, the noise introduces strongly nonlinear gradient terms and the total energy lacks convexity. These challenges are overcome in the present work by a novel variable transformation, and novel entropy dissipation estimates. Thereby, the existence of global-in-time weak solutions for $L_x^2$ initial data, the existence of local-in-time strong solutions for regular initial data, and weak-strong uniqueness are obtained.

math.PR↗

arXiv2025

A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise

The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and H\"older continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, boundedness has recently been established, but H\"older continuity remains a key open problem in the regularity theory of parabolic SPDEs. In this work, we resolve this question under the assumption that the noise coefficients are sufficiently regular in space. Our approach relies on Kunita's stochastic method of characteristics, which allows us to transform the original SPDE-via a stochastic flow of diffeomorphisms-into a random PDE to which the classical De Giorgi-Nash-Moser estimates apply. This program is accomplished through new a-priori estimates for the inverse of stochastic flows of diffeomorphisms, and a novel version of the It\^o-Wentzell formula adapted to rough random fields. To demonstrate the applicability of our results, we establish the existence of global, regular solutions to quasilinear SPDEs with transport noise.

math.PR↗

arXiv2024

Well-posedness of the stochastic thin-film equation with an interface potential

We consider strictly positive solutions to a class of fourth-order conservative quasilinear SPDEs on the $d$-dimensional torus modeled after the stochastic thin-film equation. We prove local Lipschitz estimates in Bessel potential spaces under minimal assumptions on the parameters and corresponding stochastic maximal $L^p$-regularity estimates for thin-film type operators with measurable in-time coefficients. As a result, we deduce local well-posedness of the stochastic thin-film equation as well as blow-up criteria and instantaneous regularization for the solution. In dimension one, we additionally close $\alpha$-entropy estimates and subsequently an energy estimate for the stochastic thin-film equation with an interface potential so that global well-posedness follows. We allow for a wide range of mobility functions including the power laws $u^n$ for $n\in [0,6)$ as long as the interface potential is sufficiently repulsive.

math.AP↗

arXiv2023

Solutions to the stochastic thin-film equation for the range of mobility exponents $n\in (2,3)$

Recently, many existence results for the stochastic thin-film equation were established in the case of a quadratic mobility exponent $n=2$, in which the noise term $\partial_x(u^\frac{n}{2}\mathcal{W})$ becomes linear. In the case of a non-quadratic mobility exponent, results are only available in the situation that $n\ge \frac{8}{3}$ leaving the interval of mobility exponents $n\in (2,\frac{8}{3})$ untreated. In this article we resolve the current gap in the literature by presenting a proof, which works under the assumption $n\in (2,3)$, i.e., the regime of weak slippage. The key idea is to use that the $\log$-entropy dissipation coincides with the energy production due to the noise. To realize this idea, we approximate the stochastic thin-film equation by stochastic thin-film equations with inhomogeneous mobility functions, which behave like a higher power near $0$. As a consequence the approximate solutions are non-negative, which is vital to use the $\log$-entropy estimate.

math.PR↗

arXiv2023

Solutions to the stochastic thin-film equation for initial values with non-full support

The stochastic thin-film equation with mobility exponent $n\in [\frac{8}{3},3)$ on the one-dimensional torus with multiplicative Stratonovich noise is considered. We show that martingale solutions exist for non-negative initial values. This advances on existing results in three aspects: (1) Non-quadratic mobility with not necessarily strictly positive initial data, (2) Measure-valued initial data, (3) Less spatial regularity of the noise. This is achieved by carrying out a compactness argument based solely on the control of the $\alpha$-entropy dissipation and the conservation of mass.

math.AP↗

arXiv2021

Dirichlet form analysis of the Jacobi process

We construct and analyze the Jacobi process - in mathematical biology referred to as Wright-Fisher diffusion - using a Dirichlet form. The corresponding Dirichlet space takes the form of a Sobolev space with different weights for the function itself and its derivative. Depending on the parameters we characterize the boundary behavior of the functions in the Dirichlet space, show density results, derive Sobolev embeddings and verify functional inequalities of Hardy type. Since the generator is a hypergeometric differential operator, many of the proofs can be carried out by explicit calculations involving hypergeometric functions. We deduce corresponding properties for the associated semigroup and Markov process and show that the latter is up to minor technical modifications a solution to the Jacobi SDE.

math.PR↗

arXiv2021

Martingale solutions to the stochastic thin-film equation in two dimensions

We construct solutions to the stochastic thin-film equation with quadratic mobility and Stratonovich gradient noise in the physically relevant dimension $d=2$ and allow in particular for solutions with non-full support. The construction relies on a Trotter-Kato time-splitting scheme, which was recently employed in $d=1$. The additional analytical challenges due to the higher spatial dimension are overcome using $\alpha$-entropy estimates and corresponding tightness arguments.

math.PR↗