Nonlinear PDEs

Mathematical challenges

Nonlinear partial differential equations provide the continuum language for systems whose response depends on their current state. We study nonlinear diffusion, conservation laws and gradient-flow structures, especially when diffusion is degenerate or coefficients are irregular.

Central questions concern existence, uniqueness and regularity of solutions, the propagation of singularities, and the way qualitative structures survive approximation and limiting procedures. Variational methods and Γ-convergence help us pass between microscopic models, effective energies and macroscopic equations. These ideas also prepare the analytical foundations for stochastic models: identifying robust estimates and compactness principles is essential when deterministic equations are perturbed by fluctuations.

Current directions

  • Gradient flows and variational structures
  • Conservation laws and degenerate diffusion
  • PDEs with irregular coefficients
  • Regularity and singular limits
  • Γ-convergence and effective equations

Related people

Related projects

Selected publications

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

    Preprint · arXiv:2609.11468 (2026). Max Sauerbrey and Joshua Utley study support propagation and waiting-time phenomena for stochastic porous media equations with conservative noise.

  • The stochastic Cahn-Hilliard equation in critical spaces

    We study stochastic Cahn-Hilliard equations in bounded smooth domains with a double-well potential, transport-type noise, and natural Neumann boundary conditions in dimensions $d\le 4$. By employing stochastic maximal regularity techniques and deriving suitable energy estimates, we prove local and global well-posedness. The…

  • The stochastic Keller–Segel system in critical spaces

    Accepted / forthcoming · Oberwolfach Report for the seminar Stochastic Partial Differential Equations 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.,…

  • The Porous Medium Equation: Multiscale Integrability in Large Deviations

    We consider a zero-range process $η^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12Δu^α, α>1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing…

  • Mixing at the Batchelor Scale for White-In-Time Flows

    We consider the mixing properties of solutions to the advection-diffusion equation of a white-in-time velocity field on the 2-dimensional torus with four forced modes. As the diffusivity parameter goes to zero, we show that the almost-sure exponential dissipation rate stays bounded from…

  • 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 Hölder continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, boundedness has recently been established, but Hölder continuity remains a key…

  • A Dynamical Systems Perspective on the Analysis of Neural Networks

    In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements.…

  • Non-Asymptotic Analysis of Projected Gradient Descent for Physics-Informed Neural Networks

    Published · Scientific Machine Learning: Emerging Topics, SEMA SIMAI Springer Series (2026). In this work, we provide a non-asymptotic convergence analysis of projected gradient descent for physics-informed neural networks for the Poisson equation. Under suitable assumptions, we show that the optimization error can be…