arXiv subjects
Publications and source records attributed to Erika Hausenblas.
One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations
We study the controllability and reachability of sphere-constrained evolution equations under degenerate (low-mode) forcing, with the harmonic map heat flow as the principal application. Exploiting the underlying geometric structure, we reformulate the problem as an infinite-dimensional control-affine system in Fourier variables and analyze the Lie algebra generated by the controlled vector fields. We prove that iterated Lie brackets generate new admissible directions, providing a mechanism through which finitely many control modes propagate their influence across infinitely many Fourier components. The results provide a Lie-algebraic framework for controlling manifold-valued evolution equations.
The goal of this work is to analyze the long-term behavior of reaction-diffusion systems arising in two-species chemical models and to identify finite sets of modes that determine their asymptotic dynamics. The models considered include, as particular cases the Gray--Scott and the Glycolysis models. These systems are described by coupled reaction-diffusion equations and admit a finite-dimensional representation based on a limited number of spatial Fourier modes that capture their essential reduced dynamics. The concept of determining modes, introduced in this context, is closely related to other approaches that seek finite-dimensional representations of infinite-dimensional dynamics, such as the Proper Orthogonal Decomposition and the construction of Approximate Inertial Manifolds. We prove that the dynamics of the system can be completely characterized by a finite number of low modes, since all higher modes are asymptotically determined by them, thus providing an analytical foundation for reduced dynamics in models of pattern formation.
We show a stochastic version of the Schauder-Tychonoff fixed point theorem which yields a solution of the martingale problem for a class of systems of nonlinear reaction-diffusion equations driven by a cylindrical Wiener process and a Poisson random measure with certain moments. By this type of theorem one can solve systems by linearization which have a possibly unbounded, non-dissipative and non-coercive nonlinearity.
Chemical and biochemical reactions can exhibit surprisingly different behaviours, ranging from multiple steady-state solutions to oscillatory solutions and chaotic behaviours. These types of systems are often modelled by a system of reaction-diffusion equations coupled by a nonlinearity. In the article, we study the existence of stochastically perturbed equations of this type. In particular, we show the existence of a probabilitic weak solution of the following stochastic system \begin{align*} \dot {u} & = r_1\,Δu+ a_1\, u + b_1 -c_1\, u\cdot v^q+σ_1\, g_1(u)\circ \dot W_1, \\ \dot{v} & = r_2 \,A v + a_2\, v + b_2 +c_2\, u\cdot v^q + σ_2\, g_2(v)\circ \dot W_2, \end{align*} where $r_i,b_i,c_i, σ_i>0$, $a_i\in\mathbb{R}$, and $g_i$ are linear, $i=1,2$, and the exponent $q\geq 1$. The operator $A=-(-Δ)^{\aleph/2}$ is a fractional power of the Laplacian, $1<\aleph \le2$. The main result is obtained by a Schauder-Tychonoff type fixed point theorem for the controlled versions of the laws of the respective (infinite dimensional) Ornstein-Uhlenbeck system, from which we infer the existence of a weak solution of the coupled system.
This work is devoted to the convergence of a time-discrete numerical scheme of a semi-discretization model arising from biology, consisting of a chemotaxis equation coupled with a Galerkin approximation of Navier-Stokes system driven by transport noise in a three-dimensional bounded and convex domain. We propose a semi-implicit Euler numerical scheme approximating the infinite dimensional model, for which we study the well-posedness and derive some uniform estimates for the discrete variables
The Yamada-Watanabe theory provides a robust framework for understanding stochastic equations driven by Wiener processes. Despite its comprehensive treatment in the literature, the applicability of the theory to SPDEs driven by Poisson random measures or, more generally, Lévy processes remains significantly less explored, with only a handful of results addressing this context. In this work, we leverage a result by Kurtz to demonstrate that the existence of a martingale solution combined with pathwise uniqueness implies the existence of a unique strong solution for SPDEs driven by both a Wiener process and a Poisson random measure. Our discussion is set within the variational framework, where the SPDE under consideration may be nonlinear. This work is influenced by earlier research conducted by the second author alongside de Bouard and Ondreját.
The purpose of the paper is twofold. Firstly, we want to present a Meta Theorem to show the existence of a martingale solution for coupled systems of non-linear stochastic differential equations. The idea is first to split the system by rewriting the non-linear part in a linear part acting on a given process $ξ$. This is done in such a way that the fixpoint with respect to $ξ$ would be the solution. However, to show the well posedness of the {\sl linearized} system, one needs a cut-off argument. Under which conditions one can handle the limits of the cut-off parameter to get in the end a martingale solution of the original system is given in the Meta-Theorem. Secondly, we want to verify the full applicability of the Meta Theorem by showing the existence of a martingale solution of a highly nonlinear chemotaxis system with underlying fluid dynamic.
On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.
In this article, we study a mathematical system which models the dynamic of the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in a two dimensional bounded domain under stochastic perturbation. This model can be seen as a stochastic version of Chemotaxis-Navier-Stokes model. We prove the existence of a unique (probabilistic) strong solution. In addition, we establish some properties of the strong solution. More precisely, we prove that the unique solution is non-negative and satisfies the mass conservation property and an energy inequality.
In this article, we study the controllability issues of the Landau-Lifshitz-Gilbert Equations (LLGEs), accompanied with non-zero exchange energy only, in an interval in one spatial dimension with Neumann boundary conditions. The paper is of twofold. In the first part of the paper, we study the controllability issues of the LLGEs. The control force acting here is degenerate i.e., it acts through a few numbers of low mode frequencies. We exploit the Fourier series expansion of the solution. We borrow methods of differential geometric control theory (Lie bracket generating property) to establish the global controllability of the finite-dimensional Galerkin approximations of LLGEs. We show $L^2$ approximate controllability of the full system. In the second part, we consider the LLGEs with lower-dimensional degenerate random forcing (finite-dimensional Brownian motions) and study support theorems.
In a recent paper (J. Differential Equations, 310: 506-554, 2022), the authors proved the existence of martingale solutions to a stochastic version of the classical Patlak-Keller-Segel system in 1 dimension (1D), driven by time-homogeneous spatial Wiener processes. The current paper is a continuation and consists of two results about the stochastic Patlak-Keller-Segel system in 1D. First, we establish some additional regularity results of the solutions. The additional regularity is, e.g. important for its numerical modeling. Then, as a second result, we obtain the pathwise uniqueness of the solutions to the stochastic Patlak-Keller-Segel system in 1D. Finally, we conclude the paper with the existence of the strong solution to this system in 1D.
In this paper, we study the classical Keller - Segel system on a two-dimensional domain perturbed by a pair of Wiener processes, where the leading diffusion term is replaced by a porous media term. Since the randomness is intrinsic, the interpretation of the stochastic integral in the Stratonovich sense is natural. We construct a solution (integral) operator and establish its continuity and compactness properties in an appropriately chosen Banach space. In this manner, we formulate a stochastic version of the Schauder - Tychonoff Type Fixed Point Theorem which is specific to our problem to obtain a solution. In-kind, we achieve the existence of a martingale solution.
We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.
The Gierer-Meinhardt system occurs in morphogenesis, where the development of an organism from a single cell is modelled. One of the steps in the development, is the formation of spatial patterns of the cell structure, starting from an almost homogeneous cell distribution. Turing proposed in his pioneering work different activator-inhibitor systems with different diffusion rates, which could trigger the emergence of such cell structures. Mathematically, one describes these activator-inhibitor systems as a coupled systems of reaction-diffusion equations with hugely different diffusion coefficients and highly nonlinear interaction. One famous example of these systems is the Gierer-Meinhardt system. These systems usually are not of monotone type, such that one has to apply other techniques. The purpose of this article is to study the stochastic reaction-diffusion Gierer-Meinhardt system with homogeneous Neumann boundary condition on a one or two-dimensional bounded spatial domain. To be more precise, we perturb the original Gierer-Meinhardt system by an infinite-dimensional Wiener process and show under which conditions on the Wiener process and the system, a solution exists. In dimension one, we even show the pathwise uniqueness. In dimension two, uniqueness is still an open question.
We analyse analytic properties of nonlocal transition semigroups associated with a class of stochastic differential equations (SDEs) in $\mathbb{R}^d$ driven by pure jump--type Lévy processes. First, we will show under which conditions the semigroup will be analytic on the Besov space $B_{p,q}^ m(\mathbb{R}^d)$ with $1\le p, q<\infty$ and $m\in\mathbb{R}$. Secondly, we present some applications by proving the strong Feller property and give weak error estimates for approximating schemes of the SDEs over the Besov space $B_{\infty,\infty}^ m(\mathbb{R}^d)$.
In this note we prove the existence and uniqueness of local maximal smooth solution of the stochastic simplified Ericksen-Leslie systems modelling the dynamics of nematic liquid crystals under stochastic perturbations.
Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak--Keller--Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak--Keller--Segel system perturbed by random processes. We study a stochastic version of the classical Patlak--Keller--Segel system under homogeneous Neumann boundary conditions on an interval $\mathcal{O}=[0,1]$. In particular, let $\mathcal{W}_1$, $\mathcal{W}_2$ be two time-homogeneous spatial Wiener processes over a filtered probability space $\mathfrak{A}$. Let $u$ and $v$ denote the cell density and concentration of the chemical signal. We investigate the coupled system \begin{align*} & d {u} - ( r_uΔu- χ{\rm div }( u\nabla v) )\, dt =u\circ d\mathcal{W}_1, \\ & d{v} -(r_v Δv -αv)\, dt = βu \, dt+ v\circ d\mathcal{W}_2, \end{align*} with initial conditions $(u(0),v(0))=(u_0,v_0)$. The positive terms $r_u$ and $r_v$ are the diffusivity of the cells and chemoattractant, respectively, the positive value $χ$ is the chemotactic sensitivity, $α\ge0$ is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given $T>0$, we will prove the existence of a martingale solution on $[0,T]$.