We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by the total variation flow, respectively, by a singular $p$-Laplace-type operator. We impose nonlinear zero Neumann boundary conditions and precisely investigate their connection with the coefficient fields of the noise. This solves an open problem posed in [Barbu, Brzeźniak, Hausenblas, Tubaro; Stoch. Proc. Appl., 123 (2013)] and [Barbu, Röckner; J. Eur. Math. Soc., 17 (2015)].
We consider the problem of constructing weak solutions to the Itô and to the Stratonovich stochastic differential equations having critical-order singularities in the drift and critical-order discontinuities in the dispersion matrix.
In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including one-Laplacian, and is perturbed by a $p$-Laplacian-type diffusion operator with $1<p<\infty$. This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use st…
We consider a nonlinear reaction--diffusion equation in a domain consisting of two bulk regions connected via small channels periodically distributed within a thin layer. The height and the thickness of the channels are of order $ε$, and the equation inside the layer depends on the parameter $ε$. We consider the critical scaling of the diffusion coefficients in the channels and nonlinear Neumann-boundary condition on the channels' lateral boundaries. We derive effective models in the limit $ε\to 0 $, when the channel-domain is replaced by an interface $Σ$ between the two bulk-domains. Due to t…
We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the $d$-dimensional torus with singular $p$-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with divergence-free coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative Itô-formula.
We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profil…
A notion of measure solution is formulated for a coagulation-diffusion equation, which is the natural counterpart of Smoluchowski's coagulation equation in a spatially inhomogeneous setting. Some general properties of such solutions are established. Sufficient conditions are identified on the diffusivity, coagulation rates and initial data for existence, uniqueness and mass conservation of solutions. These conditions impose no form of monotonicity on the coagulation kernel, which may depend on complex characteristics of the particles. They also allow singular behaviour in both diffusivity and …
We study mass preserving transport of passive tracers in the low-diffusivity limit using Lagrangian coordinates. Over finite-time intervals, the solution-operator of the nonautonomous diffusion equation is approximated by that of a time-averaged diffusion equation. We show that leading order asymptotics that hold for functions [Krol, 1991] extend to the dominant nontrivial singular value. This answers questions raised in [Karrasch & Keller, 2020]. The generator of the time-averaged diffusion/heat semigroup is a Laplace operator associated to a weighted manifold structure on the material manifo…
Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby--Gawlinski model \[ \partial_t U = U\{f(U)-dV\}, \qquad \partial_t V = \partial_x \{f(U)\,\partial_x V\} + r V f(V), \] where $f(u) = 1-u$ and the parameters $d,r$ are positive. Denoting by $(\mathcal{U},\mathcal{V})$ the traveling wave profile and by $(\mathcal{U}_\pm,\mathcal{V}_\pm)$ its asymptotic states at $\pm\infty$, we investigate existence in the regimes i) $d > 1$ (homogeneous invasion) : $(\mathcal{U}_-,\mathcal{V}_-) = (0,1)$, $(\math…
We consider stochastic non-linear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational inequalities. We also show extinction in finite time with probability one. These kind of equations arise, e.g. in the use for simulation of image restoring techniques or for modeling turbulence.
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 …
