Stochastic Chemotaxis Model with Fractional Derivative Driven by Multiplicative Noise

We introduce stochastic model of chemotaxis by fractional Derivative generalizing the deterministic Keller Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. In this work, we study of nonlinear stochastic chemotaxis model with Dirichlet boundary conditions, fractional Derivative and disturbed by multiplicative noise. The required results prove the existence and uniqueness of mild solution to time and space-fractional, for this we use analysis techniques and fractional calculus and semigroup theory, also studying the regularity properties of mild solution for this model.

Mixed-Integer Convex Representability

Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first nonrepresentability results for various nonconvex sets, such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear representability by Jeroslow and Lowe, we study the representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties, such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets.

A pointwise differential inequality and second-order regularity for nonlinear elliptic systems

A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $${\mathbb R^n}$$ R n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known.

Global continuity of variational solutions weakening the one-sided bounded slope condition

We study regularity properties of variational solutions to a class of Cauchy–Dirichlet problems of the form { ∂ t ⁡ u - div x ⁡ ( D ξ ⁢ f ⁢ ( D ⁢ u ) ) = 0 in ⁢ Ω T , u = u 0 on ⁢ ∂ 𝒫 ⁡ Ω T . \left\{\begin{aligned} \displaystyle\partial_{t}u-\operatorname{div}_{x}(D_{% \xi}f(Du))&\displaystyle=0&&\displaystyle\phantom{}\text{in }\Omega_{T},\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on }\partial% _{\mathcal{P}}\Omega_{T}.\end{aligned}\right. We do not impose any growth conditions from above on f : ℝ n → ℝ {f\colon\mathbb{R}^{n}\to\mathbb{R}} , but only require it to be convex and coercive. The domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is mainly supposed to be bounded and convex, and for the time-independent boundary datum u 0 : Ω ¯ → ℝ {u_{0}\colon\overline{\Omega}\to\mathbb{R}} we only require continuity. These requirements are weaker than a one-sided bounded slope condition. We prove global continuity of the unique variational solution u : Ω T → ℝ {u\colon\Omega_{T}\to\mathbb{R}} . If the boundary datum is Lipschitz continuous, we obtain global Hölder continuity of the solution.

Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels

We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

Fractional diffusion-wave equations: Hidden regularity for weak solutions

We prove a “hidden” regularity result for weak solutions of time fractional diffusion-wave equations where the Caputo fractional derivative is of order α ∈ (1, 2). To establish such result we analyse the regularity properties of the weak solutions in suitable interpolation spaces.

Non-uniformly parabolic equations and applications to the random conductance model

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on $$\mathbb Z^d$$ Z d . In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.