Mathematics

We study the weak solvability of a quasilinear reaction-diffusion system nonlinearly coupled with an linear elliptic system posed in a domain with distributed microscopic balls in 2D . The size of these balls are governed by an ODE with direct feedback on the overall problem. The system describes the diffusion, aggregation, fragmentation, and deposition of populations of colloidal particles of various sizes inside a porous media made of prescribed arrangement of balls. The mathematical analysis of the problem relies on a suitable application of Schauder's fixed point theorem which also provides a convergent algorithm for an iteration method to compute finite difference approximations of smooth solutions to our multiscale model. Numerical simulations illustrate the behavior of the local concentration of the colloidal populations close to clogging situations.

In this work, we numerically study the higher-ordered/extended Boussinesq system describing the propagation of water-waves over flat topography. An equivalent suitable reformulation is proposed, making the model more appropriate for the numerical implementation and significantly improved in terms of linear dispersive properties in high frequency regimes due to the suitable adjustment of a dispersion correction parameter. Moreover, we show that a significant interest is behind the derivation of a new formulation of the higher-ordered/extended Boussinesq system that avoids the calculation of high order derivatives existing in the model. We show that this formulation enjoys an extended range of applicability while remaining stable. We develop a second order splitting scheme where the hyperbolic part of the system is treated with a high-order finite volume scheme and the dispersive part is treated with a finite difference approach. Numerical simulations are then performed to validate the model and the numerical methods.

A strong inspiration for studying Sobolev type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices.

In this paper, we investigate the following critical elliptic equation -\Delta u+V(y)u=u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,\R^{N},\,\,u\in H^{1}(\R^{N}), where V(y) is a bounded non-negative function in $\R^{N}.$ Assuming that $V(y)=V(|\hat{y}|,y^{*}),y=(\hat{y},y^{*})\in \R^{4}\times \R^{N-4}$ and gluing together bubbles with different concentration rates, we obtain new solutions provided that N??, whose concentrating points are close to the point ( r 0 , y ??0 ) which is a stable critical point of the function r 2 V(r, y ??) satisfying r 0 >0 and V( r 0 , y ??0 )>0. In order to construct such new bubble solutions for the above problem, we first prove a non-degenerate result for the positive multi-bubbling solutions constructed in \cite{PWY-18-JFA} by some local Pohozaev identities, which is of great interest independently. Moreover, we give an example which satisfies the assumptions we impose.

Structured equations are a standard modeling tool in mathematical biology. They areintegro-differential equations where the unknown depends on one or several variables, representing the state or phenotype of individuals. A large literature has been devoted to many aspects of these equations and in particular to the study of measure solutions.Here we introduce a transport distance closely related to the Monge-Kantorovich distance,which appears to be non-expanding for several (mainly linear) examples of structured equations.

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.

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 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.

We give a proof by foliation that the cones over S k ? S l minimize parametric elliptic functionals for each k,l?? . We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.

In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on R 2 . We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi and Hundertmark-Zharnitsky.

Assume M is a closed, connected and smooth Riemannian manifold. We consider the following two forms of Hamilton-Jacobi equations { ??t u(x,t)+H(x,u(x,t), ??x u(x,t))=0,(x,t)?�M?(0,+??. u(x,0)=?(x),x?�M, ??�C(M,R). and H(x,u(x), ??x u(x))=0, where H(x,u,p) is continuous, convex and coercive in p , uniformly Lipschitz in u . By introducing a solution semigroup, we provide a representation formula of the viscosity solution of the evolutionary equation. As its applications, we obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations. Especially, we prove a new comparison result with a necessary neighborhood of projected Aubry set for a special class of Hamilton-Jacobi equations that do not satisfy the "proper" condition introduced in the seminal paper \cite{CHL2}.