arXiv subjects

Gopikrishnan Chirappurathu Remesan

Publications and source records attributed to Gopikrishnan Chirappurathu Remesan.

3 recordsLinked to original sources

arXiv2026

Well-posedness and regularity for seminlinear time-dependent second and fourth order in space equations

This article discusses a unified convergence analysis of the semilinear time-dependent equation $\partial_t u + (-1)^\mathrm{m}Δ^{\mathrm{m}}u + u^3 - u = f$ with $\mathrm{m} \in \{1,2\}$ and homogeneous Dirichlet boundary conditions. The analysis relies on Faedo-Galerkin approximation and convergence via compactness estimates. The existence and uniqueness of the weak solution is proved when the initial data is smooth. A refined and novel analysis extends the existence result to problems with rough initial data also.

math.AP↗

arXiv2020

Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth model

A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law $\partial_t α+ \mathrm{div}(\boldsymbol{u}f(α)) = 0$ in two and three spatial dimensions with an initial data of bounded variation is established. We assume that the divergence of the velocity $\mathrm{div}(\boldsymbol{u})$ is of bounded variation instead of the classical assumption that $\mathrm{div}(\boldsymbol{u})$ is zero. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law $\partial_t α+ \mathrm{div}(\boldsymbol{F}(t,\boldsymbol{x},α)) = 0$, where $\mathrm{div}_{\boldsymbol{x}}\boldsymbol{F} \not=0$ on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in $L^p$ spaces, which is essential to prove the existence of a solution for a partial differential equation with nonlinear terms in $α$, when the uniqueness of the solution is not available. This application is demonstrated by establishing the existence of a weak solution for a model that describes the evolution of initial stages of breast cancer proposed by S. J. Franks et al. [14]. The model consists of four coupled variables: tumour cell concentration, tumour cell velocity--pressure, and nutrient concentration, which are governed by a hyperbolic conservation law, viscous Stokes system, and Poisson equation, respectively. Results from numerical tests are provided and they complement theoretical findings.

math.NA↗

arXiv2020

Convergence analysis of a numerical scheme for a tumour growth model

We consider a one--spatial dimensional tumour growth model [2, 3, 4] that consists of three dependent variables of space and time: volume fraction of tumour cells, velocity of tumour cells, and nutrient concentration. The model variables satisfy a coupled system of semilinear advection equation (hyperbolic), simplified linear Stokes equation (elliptic), and semilinear diffusion equation (parabolic) with appropriate conditions on the time-dependent boundary, which is governed by an ordinary differential equation. We employ a reformulation of the model defined in a larger, fixed time-space domain to overcome some theoretical difficulties related to the time-dependent boundary. This reformulation reduces the complexity of the model by removing the need to explicitly track the time-dependent boundary, but nonlinearities in the equations, noncoercive operators in the simplified Stokes equation, and interdependence between the unknown variables still challenge the proof of suitable a priori estimates. A numerical scheme that employs a finite volume method for the hyperbolic equation, a finite element method for the elliptic equation, and a backward Euler in time--mass lumped finite element in space method for the parabolic equation is developed. We establish the existence of a time interval $(0,T_{\ast})$ over which, using compactness techniques, we can extract a convergent subsequence of the numerical approximations. The limit of any such convergent subsequence is proved to be a weak solution of the continuous model in an appropriate sense, which we call a threshold solution. Numerical tests and justifications that confirm the theoretical findings conclude the paper.

math.NA↗