arXiv subjects

Manas R. Sahoo

Publications and source records attributed to Manas R. Sahoo.

6 recordsLinked to original sources

arXiv2026

Existence and Uniqueness of a Transport Equation coupled with a Balance law

In this article, we study the existence and uniqueness of a transport equation with discontinuous coefficients that comes from the solution of a balance law with a time-dependent source term. We also give a simplified proof providing an explicit formula for the solution of the balance law using a variational formulation. This system can be considered as a generalized version of the one-dimensional model for the large-scale structure of the formation of the universe.

math.AP↗

arXiv2026

Viscous Burgers equation driven by point source: a formula for the weak limit

In this article, we obtain the weak limit of the solutions of the viscous Burgers equation driven by a point source term, as the coefficient of viscosity tends to zero. The weak limit is related to the variational problem that consists of three types of functional, which is not usual in the absence of the source term.

math.AP↗

arXiv2023

Initial boundary value problem for 1D scalar balance laws with strictly convex flux

A Lax-Oleinik type explicit formula for 1D scalar balance laws has been recently obtained for the pure initial value problem by Adimurthi et al. in [1]. In this article, by introducing a suitable boundary functional, we establish a Lax-Oleinik type formula for the initial boundary value problem. For the pure initial value problem, the solution for the corresponding Hamilton-Jacobi equation turns out to be the minimizer of a functional on the set of curves known as h-curves. In the present situation, part of the h-curve joining any two points in the quarter plane may cross the boundary $x = 0.$ This phenomenon breaks the simplicity of the minimization process through the boundary functional compared to the case of conservation laws. Moreover, this complicates the verification of the boundary condition in the sense of Bardos, le Roux, and Nedelec [2]. To verify the boundary condition, the boundary points are classified into three types depending on the structure of the minimizers at those points. Finally, by introducing characteristic triangles, we construct generalized characteristics and show that the explicit solution is entropy admissible.

math.AP↗

arXiv2016

Density property of certain sets and their applications

In this paper we show that certain sets are dense in $\mathbb{R}$. We give some applications. For example, we show an analytical proof that $q^{\frac{1}{n}}$, $q$ is a prime number and $e$; are irrational numbers. As another application we show: If $f$ is an locally integrable function on $\mathbb{R}-\{0\}$ satisfying $\int_x ^{px}f(t)dt$ and $\int_x ^{qx}f(t)dt$ are constant with $\frac{\ln p}{\ln q}$ is an irrational number; implies $f(t)=\frac{c}{t}\,\,\ a.e.$, where $c$ is constant; which is already considered in \cite{b1} for the case when $f$ is continuous.

math.CA↗

arXiv2014

Generalized solution to a system of conservation law which is not strictly hyperbolic

In this paper we study a non strictly system of conservation law when viscosity is present and viscosity is zero, which is studied in [10]. We show the existence and uniqueness of the solution in the space of generalized functions of Colombeau for the viscous problem and construct a solution to the inviscid system in the sense of association. Also we construct a solution using shadow wave approach [5] and Volpert product which was partly determined as vanishing viscosity limit in [10].

math.AP↗

arXiv2014

Spherically Symmetric Solutions of Multi-dimensional Zero-pressure Gas Dynamics System

In this article we find explicit formulae for spherically symmetric solutions of the multidimensional zero-pressure gas dynamics system and its adhesion approximation. The asymptotic behaviour of the explicit solutions of the adhesion approximation has been studied. The radial components of the velocity and density satisfy a simpler equation which enables us to get explicit formulae for different types of domains and study its asymptotic behaviour. A class of solutions for the inviscid system with conditions on the mass instead of conditions at origin has also been analyzed.

math.AP↗