arXiv subjects

Manas Bhatnagar

Publications and source records attributed to Manas Bhatnagar.

9 recordsLinked to original sources

arXiv2024

A Convergent Front Tracking Scheme

We present a modified Front Tracking (mFT) scheme for hyperbolic systems of conservation laws in one space dimension, in which we allow arbitrarily large nonlinear waves. We build the scheme by introducing and solving a ``generalized Riemann Problem'', which yields exact solutions for finite times. This allows us to treat the states adjacent to all waves exactly, and approximate compressive simple waves in addition to rarefactions, contacts and shocks. In particular, we require exact expression of the various wave curves and avoid the use of Taylor expansions. After construction of the scheme, under reasonable assumptions, we show that the mFT approximations converge to a weak* solution of the system. This essentially reduces existence of solutions with large amplitude data to obtaining uniform bounds on the total variation of the approximations. We then apply the scheme to the Euler equations of gas dynamics, for which we exactly solve the generalized Riemann Problem and define the scheme for both 3x3 and 2x2 systems, and prove the equivalence of Eulerian and Lagrangian frames. For the $p$-system, modeling isentropic gas dynamics in a Lagrangian frame, we show that there is no finite accumulation of interaction times. This means that the last remaining obstacle to global existence of large data, large amplitude solutions is the construction of a decreasing Glimm potential.

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arXiv2023

On critical thresholds for hyperbolic balance law systems

We review the theoretical development in the study of critical thresholds for hyperbolic balance laws. The emphasis is on two classes of systems: Euler-Poisson-alignment (EPA) systems and hyperbolic relaxation systems. We start with an introduction to the `Critical Threshold Phenomena' and study some nonlocal PDE systems, which are important from modeling point of view.

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arXiv2023

A complete characterization of sharp thresholds to spherically symmetric multidimensional pressureless Euler-Poisson systems

The Euler-Poisson (EP) system models the dynamics of a variety of physical processes, including charge transport, collisional plasmas, and certain cosmological wave phenomena. In this work, we establish sharp critical threshold conditions that distinguish global-in-time regularity from finite-time breakdown for solutions of the radially symmetric, multidimensional pressureless EP system. Overall, there are two cases: with and without background ($c>0, c=0$ respectively). For $c>0$, we obtain precise thresholds assuming a periodicity condition. A key feature of our approach is that it extends seamlessly to the zero background case, where we obtain sharp thresholds without imposing any additional assumptions. In particular, the framework accommodates initial velocities that may be negative, allowing the flow to be directed toward the origin. The main analytical challenge of deriving threshold conditions for EP systems stems from the intricate coupling of various local/nonlocal forces. To overcome this, we identify a novel nonlinear quantity that plays a decisive role in the analysis and enables a unified treatment of all relevant scenarios. Our results provide a comprehensive characterization of critical thresholds for the pressureless EP system in multiple dimensions.

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arXiv2021

Critical thresholds in the Euler-Poisson-alignment system

This paper is concerned with the global wellposedness of the Euler-Poisson-alignment (EPA) system. This system arises from collective dynamics, and features two types of nonlocal interactions: the repulsive electric force and the alignment force. It is known that the repulsive electric force generates oscillatory solutions, which is difficult to be controlled by the nonlocal alignment force using conventional comparison principles. We construct \emph{invariant regions} such that the solution trajectories cannot exit, and therefore obtain global wellposedness for subcritical initial data that lie in the invariant regions. Supercritical regions of initial data are also derived which leads to finite-time singularity formations. To handle the oscillation and the nonlocality, we introduce a new way to construct invariant regions piece by piece in the phase plane of a reformulation of the EPA system. Our result is extended to the case when the alignment force is weakly singular. The singularity leads to the loss of a priori bounds crucial in our analysis. With the help of improved estimates on the nonlocal quantities, we design non-trivial invariant regions that guarantee global wellposedness of the EPA system with weakly singular alignment interactions.

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arXiv2021

Global dynamics of the Euler-alignment system with weakly singular kernel

This letter studies the Euler-alignment system with weakly singular influence functions by introducing a novel technique to bound the density. Instead of resorting to a nonlinear maximum principle used in [C. Tan, Nonlinearity, 33: 1907--1924, 2020] to bound the interaction term $\psi\ast\rho$ by $\rho^s$ with $s\in (0, 1)$ for $\psi$ with an algebraic singularity at origin, we bound $\psi*\rho$ by a relaxed constant for any $\psi \in L^1$. We thus establish the global-in-time existence results with weaker assumptions on $\psi$ and refined solution bounds, characterized by the structure of $\psi$.

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arXiv2020

Sharp critical thresholds in a hyperbolic system with relaxation

We propose and study a one-dimensional $2\times 2$ hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic.

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arXiv2020

Critical thresholds in a nonlocal Euler system with relaxation

We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.

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arXiv2020

Critical thresholds in 1D pressureless Euler-Poisson systems with varying background

The Euler Poisson equations describe important physical phenomena in many applications such as semiconductor modeling and plasma physics. This paper is to advance our understanding of critical threshold phenomena in such systems in the presence of different forces. We identify critical thresholds in two damped Euler Poisson systems, with and without alignment, both with attractive potential and spatially varying background state. For both systems, we give respective bounds for subcritical and supercritical regions in the space of initial configuration, thereby proving the existence of a critical threshold for each scenario. Key tools include comparison with auxiliary systems, phase space analysis of the transformed system.

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arXiv2019

Critical Thresholds in One Dimensional Damped Euler-Poisson Systems

This paper is concerned with the critical threshold phenomenon for one dimensional damped, pressureless Euler-Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J., 50:109--157, 2001]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line.

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