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Prerona Dutta

Publications and source records attributed to Prerona Dutta.

8 recordsLinked to original sources
arXiv2026

Weak Diffeomorphisms and Extremals for Scalar Conservation Laws

Scalar conservation laws in one space variable allow a Lagrangian (particle path) formulation. The Lagrangian trajectory in the infinite-dimensional group of diffeomorphisms on the physical space can be written as a system of conservation laws. The relation between solutions of the Cauchy problem for the conservation law and solutions of the corresponding Cauchy problem on the diffeomorphism group extends to weak solutions of the coresponding problems. The correspondence between particle paths and transport equations is analogous to that between a Lie group and the corresponding Lie algebra. This paper establishes that for scalar conservation laws the particle paths are extremals of an action functional on the space of diffeomorphisms; that is, they are geodesics in some metric. In some examples of systems of conservation laws, including the physical example of isentropic gas dynamics in one space dimension, diffeomorphism representations also exist and may be interpreted as extremals of action functionals.

math.AP↗
arXiv2025

On the rate of convergence in superquadratic Hamilton--Jacobi equations with state constraints

In this paper, we investigate the convergence rate in the vanishing viscosity limit for solutions to superquadratic Hamilton--Jacobi equations with state constraints. For every $p>2$, we establish the rate of convergence for nonnegative Lipschitz data vanishing on the boundary to be of order $ \mathcal{O}(\varepsilon^{1/2}) $ and obtain an improved upper rate of order $ \mathcal{O}\big(\varepsilon^{\frac{p}{2(p-1)}}\big)$ for semiconcave data.

math.AP↗
arXiv2024

Non-uniform dependence on periodic initial data for the two-component Fornberg-Whitham system in Besov spaces

This paper establishes non-uniform continuity of the data-to-solution map in the periodic case, for the two-component Fornberg-Whitham system in Besov spaces $B^s_{p,r}(\mathbb{T}) \times B^{s-1}_{p,r}(\mathbb{T})$ for $s> \max\{2+\frac{1}{p}, \frac{5}{2}\}$. In particular, when $p=2$ and $r=2$, this proves the non-uniform dependence on initial data for the system in Sobolev spaces $H^s(\mathbb{T})\times H^{s-1}(\mathbb{T})$ for $s> \frac{5}{2}$.

math.AP↗
arXiv2022

Extending Lagrangian transformations to nonconvex scalar conservation laws

The present paper studies a method of finding Lagrangian transformations, in the form of particle paths, for all scalar conservation laws having a smooth flux. These are found using the notion of weak diffeomorphisms. More precisely, from any given scalar conservation law, we derive a Temple system having one linearly degenerate and one genuinely nonlinear family. We modify the system to make it strictly hyperbolic and prove an existence result for it. Finally we establish that entropy admissible weak solutions to this system are equivalent to those of the scalar equation. This method also determines the associated weak diffeomorphism.

math.AP↗
arXiv2022

Metric entropy for Hamilton-Jacobi equation with uniformly directionally convex Hamiltonian

The present paper first aims to study the BV-type regularity for viscosity solutions of the Hamilton-Jacobi equation \[ u_t(t,x)+H\big(D_{x} u(t,x)\big)~=~0\qquad\forall (t,x)\in ]0,\infty[\times\mathbb{R}^d \] with a coercive and uniformly directionally convex Hamiltonian $H\in\mathcal{C}^{1}(\mathbb{R}^d)$. More precisely, we establish a BV bound on the slope of backward characteristics $DH(u(t,\cdot))$ starting at a positive time $t>0$. Relying on the BV bound, we quantify the metric entropy in ${\bf W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^d)$ for the map $S_t$ that associates to every given initial data $u_0\in{\bf Lip}\big(\mathbb{R}^d\big)$, the corresponding solution $S_tu_0$. Finally, a counter example is constructed to show that both $D_xu(t,\cdot)$ and $DH(D_xu(t,\cdot))$ fail to be in $BV_{\mathrm{loc}}$ for a general strictly convex and coercive $H\in\mathcal{C}^2(\mathbb{R}^d)$.

math.AP↗
arXiv2020

Metric entropy for functions of bounded total generalized variation

We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space $(E,ρ)$ up to an accuracy of $\varepsilon>0$ with respect to the ${\bf L}^1$-distance. Such an estimate is explicitly computed in terms of doubling and packing dimensions of $(E,ρ)$. The obtained result is applied to provide an upper bound on the metric entropy for a set of entropy admissible weak solutions to scalar conservation laws in one-dimensional space with weakly genuinely nonlinear fluxes.

math.FA↗
arXiv2018

Covering numbers for bounded variation functions

In this paper, we provide upper and lower estimates for the minimal number of functions needed to represent a bounded variation function with an accuracy of epsilon with respect to ${\bf L}^1$-distance.

math.FA↗