arXiv subjects

Abhrojyoti Sen

Publications and source records attributed to Abhrojyoti Sen.

15 recordsLinked to original sources

arXiv2026

Regularity for the fractional logarithmic $p$-Laplacian

We prove the Harnack inequality (with tails) and local H\"older regularity for the fractional logarithmic $p$-Laplace operator, which is derived by differentiating the fractional $p$-Laplace operator with respect to its order. To be more precise, for a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-\Delta_p)^{s+\log} u:= \frac{{\rm d}}{{\rm d}t}(-\Delta_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order $s\in (0, 1).$ The kernel of this operator involves a logarithmic factor. As a consequence, it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional $p$-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in this setting. We also construct an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear setup $p=2$.

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arXiv2026

Higher integrability for parabolic double phase equations with an improved gap bound

We prove a local higher integrability result for the gradient of H\"older continuous weak solutions to the parabolic double phase equation \[ \partial_t u - \operatorname{div} \left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right) = 0 \qquad \text{in } \Omega_T. \] We work under a relaxed gap condition on the exponents $p$ and $q$. The coefficient $a$ is assumed to belong to the class $\mathcal{Z}^{\kappa}(\Omega_T)$ for some $\kappa \in (0,\infty)$. The functions in this class satisfy a one-sided pointwise bound that controls how fast $a$ can grow away from its zero set, and the class contains the H\"older continuous functions. We also impose a mild almost increasing condition on $a$, which motivates the introduction of a new mollification, which we call the slanted Steklov average. For $u \in C^{0,\gamma,\gamma/q}_{\mathrm{loc}}(\Omega_T)$ with $\gamma \in [0,1)$, our main result holds under the gap bound \begin{equation}\tag{G}\label{eq:G} 2 \le p \le q \le p + \frac{q\kappa}{q - 2\gamma}. \end{equation} The new gap condition \eqref{eq:G} is purely parabolic in nature and is stricter than the optimal gap relation associated with the Lavrentiev phenomenon for the elliptic double phase functional.

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arXiv2026

$s$-harmonic functions in the small order limit

We study families $u_s$ of functions satisfying the equations $(-\Delta)^s u_s=0$, $s \in (0,1)$ in a smooth bounded open set $\Omega \subset \mathbb{R}^N$. The main purpose of this paper is twofold. First, we provide a detailed analysis of the asymptotics of these families in the zero order limit $s \to 0^+$. Second, we study the differentiability of $u_s$ as a function of $s$. Most of our results are devoted to the associated Poisson problem, where the family $u_s$ is determined by the exterior condition $u_s = g$ in $\mathbb{R}^N \setminus \Omega$ for some fixed function $g \in L^\infty(\mathbb{R}^N \setminus \Omega)$. Our results show that both the zero order asymptotics and the differentiability properties of $u_s$ can be expressed in terms of the logarithmic Laplacian of suitable extensions of $g$. This allows to deduce pointwise monotonicity properties of $u_s$ in the order parameter $s$ for a large class of functions $g$.

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arXiv2025

Lipschitz regularity for parabolic double phase equations with gradient nonlinearity

We establish the local Lipschitz regularity in space for the viscosity solutions to the parabolic double phase equation of the form \[ \smash{\partial_{t}u-\operatorname{div} \left(|Du|^{p-2}D u+a(z)|D u|^{q-2}D u\right)=f(z, Du)} \] by employing the Ishii-Lions method. In addition, we obtain H\"{o}lder estimate in time which turns out to be sharp in the degenerate regime. Here, $1< p\leq q<\infty,$ and the coefficient $a\geq 0$ is assumed to be bounded, locally Lipschitz continuous in space, and continuous in time. Furthermore, the non-homogeneity $f$ is assumed to be continuous on $\Omega\times \mathbb{R}\times \mathbb{R}^N,$ and to satisfy a suitable gradient growth condition. We also establish the equivalence between bounded viscosity solutions and weak solutions, under appropriate additional regularity assumption on the coefficient $a.$

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arXiv2025

Gradient higher integrability for degenerate parabolic double phase systems with two modulating coefficients

We establish an interior gradient higher integrability result for weak solutions to degenerate parabolic double phase systems involving two modulating coefficients. To be more precise, we study systems of the form \[ u_t-\operatorname{div} \left(a(z)|Du|^{p-2}Du+ b(z)|Du|^{q-2}Du\right)=-\operatorname{div} \left(a(z)|F|^{p-2}F+ b(z)|F|^{q-2}F\right), \] where $2\leq p\leq q < \infty$ and the modulating coefficients $a(z)$ and $b(z)$ are non-negative, with $a(z)$ being uniformly continuous and $b(z)$ being H\"{o}lder continuous. We further assume that the sum of two modulating coefficients is bounded from below by some positive constant. To establish the gradient higher integrability result, we introduce a suitable intrinsic geometry and develop a delicate comparison scheme to separate and analyze the different phases--namely, the $p$-phase, $q$-phase and $(p,q)$-phase. To the best of our knowledge, this is the first regularity result in the parabolic setting that addresses general double phase systems within the framework of weak solutions.

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arXiv2025

Parabolic Lipschitz truncation for multi-phase problems: the degenerate case

This article is devoted to exploring the Lipschitz truncation method for parabolic multi-phase problems. The method is based on Whitney decomposition and covering lemmas with a delicate comparison scheme of appropriate alternatives to distinguish phases, as introduced by the first and the second author in [24].

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arXiv2025

1D pressureless gas dynamics systems in a strip

We construct explicit measure-valued solutions to the one-dimensional pressureless gas dynamics system in a strip-like domain by introducing a new boundary potential. The constructed solutions satisfy an entropy condition, and depending on the boundary data and the behavior of the potentials, mass accumulation can occur at the boundaries. The approach relies on a systematic treatment of boundary potentials and their interactions with the initial data, providing a more precise understanding of the formation and propagation of singularities in measure-valued solutions.

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arXiv2024

Gradient higher integrability for degenerate/ singular parabolic multi-phase problems

This article establishes an interior gradient higher integrability result for weak solutions to parabolic multi-phase problems. The prototype equation for the parabolic multi-phase problem of $p$-Laplace type is given by \[ u_t - \operatorname{div} \left(|\nabla u|^{p-2} \nabla u + a(z) |\nabla u|^{q-2} \nabla u + b(z) |\nabla u|^{s-2} \nabla u \right) = 0, \] where $\frac{2n}{n+2} < p \leq q \leq s < \infty$, and the coefficients $a(z)$ and $b(z)$ are non-negative Hölder continuous functions on $Ω_T = Ω\times (0, T)$, with $Ω\subset \mathbb{R}^n$. We introduce a novel intrinsic scaling to address the problem in both the degenerate regime ($p \geq 2$) and the singular regime $\left(\frac{2n}{n+2} < p < 2\right),$ providing a unified framework. Our approach involves proving uniform parabolic Sobolev-Poincaré inequalities, which are key to establishing reverse Hölder type inequalities, along with covering lemmas for the $p$, $(p,q)$, $(p,s)$, and $(p,q,s)$-phases without distinguishing between the regimes of $p$, $q$, and $s$. In the end, we also discuss the gradient higher integrability for general parabolic multi-phase problem involving a finite number of phases.

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arXiv2023

Existence of global entropy solution for Eulerian droplet models and two-phase flow model with non-constant air velocity

This article addresses the question concerning the existence of global entropy solution for generalized Eulerian droplet models with air velocity depending on both space and time variables. When $f(u)=u,$ $κ(t)=const.$ and $u_a(x,t)=const.$ in (1.1), the study of the Riemann problem has been carried out by Keita and Bourgault [42] & Zhang et al. [38]. We show the global existence of the entropy solution to (1.1) for any strictly increasing function $f(\cdot)$ and $u_a(x,t)$ depending only on time with mild regularity assumptions on the initial data via shadow wave tracking approach. This represents a significant improvement over the findings of Yang [26]. Next, by using the generalized variational principle, we prove the existence of an explicit entropy solution to (1.1) with $f(u)=u,$ for all time $t>0$ and initial mass $v_0>0,$ where $u_a(x,t)$ depends on both space and time variables, and also has an algebraic decay in the time variable. This improves the results of many authors such as Ha et al. [40], Cheng and Yang [27] & Ding and Wang [50] in various ways. Furthermore, by employing the shadow wave tracking procedure, we discuss the existence of global entropy solution to the generalized two-phase flow model with time-dependent air velocity that extends the recent results of Shen and Sun [9].

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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.

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arXiv2023

A note on Hopf's lemma and strong minimum principle for nonlocal equations with non-standard growth

Let $Ω\subset \mathbb{R}^n $ be any open set and $u$ be a weak supersolution of $\mathcal{L}u=c(x)g(|u|)\frac{u}{|u|}$ where \[\mathcal{L}u(x)=\text{p.v.} \int_{\mathbb{R}^n} g\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right) \frac{u(x)-u(y)}{|u(x)-u(y)|} K(x,y)\frac{dy}{|x-y|^s}\] and $g=G^{\prime}$ for some Young function $G.$ This note imparts a Hopf's type lemma and strong minimum principle for $u$ when $c(x)$ is continuous in $\barΩ$ that extend the results of Del Pezzo and Quaas (JDE-2017) in fractional Orlicz-Sobolev setting.

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arXiv2023

Fine boundary regularity for fully nonlinear mixed local-nonlocal problems

We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. For a bounded $C^2$ domain $\Omega \subset \mathbb{R}^d,$ let $u\in C(\mathbb{R}^d)$ be a viscosity solution of such Dirichlet problem. We obtain global Lipschitz regularity and fine boundary regularity for $u$ by constructing appropriate sub and supersolutions coupled with a Harnack type inequality. We apply these results to obtain H\"{o}lder regularity of $Du$ up to the boundary.

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arXiv2022

Boundary regularity of mixed local-nonlocal operators and its application

Let $Ω$ be a bounded $C^2$ domain in $\mathbb{R}^n$ and $u\in C(\mathbb{R}^n)$ solves \begin{equation*} \begin{aligned} Δu + a Iu + C_0|Du| \geq -K\quad \text{in}\; Ω, \quad Δu + a Iu - C_0|Du|\leq K \quad \text{in}\; Ω, \quad u=0\quad \text{in}\; Ω^c, \end{aligned} \end{equation*} in the viscosity sense, where $0\leq a\leq A_0$, $C_0, K\geq 0$, and $I$ is a suitable nonlocal operator. We show that $u/δ$ is in $C^κ(\bar Ω)$ for some $κ\in (0,1)$, where $δ(x)={\rm dist}(x, Ω^c)$. Using this result, we also establish that $u\in C^{1, γ}(\barΩ)$. Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.

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arXiv2019

Limiting behavior of scaled general Euler equations of compressible fluid flow

The aim of this article is to study the limiting behavior of the solutions for the scaled generalized Euler equations of compressible fluid flow. When the initial data is of Riemann type, we showed the existence of solution which consists of shock waves and rarefaction waves and that the distributional limit of the solutions for this system converges to the solution of a non-strictly hyperbolic system, called one dimensional model for large scale structure formation of universe as the scaling parameter vanishes. An explicit entropy and entropy flux pair are also constructed for the particular flux function (Brio system) and it is shown that the solution constructed is entropy admissible. This is a continuation of our work[23].

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arXiv2018

Limiting behavior of solutions for Euler equations of compressible fluid flow

We study the limiting behavior of the solutions of Euler equations of one-dimensional compressible fluid flow as the pressure like term vanishes. This system can be thought of as an approximation for the one dimensional model for large scale structure formation of universe. We show that the solutions of former equation converges to the solution of later in the sense of distribution and agrees with the vanishing viscosity limit when the initial data is of Riemann type. A different approximation for the one dimensional model for large scale structure formation of universe is also studied.

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