arXiv subjects

Divyang G. Bhimani

Publications and source records attributed to Divyang G. Bhimani.

35 records · Page 2Linked to original sources

arXiv2023

Remark on the ill-posedness for KdV-Burgers equation in Fourier amalgam spaces

We have established (a weak form of) ill-posedness for the KdV-Burgers equation on a real line in Fourier amalgam spaces $\widehat{w}_s^{p,q}$ with $s<-1$. The particular case $p=q=2$ recovers the result of L. Molinet and F. Ribaud [Int. Math. Res. Not., (2002), pp. 1979-2005]. The result is new even in Fourier Lebesgue space $\mathcal{F}L_s^q$ which corresponds to the case $p=q(\neq 2)$ and in modulation space $M_s^{2,q}$ which corresponds to the case $p=2,q\neq 2$.

math.AP↗

arXiv2022

The Blow-up solutions for fractional heat equations on torus and Euclidean space

We produce a finite time blow-up solution for nonlinear fractional heat equation ($\partial_t u + (-Δ)^{β/2}u=u^k$) in modulation and Fourier amalgam spaces on the torus $\mathbb T^d$ and the Euclidean space $\mathbb R^d.$ This complements several known local and small data global well-posedness results in modulation spaces on $\mathbb R^d.$ Our method of proof rely on the formal solution of the equation. This method should be further applied to other non-linear evolution equations.

math.AP↗

arXiv2022

On inhomogeneous heat equation with inverse square potential

We study inhomogeneous heat equation with inverse square potential, namely, \[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^αu,\] where $\mathcal{L}_a=-Δ+ a |x|^{-2}.$ We establish some fixed-time decay estimate for $e^{-t\mathcal{L}_a}$ associated with inhomogeneous nonlinearity $|\cdot|^{-b}$ in Lebesgue spaces. We then develop local theory in $L^q-$ scaling critical and super-critical regime and small data global well-posedness in critical Lebegue spaces. We further study asymptotic behaviour of global solutions by using self-similar solutions, provided the initial data satisfies certain bounds. Our method of proof is inspired from the work of Slimene-Tayachi-Weissler (2017) where they considered the classical case, i.e. $a=0$.

math.AP↗

arXiv2022

On heat equations associated with fractional harmonic oscillators

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator $e^{-tH^β}$, $t, β>0$, associated with the harmonic oscillator $H=-Δ+ |x|^2$. We then prove some local and global wellposedness results for nonlinear fractional heat equations.

math.AP↗

arXiv2022

The Hartree and Hartree-Fock equations in Lebesgue $L^p$ and Fourier-Lebesgue $\hat{L}^p$ spaces

We establish some local and global well-posedness for Hartree-Fock equations of $N$ particles (HFP) with Cauchy data in Lebesgue spaces $L^p \cap L^2 $ for $1\leq p \leq \infty$. Similar results are proven for fractional HFP in Fourier-Lebesgue spaces $ \hat{L}^p \cap L^2 \ (1\leq p \leq \infty).$ On the other hand, we show that the Cauchy problem for HFP is ill-posed if we simply work in $\hat{L}^p \ (2<p\leq \infty).$ Analogue results hold for reduced HFP. In the process, we prove the boundedeness of various trilinear estimates for Hartree type non linearity in these spaces which may be of independent interest. As a consequence, we get natural $L^p$ and $\hat{L}^p$ extension of classical well-posedness theories of Hartree and Hartree-Fock equations with Cauchy data in just $L^2-$based Sobolev spaces.

math.AP↗

arXiv2021

Norm inflation for BBM equation in Fourier amalgam and Wiener amalgam spaces with negative regularity

We consider Benjamin-Bona-Mahony (BBM) equation of the form $$ u_t+u_x+uu_x-u_{xxt}=0, \quad (x, t)\in \mathcal{M}\times \mathbb R $$ where $\mathcal{M}= \mathbb T$ or $\mathbb R.$ We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthen several known NI results at zero initial data in $H^s(\mathbb T)$ established by Bona-Dai (2017) and ill-posedness result established by Bona-Tzvetkov (2008) and Panthee (2011) in $H^s(\mathbb R).$ Our result is sharp with respect to local well-posedness result of Banquet-Villamizar-Roa (2021) in modulation spaces $M^{2,1}_s(\mathbb R)$ for $s\geq 0$.

math.AP↗

arXiv2021

Norm inflation with infinite loss of regularity at general initial data for nonlinear wave equations in Wiener amalgam and Fourier amalgam spaces

We study the strong ill-posedness (norm inflation with infinite loss of regularity) for the nonlinear wave equation at every initial data in Wiener amalgam and Fourier amalgam spaces with negative regularity. In particular these spaces contain Fourier-Lebesgue, Sobolev and some modulation spaces. The equations are posed on $\mathbb R^d$ and on torus $\mathbb T^d$ and involve a smooth power nonlinearity. Our results are sharp with respect to well-posedness results of Bényi and Okoudjou (2009) and Cordero and Nicola (2009) in the Wiener amalgam and modulation space cases. In particular, we also complement norm inflation result of Christ, Colliander and Tao (2003) and Forlano and Okamoto (2020) by establishing infinite loss of regularity in the aforesaid spaces.

math.AP↗

arXiv2020

Norm inflation for nonlinear Schrodinger equations in Fourier-Lebesgue and modulation spaces of negative regularity

We consider nonlinear Schr{ö}dinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schr{ö}dinger equations in a general abstract functional setting.

math.AP↗

arXiv2020

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

We study the Hermite operator $H=-Δ+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^β$, $β>0$ in phase space. Namely, we represent functions $f$ via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform $V_g f$ ($g$ being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of $V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0< p,q\leq \infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^β}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$, exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for $H^β$ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay $e^{-c t}$ as the solution of the corresponding linear equation, where $c=d^β$ is the bottom of the spectrum of $H^β$. This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data - hence in $M^{\infty,1}$.

math.FA↗

arXiv2019

The Hartree-Fock equations in modulation spaces

We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on $\mathbb R^d$. In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest.

math.AP↗

arXiv2018

Bimodal Wilson systems in $L^2(\mathbb R)$

Given a window $ϕ\in L^2(\mathbb R),$ and lattice parameters $α, β>0,$ we introduce a bimodal Wilson system $\mathcal{W}(ϕ, α, β)$ consisting of linear combinations of at most two elements from an associated Gabor $\mathcal{G}(ϕ, α, β)$. For a class of window functions $ϕ,$ we show that the Gabor system $\mathcal{G}(ϕ, α, β)$ is a tight frame of redundancy $β^{-1}$ if and only if the Wilson system $\mathcal{W}(ϕ, α, β)$ is Parseval system for $L^2(\mathbb R).$ Examples of smooth rapidly decaying generators $ϕ$ are constructed. In addition, when $3\leq β^{-1}\in \mathbb N$, we prove that it is impossible to renormalize the elements of the constructed Parseval Wilson frame so as to get a well-localized orthonormal basis for $L^2(\mathbb R)$.

math.FA↗

arXiv2018

Global Cauchy problems for the Klein-Gordon, wave and fractional Schrödinger equations with Hartree nonlinearity on modulation spaces

We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schrödinger (HNLS) equations with cubic convolution (Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrödinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. This improves known results in Sobolev spaces in some sense. As a consequence, we get local and global well-posedness and scattering in larger than usual $L^p-$Sobolev spaces and we could include wider class of Hartree type nonlinarity.

math.AP↗

arXiv2018

The nonlinear Schrödinger equations with harmonic potential in modulation spaces

We study nonlinear Schrödinger $i\partial_tu-Hu=F(u)$ (NLSH) equation associated to harmonic oscillator $H=-Δ+|x|^2$ in modulation spaces $M^{p,q}.$ When $F(u)= (|x|^{-γ}\ast |u|^2)u, $ we prove global well-posedness for (NLSH) in modulation spaces $M^{p,p}(\mathbb R^d)$ $ (1\leq p < 2d/(d+γ), 0<γ< \min \{ 2, d/2\}).$ When $F(u)= (K\ast |u|^{2k})u$ with $K\in \mathcal{F}L^q $ (Fourier-Lebesgue spaces) or $M^{\infty,1}$ (Sjöstrand's class) or $M^{1, \infty},$ some local and global well-posedness for (NLSH) are obtained in some modulation spaces. When $F$ is real entire and $F(0)=0$, we prove local well-posedness for (NLSH) in $M^{1,1}.$ As a consequence, we can get local and global well-posedness for (NLSH) in a function spaces$-$which are larger than usual $L^p_s-$Sobolev spaces.

math.AP↗

arXiv2018

Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra

We study the Cauchy problem for fractional Schrödinger equation with cubic convolution nonlinearity ($i\partial_t u - (-Δ)^{\fracα{2}}u\pm (K\ast |u|^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb R^{d}).$ For $K(x)= |x|^{-γ}$ $ (0< γ< \text{min} \{α, d/2\})$, we establish global well-posedness results in $M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+γ))$ when $α=2, d\geq 1$, and with radial Cauchy data when $d\geq 2, \frac{2d}{2d-1}< α< 2. $ Similar results are proven in Fourier algebra $\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$

math.AP↗

arXiv2018

Composition Operators on Wiener amalgam Spaces

For a complex function $F$ on $\mathbb C$, we study the associated composition operator $T_{F}(f):=F\circ f= F(f)$ on Wiener amalgam $W^{p,q}(\mathbb R^d) \ (1\leq p< \infty, 1\leq q<2).$ We have shown $T_{F} $ maps $W^{p, 1}(\mathbb R^d)$ to $W^{p,q}(\mathbb R^d)$ if and only if $F$ is real analytic on $\mathbb R^2$ and $F(0)=0.$ Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb R^d).$ In particular, this gives an affirmative answer to the open question proposed by Bhimani-Ratnakumar (J. Funct. Anal. 270 (2016), p.621-648).

math.AP↗

arXiv2017

Hermite Multipliers on Modulation Spaces

We study multipliers associated to the Hermite operator $H=-Δ+ |x|^2$ on modulation spaces $M^{p,q}(\mathbb R^d)$. We prove that the operator $m(H)$ is bounded on $M^{p,q}(\mathbb R^d)$ under standard conditions on $m,$ for suitable choice of $p$ and $q$. As an application, we point out that the solutions to the free wave and Schrödinger equations associated to $H$ with initial data in a modulation space will remain in the same modulation space for all times. We also point out that Riesz transforms associated to $H$ are bounded on some modulation spaces.

math.AP↗

arXiv2014

Functions operating on modulation spaces and nonlinear dispersive equations

The aim of this paper is two fold. We show that if a complex function $F$ on $\C$ operates in the modulation spaces $M^{p,1}(\R^n)$ by composition, then $F$ is real analytic on $\R^2 \approx \C$. This answers negatively, the open question posed in [M. Ruzhansky, M. Sugimoto, B. Wang, Modulation Spaces and Nonlinear Evolution Equations, arXiv:1203.4651], regarding the general power type nonlinearity of the form $|u|^αu$. We also characterise the functions that operate in the modulation space $M^{1,1}(\R^n)$. The local well-posedness of the NLS, NLW and NLKG equations for the `real entire' nonlinearities are also studied in some weighted modulation spaces $M^{p,q}_s(\R^n)$.

math.AP↗