arXiv subjects

Pham Truong Xuan

Publications and source records attributed to Pham Truong Xuan.

At least 19 recordsLinked to original sources

arXiv2026

Peeling of Dirac fields on Kerr spacetimes

In a recent paper with J.-P. Nicolas [J.-P. Nicolas and P.T. Xuan, Annales Henri Poincare 2019], we studied the peeling for scalar fields on Kerr metrics. The present work extends these results to Dirac fields on the same geometrical background. We follow the approach initiated by L.J. Mason and J.-P. Nicolas [L. Mason and J.-P. Nicolas, J.Inst.Math.Jussieu 2009; L. Mason and J.-P. Nicolas, J.Geom.Phys 2012] on the Schwarzschild spacetime and extended to Kerr metrics for scalar fields. The method combines the Penrose conformal compactification and geometric energy estimates in order to work out a definition of the peeling at all orders in terms of Sobolev regularity near $\mathscr{I}$, instead of ${\mathcal C}^k$ regularity at $\mathscr{I}$, then provides the optimal spaces of initial data such that the associated solution satisfies the peeling at a given order. The results confirm that the analogous decay and regularity assumptions on initial data in Minkowski and in Kerr produce the same regularity across null infinity. Our results are local near spacelike infinity and are valid for all values of the angular momentum of the spacetime, including for fast Kerr metrics.

gr-qc↗

arXiv2026

Peeling for tensorial wave equations on Schwarzschild spacetime

In this paper, we establish the asymptotic behaviour along outgoing and incoming radial geodesics, i.e., the peeling property for the tensorial Fackrell-Ipser and spin $\pm 1$ Teukolsky equations on Schwarzschild spacetime. Our method combines a conformal compactification with vector field techniques to prove the two-side estimates of the energies of tensorial fields through the future and past null infinity $\mathscr{I}^\pm$ and the initial Cauchy hypersurface $Σ_0 = \left\{ t=0 \right\}$ in a neighbourhood of spacelike infinity $i_0$ far away from the horizon and future timelike infinity. Our results obtain the optimal initial data which guarantees the peeling at all orders.

gr-qc↗

arXiv2026

Interpolation scattering for wave equations with singular potentials and singular data

In this paper we investigate a construction of scattering for wave-type equations with singular potentials on the whole space $\mathbb{R}^n$ in a framework of weak-$L^p$ spaces. First, we use an Yamazaki-type estimate for wave groups on Lorentz spaces and fixed point arguments to prove the global well-posedness for wave-type equations on weak-$L^p$ spaces. Then, we provide a corresponding scattering results in such singular framework. Finally, we use also the dispersive estimates to establish the polynomial stability and improve the decay of scattering in weak-$L^p$ spaces.

math.AP↗

arXiv2026

Geometric scattering for nonlinear wave equations on the Schwarzschild metric

In this paper, we establish a conformal scattering theory for defocusing semilinear wave equations on Schwarzschild spacetime. We combine the energy and pointwise decay results for solutions obtained in \cite{Yang} with a Sobolev embedding on spacelike hypersurfaces to derive two-sided energy estimates between the energy flux of solutions through the Cauchy initial hypersurface $Σ_0 = \{ t = 0 \}$ and that through the null conformal boundaries $\mathfrak{H}^+ \cup \scri^+$ (respectively, $\mathfrak{H}^- \cup \scri^-$). By combining these estimates with the well-posedness of the Cauchy and Goursat problems for nonlinear wave equations, we construct a bounded linear and locally Lipschitz scattering operator that maps past scattering data to future scattering data.

math.AP↗

arXiv2025

Well-posedness and exponential stability for Boussinesq systems on real hyperbolic Manifolds and application

We investigate the global existence and exponential decay of mild solutions for the Boussinesq systems in $L^p$-phase spaces on the framework of real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$, where $d \geqslant 2$ and $1<p\leq d$. We consider a couple of Ebin-Marsden's Laplace and Laplace-Beltrami operators associated with the corresponding linear system which provides a vectorial matrix semigoup. First, we show the existence and the uniqueness of the bounded mild solution for the linear system by using dispersive and smoothing estimates of the vectorial matrix semigroup. Next, using the fixed point arguments, we can pass from the linear system to the semilinear system to establish the existence of the bounded mild solutions. By using Gronwall's inequality, we establish the exponential stability of such solutions. Finally, we give an application of stability to the existence of periodic mild solutions for the Boussinesq systems.

math.AP↗

arXiv2025

Asymptotically almost Periodic Solutions of the parabolic-parabolic Keller-Segel systems on bounded domains

In this paper, we investigate the existence, uniqueness, and exponential decay of asymptotically almost periodic (AAP-) mild solutions for the parabolic-parabolic Keller-Segel systems on a bounded domain $Ω\subset \mathbb{R}^n$ with a smooth boundary. First, we establish the well-posedness of mild solutions for the corresponding linear systems by utilizing the dispersive and smoothing estimates of the Neumann heat semigroup on the bounded domain $Ω$. We then prove the existence and uniqueness of AAP-mild solutions for the linear systems by providing a Massera-type principle. Next, using results of the linear systems and fixed-point arguments, we derive the well-posedness of such solutions for the Keller-Segel systems. Finally, the exponential decay of these solutions is demonstrated through a Gronwall-type inequality.

math.AP↗

arXiv2024

Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds

We study the global existence, uniqueness and exponential stability of mild solutions to the Boussinesq systems equipped with a generalized gravitational field on the framework of non-compact Riemannian manifolds. We work on some manifolds satisfying some bounded and negative conditions on curvature tensors. We consider a couple of Stokes and heat semigroups associated with the corresponding linear system which provides a vectorial matrix semigoup. By using dispersive and smoothing estimates of the vectorial matrix semigroup we establish the global-in-time existence and uniqueness of mild solutions for linear systems. Next, we can pass from the linear system to the semilinear systems to obtain the well-posedness by using fixed point arguments. Moreover, we will prove the exponential stability of such solutions by using Gronwall's inequality.

math.AP↗

arXiv2024

On asymptotically almost periodic mild solutions for Navier-Stokes equations on non-compact Riemannian manifolds

In this paper, we study the existence, uniqueness and asymptotic behaviour of almost periodic and asymptotically almost periodic mild solutions to the incompressible Navier-Stokes equations on $d$-dimensional non-compact manifold $(\mathcal{M},g)$ which satisfies some bounded conditions on curvature tensors. First, we use the $L^p-L^q$-dipsersive and smoothing estimates of the Stokes semigroup to prove Massera-type principles which guarantees the well-posedness of almost periodic and asymptotically almost periodic mild solutions for the inhomogeneous Stokes equations. Then, by using fixed point arguments and Gronwall's inequality we establish the well-posedness and exponential decay for global-in-time of such solutions of Navier-Stokes equations. Our results extend the previous ones \cite{Xuan2022,Xuan2023} to the generalized non-compact Riemannian manifolds.

math.AP↗

arXiv2024

On pseudo almost Periodic Solutions of the parabolic-elliptic Keller-Segel systems

In this paper we investigate the existence, uniqueness and exponential stability of pseudo almost periodic (PAP-) mild solutions of the parabolic-elliptic (P-E) Keller-Segel system on a bounded domain $Ω\in \mathbb{R}^n$ with smooth boundary. First, the well-posedness of the corresponding linear system is established by using the smoothing estimates of the Neumann heat semigroup on $Ω$. Then, the existence of PAP-mild solution of linear system is done by proving a Massera-type principle. Next, we obtain the well-posedness of such solutions for semilinear system by using the results of linear system and fixed point arguments. The exponential stability is proven by using again the estimates of the Neumann heat semigroup. Finally, we discuss also such results for the case of the Keller-Segel system on the framework of real hyperbolic manifolds.

math.AP↗

arXiv2024

On certain classes of mild solutions of the scalar Liénard equation revisited

In this work we revisit the existence, uniqueness and exponential decay of some classes of mild solutions which are almost periodic (AP-), asymptotically almost periodic (AAP-) and pseudo almost periodic (PAP-) of the scalar Linéard equation by employing the notion of Green function and Massera-type principle. First, by changing variable we convert this equation to a system of first order differential equations. Then, we transform the problem into a framework of an abstract parabolic evolution equation which associates with an evolution family equipped an exponential dichtonomy and the corresponding Green function is exponentially almost periodic. After that, we prove a Massera-type principle that the corresponding linear equation has a uniqueness AP-, AAP- and PAP- mild solution if the right hand side and the coefficient functions are AP-, AAP- and PAP- functions, respectively. The well-posedness of semilinear equation is proved by using fixed point arguments and the exponential decay of mild solutions are obtained by using Gronwall's inequality. Although our work revisits some previous works on well-posedness of the Linéard equation on these type solutions but provide a difference view by using Green function and go farther on the aspects of asymptotic behaviour of solutions and the construction of abstract theory. Our abstract results can be also applied to other parabolic evolution equations.

math.AP↗

arXiv2024

Well-posedness and stability for a class of solutions of semi-linear diffusion equations with rough coefficients

In this work we study the existence, uniqueness and polynomial stability of the pseudo almost periodic mild solutions of semi-linear diffusion equations with rough coefficients in certain interpolation spaces. First, we rewirte the equations in abstract parabolic equation. Then, we use the polynomial stability of the semigroups of the corresponding linear equations to prove the boundedness of the solution operator for the linear equations in appropriate interpolation spaces. We show that this operator preserves the pseudo almost periodic property of functions. We will use the fixed point argument to obtain the existence and stability of the pseudo almost periodic mild solutions for the semi-linear equations. The abstract results will be applied to the semi-linear diffusion equations with rough coefficients to obtain our desired results.

math.AP↗

arXiv2024

Well-posedness and scattering for wave equations on hyperbolic spaces with singular data

We consider the wave and Klein-Gordon equations on the real hyperbolic space $\mathbb{H}^{n}$ ($n \geq2$) in a framework based on weak-$L^{p}$ spaces. First, we establish dispersive estimates on Lorentz spaces in the context of $\mathbb{H}^{n}$. Then, employing those estimates, we prove global well-posedness of solutions and an exponential asymptotic stability property. Moreover, we develop a scattering theory and construct wave operators in such singular framework.

math.AP↗

arXiv2024

On a nonhomogeneous heat equation on the complex plane

In this article, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of a parabolic evolution equations on complex plane, in which the diffusion operator has the form $\overline{\Box}_φ = \overline{D}\, \overline{D}^{\ast}$, where $\overline{D} f = \bar{\partial}f + φ_{\bar{z}} f$, the function $φ$ is smooth and subharmonic on $\mathbb{C}$, and $\overline{D}^{\ast}$ is the formal adjoint of $\overline{D}$. Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich \cite{raich06} and a fixed point argument.

math.AP↗

arXiv2024

Stability of solutions of stationary Boussinesq systems on weak-Morrey spaces

In this paper we establish the asymptotic stability of steady solutions for the Boussinesq systems in the framework of Cartesian product of critical weak-Morrey spaces on $\mathbb{R}^n$, where $n \geqslant 3$. In our strategy, we first establish the continuity for the long time of the bilinear terms associated with the mild solutions of the Boussinesq systems, i.e., the bilinear estimates by using only the norm of the present spaces. As a direct consequence, we obtain the existence of global small mild solutions and asymptotic stability of steady solutions of the Boussinesq systems in the class of continuous functions from $[0, \infty)$ to the Cartesian product of critical weak-Morrey spaces. Our techniques consist interpolation of operators, duality, heat semigroup estimates , Holder and Young inequalities in block spaces (based on Lorentz spaces) that are preduals of Morrey-Lorentz spaces.

math.AP↗

arXiv2024

On Asymptotically Almost Periodic Solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic Manifolds

In this article we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of such solutions in the linear equation case by using the dispersive and smoothing estimates of the heat semigroup. Then we pass to the well-posedness of semi-linear equation case by using the results of linear equation and fixed point arguments. The exponential decay is proven by using Gronwall's inequality.

math.AP↗

arXiv2024

Periodic solutions for Boussinesq systems in weak-Morrey spaces

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension $n\geqslant3$. Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space $\mathbb{R}^{n}$. Using certain dispersive and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as Yamazaki-type estimate on block spaces, we prove the existence of bounded mild solutions for the linear {systems} corresponding to the Boussinesq systems. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear {systems} on the half line time-axis by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear {systems} to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.

math.AP↗

arXiv2024

Periodic Solutions of the parabolic-elliptic Keller-Segel system on whole spaces

In this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic-elliptic Keller-Segel system on whole spaces detailized by Euclidean space $\mathbb{R}^n\,\,(\hbox{ where }n \geqslant 4)$ and real hyperbolic space $\mathbb{H}^n\,\, (\hbox{where }n \geqslant 2)$. We work in framework of scritical spaces such as on weak-Lorentz space $L^{\frac{n}{2},\infty}(\mathbb{R}^n)$ to obtain the results for Keller-Segel system on $\mathbb{R}^n$ and on $L^{\frac{p}{2}}(\mathbb{H}^n)$ for $n<p<2n$ to obtain the ones on $\mathbb{H}^n$. Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviours of periodic mild solutions of Keller-Segel system obtained in $\mathbb{R}^n$ and the one in $\mathbb{H}^n$.

math.AP↗

arXiv2023

Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime

In this paper, we establish the constructions of conformal scattering theories for the tensorial wave equation such as the tensorial Fackerell-Ipser and the spin $\pm 1$ Teukolsky equations on Schwarzschild spacetime. In our strategy, we construct the conformal scattering for the tensorial Fackerell-Ipser equations which are obtained from the Maxwell equation and spin $\pm 1$ Teukolsky equations. Our method combines Penrose's conformal compactification and the energy decay results of the tensorial fields satisfying the tensorial Fackerell-Ipser equation to prove the energy equality of the fields through the conformal boundary $\mathfrak{H}^+\cup \scri^+$ (resp. $\mathfrak{H}^-\cup \scri^-$) and the initial Cauchy hypersurface $Σ_0 = \left\{ t=0 \right\}$. We will prove the well-posedness of the Goursat problem by using a generalization of Hörmander's results for the tensorial wave equations. By using the results for the tensorial Fackerell-Ipser equations we will establish the construction of conformal scattering for the spin $\pm 1$ Teukolsky equations.

gr-qc↗