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Nguyen Huy Tuan

Publications and source records attributed to Nguyen Huy Tuan.

At least 19 recordsLinked to original sources
arXiv2022

Long-time asymptotics of solutions and the modified pseudo-conformal conservation law for super-critical nonlinear Schrödinger equation

In this paper, we discuss a class of nonlinear Schrödinger equations with the power-type nonlinearity: $(\mathrm{i} \frac{\partial}{\partial t} + Δ) ψ= λ|ψ|^{2η}ψ$ in $\mathbf R^N \times \mathbf R^+$. Based on the Gagliardo-Nirenberg interpolation inequality, we prove the local existence and long-time behavior (continuation, finite-time blow-up or global existence, continuous dependence) of the solutions to the $(H^q)$ super-critical Schrödinger equation. The corresponding scaling invariant space is homogeneous Sobolev $\dot H^{q_{crit}}$ with $q_{crit} > q$. Based on the estimates of the quadratic terms containing the phase derivatives used in the paper by Killip, Murphy and Visan \cite[SIAM J. Math. Anal. 50(3) (2018), 2681--2739]{KMV018} we shall study the stability with a stronger bound on the solutions to our problem. Moreover, from the arguments on virial-types presented in the paper by Killip and Visan \cite[Amer. J. Math. 132(2) (2010), 361--424]{KV010}, a modified pseudo-conformal conservation law is proposed. The Morawetz estimate for the solutions to the problem are also presented.

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arXiv2022

Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: Theoretical results and applications

The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. A general theoretical result on this topic is established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic lattice reaction-diffusion equation driven by a time-dependent nonlinear noise. A time-average method and a mild condition on the time-dependent diffusion function are used to prove that the limit of every evolution system of probability measures must be an evolution system of probability measures of the limiting equation. The theoretical results are expected to be applied to various stochastic lattice systems/ODEs/PDEs in the future.

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arXiv2021

On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative

In this work, we investigate the IVP for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn-Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $ Ξ(z)=e^{|z|^p}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.

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arXiv2021

Inverse initial problem for fractional reaction-diffusion equation with nonlinearities

The initial inverse problem of finding solutions and their initial values ($t = 0$) appearing in a general class of fractional reaction-diffusion equations from the knowledge of solutions at the final time ($t = T$). Our work focuses on the existence and regularity of mild solutions in two cases: \begin{itemize} \item[--] The first case: The nonlinearity is globally Lipschitz and uniformly bounded which plays important roles in PDE theories, and especially in numerical analysis. \item[--] The second case: The nonlinearity is locally critical which widely arises from the Navier-Stokes, Schrödinger, Burgers, Allen-Cahn, Ginzburg-Landau equations, etc. \end{itemize} Our solutions are local-in-time and are derived via fixed point arguments in suitable functional spaces. The key idea is to combine the theories of Mittag-Leffler functions and fractional Sobolev embeddings. To firm the effectiveness of our methods, we finally apply our main results to time fractional Navier-Stokes and Allen-Cahn equations.

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arXiv2020

An improved quasi-reversibility method for a terminal-boundary value multi-species model with white Gaussian noise

Upon the recent development of the quasi-reversibility method for terminal value parabolic problems in \cite{Nguyen2019}, it is imperative to investigate the convergence analysis of this regularization method in the stochastic setting. In this paper, we positively unravel this open question by focusing on a coupled system of Dirichlet reaction-diffusion equations with additive white Gaussian noise on the terminal data. In this regard, the approximate problem is designed by adding the so-called perturbing operator to the original problem and by exploiting the Fourier reconstructed terminal data. By this way, Gevrey-type source conditions are included, while we successfully maintain the logarithmic stability estimate of the corresponding stabilized operator, which is necessary for the error analysis. As the main theme of this work, we prove the error bounds for the concentrations and for the concentration gradients, driven by a large amount of weighted energy-like controls involving the expectation operator. Compared to the classical error bounds in $L^2$ and $H^1$ that we obtained in the previous studies, our analysis here needs a higher smoothness of the true terminal data to ensure their reconstructions from the stochastic fashion. Two numerical examples are provided to corroborate the theoretical results.

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arXiv2020

Backward problem for time fractional reaction-diffusion equation with nonlinear source and discrete data

In this paper, we study the problem of finding the solution of a multi-dimensional time fractional reactiondiffusion equation with nonlinear source from the final value data. We prove that the present problem is not well-posed. Then regularized problems are constructed using the truncated expansion method (in the case of two-dimensional) and the quasi-boundary value method (in the case of multi-dimensional). Finally, convergence rates of the regularized solutions are given and investigated numerically.

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arXiv2019

Existence and regularity results for terminal value problem for nonlinear fractional wave equations

We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given final value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of first and fractional orders are also derived. The effectiveness of our methods are showed by applying the results to two interesting models: Time fractional Ginzburg-Landau equation, and Time fractional Burgers equation, where time and spatial regularity estimates are obtained.

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arXiv2019

Regularization of sideways problem for a time fractional diffusion equation with nonlinear source

In this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e. the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak {\color{black} a} priori assumptions on the sought solution, we propose a new regularization method for stabili{\color{black}z}ing the ill-posed problem. We also provide a numerical example to illustrate our results.

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arXiv2019

On existence and regularity of a terminal value problem for the time fractional diffusion equation

In this paper we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain $D$ of $ \mathbb{R}^{k}$, $k\ge 1$, which includes the fractional power $\mathcal L^β$, $0<β\le 1$, of a symmetric uniformly elliptic operator $\mathcal L$ defined on $L^2(D)$. A representation of solutions is given by using the Laplace transform and the spectrum of $\mathcal L^β$. We establish some existence and regularity results for our problem in both the linear and nonlinear case.

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arXiv2018

Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements

This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value parabolic problem with noisy data. Based on data measurements, we perturb the problem by the so-called filter regularized operator to design an approximate problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semi-linear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there is likely a myriad of models derived from this simpler case, such as source localization problems for brain tumors and heat conduction problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the Faedo-Galerkin method to study the weak solvability of the approximate problem. Relying on the energy-like analysis, we provide detailed convergence rates in $L^2$-$H^1$ of the proposed method when the true solution is sufficiently smooth. Depending on the dimensions of the domain, we obtain an error estimate in $L^{r}$ for some $r>2$. Proof of the backward uniqueness for the quasi-linear system is also depicted in this work. To prove the regularity assumptions acceptable, several physical applications are discussed.

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arXiv2018

Approximation of mild solutions of a semilinear fractional elliptic equation with random noise

We study for the first time the Cauchy problem for semilinear fractional elliptic equation. This paper is concerned with the Gaussian white noise model for the initial Cauchy data. We establish the ill-posedness of the problem. Then, under some assumption on the exact solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Some convergence rates between the exact solution and the regularized solution is established in $L^2$ and $H^q$ norms.

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arXiv2018

Application of the cut-off projection to solve a backward heat conduction problem in a two-slab composite system

The main goal of this paper is applying the cut-off projection for solving one-dimensional backward heat conduction problem in a two-slab system with a perfect contact. In a constructive manner, we commence by demonstrating the Fourier-based solution that contains the drastic growth due to the high-frequency nature of the Fourier series. Such instability leads to the need of studying the projection method where the cut-off approach is derived consistently. In the theoretical framework, the first two objectives are to construct the regularized problem and prove its stability for each noise level. Our second interest is estimating the error in $L^2$-norm. Another supplementary objective is computing the eigen-elements. All in all, this paper can be considered as a preliminary attempt to solve the heating/cooling of a two-slab composite system backward in time. Several numerical tests are provided to corroborate the qualitative analysis.

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arXiv2017

The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing

Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method for the coupled elliptic sine-Gordon equations along with Cauchy data. The system of equations originates from the static case of the coupled hyperbolic sine-Gordon equations modeling the coupled Josephson junctions in superconductivity, and so far it addresses the Josephson $π$-junctions. In general, the Cauchy problem is not well-posed, and herein the Hadamard-instability occurs drastically. Generalizing the kernel-based regularization method, we propose a stable approximate solution. Confirmed by the error estimate, this solution strongly converges to the exact solution in $L^2$-norm. The main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation. We employ the proposed techniques that successfully approximated the highly oscillatory integral, and apply the Picard-like iteration to organize an efficient and reliable tool of computations. The results are viewed as the improvement as well as the generalization of many previous works. The paper is also accompanied by a numerical example that demonstrates the potential of this idea.

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arXiv2017

A random regularized approximate solution of the inverse problem for the Burgers' equation

In this paper, we find a regularized approximate solution for an inverse problem for the Burgers' equation. The solution of the inverse problem for the Burgers' equation is ill-posed, i.e., the solution does not depend continuously on the data. The approximate solution is the solution of a regularized equation with randomly perturbed coefficients and randomly perturbed final value and source functions. To find the regularized solution, we use the modified quasi-reversibility method associated with the truncated expansion method with nonparametric regression. We also investigate the convergence rate.

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arXiv2017

On a backward problem for multidimensional Ginzburg-Landau equation with random data

In this paper, we consider a backward in time problem for Ginzburg-Landau equation in multidimensional domain associated with some random data. The problem is ill-posed in the sense of Hadamard. To regularize the instable solution, we develop a new regularized method combined with statistical approach to solve this problem. We prove a upper bound, on the rate of convergence of the mean integrated squared error in $L^2 $ norm and $H^1$ norm.

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arXiv2017

Regularized solutions for some backward nonlinear parabolic equations with statistical data

In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include heat equation, extended Fisher-Kolmogorov equation, Swift-Hohenberg equation and many others. The equations with time dependent coefficients include Fisher type Logistic equations, Huxley equation, Fitzhugh-Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including 1-D Kuramoto-Sivashinsky equation, 1-D modified Swift-Hohenberg equation, strongly damped wave equation and 1-D Burger's equation with randomly perturbed operator.

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arXiv2016

Approximate solutions of inverse problems for nonlinear space fractional diffusion equations with randomly perturbed data

This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving the problem. In the case of constant coefficients, we use the truncation methods. In the case of perturbed time dependent coefficients, we apply a new quasi-reversibility method. We also show the convergence rate between the regularized solution and the sought solution under some a priori assumption on the sought solution.

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