Issue 1, Volume 48, March 2023

2023

1. Mean Oscillation Gradient Estimates for Elliptic Systems in Divergence Form with VMO Coefficients Luc Nguyen PDF

   ▸ Abstract

   We consider gradient estimates for $H^1$ solutions of linear elliptic systems in divergence form $\partial _{\alpha }(A_{ij}^{\alpha \beta } \partial _{\beta } u^{j}) = 0$. It is known that the Dini continuity of coefficient matrix $A = (A_{ij}^{\alpha \beta })$ is essential for the differentiability of solutions. We prove the following results:

   (a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the $L^2$ mean oscillation $\omega_{A,a}$ of A satisfies

   $$ X_{A,2} := \limsup\limits_{r\rightarrow 0} r {{\int \limits }_{r}^{2}} \frac {\omega _{A,2}(t)}{t^{2}} \exp \left (C_{*} {{\int \limits }_{t}^{R}} \frac {\omega _{A,2}(s)}{s} ds\right ) dt < \infty , $$

   where $C_*$ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.

   (b) If $X_{A,2}$ = 0, then ∇u ∈ V MO.

   (c) Finally, examples satisfying $X_{A,2}$ = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when $\nabla u \in L^{\infty } \cap VMO$.

   117-132
2. Hardy-Sobolev Inequalities with Dunkl Weights Dao Nguyen Anh , Nguyen Tuan Duy , Lam Hoang Nguyen , Nguyen Van Phong PDF

   ▸ Abstract

   We establish several identities and inequalities of Hardy type and Sobolev type with Dunkl weights. We also investigate the sharp constants and optimal functions of the Hardy-Sobolev inequality with Dunkl weights.

   133-149
3. General Decay and Blow-up Results of a Robin-Dirichlet Problem for a Pseudoparabolic Nonlinear Equation of Kirchhoff-Carrier Type with Viscoelastic Term Le Thi Phuong Ngoc , Nguyen Anh Triet , Phan Thi My Duyen , Nguyen Thanh Long PDF

   ▸ Abstract

   In this paper, we investigate the Robin-Dirichlet problem for a nonlinear pseudoparabolic equation of Kirchhoff-Carrier type with viscoelastic term. Under suitable assumptions on the initial data and the relaxation function included in the viscoelastic term, we obtain sufficient conditions for the existence, uniqueness, blow-up, and decay of a weak solution. The results obtained here extend the ones in a previous paper of the authors (Ngoc et al., Math. Meth. Appl. Sci. 44(11), 8697–8725, 26).

   151-191
4. Spatiotemporal Soliton Interaction of Saturable Nonlinear Schrödinger Equations in Spatial Dimensions Higher Than 1 Quan M. Nguyen , Toan T. Huynh PDF

   ▸ Abstract

   We derive an expression for the collision-induced amplitude dynamics in a fast collision between two (N + 1) −dimensional spatiotemporal solitons in saturable nonlinear media with weak perturbations in a spatial dimension of N, where N ≥ 1. The perturbed spatiotemporal soliton evolution is under a framework of the coupled saturable (N + 1 + 1) −dimensional nonlinear Schrödinger equations in the presence of weakly nonlinear loss and delayed Raman response. The perturbation approach is based on an extended perturbation technique for analyzing the collision-induced dynamics of one-dimensional temporal solitons and two-dimensional solitons. The accuracy of our theoretical calculations is validated by numerical simulations of the interaction of two 3D spatiotemporal solitons, also known as two light bullets, of the coupled nonlinear Schrödinger equations in the presence of delayed Raman response and cubic loss.

   193-208
5. Gradient Flows of Modified Wasserstein Distances and Porous Medium Equations with Nonlocal Pressure Nhan-Phu Chung , Quoc-Hung Nguyen PDF

   ▸ Abstract

   We study families of porous medium equations with nonlocal pressure. We construct their weak solutions via JKO schemes for modified Wasserstein distances. We also establish the regularization effect and decay estimates for the $L^p$ norms.

   209-235
6. Continuity of the Solution to a Stochastic Time-fractional Diffusion Equations in the Spatial Domain with Locally Lipschitz Sources Dang Duc Trong , Nguyen Dang Minh , Nguyen Nhu Lan , Nguyen Thi Mong Ngoc PDF

   ▸ Abstract

   We study the nonlinear stochastic time-fractional diffusion equation in the spatial domain $\mathbb R$ driven by a locally Lipschitz source satisfying

   $$\begin{aligned} \left( {~}_{t}D_{0^{+}}^{\alpha } - \frac{\partial ^{2} }{\partial x^{2}}\right) u(t,x) = I_{t}^{\gamma }\left( F(t,x,u)\right) , \end{aligned}$$

   where $x\in \mathbb {R},\alpha \in (0,1],\gamma \ge 1-\alpha$, the source term is defined $F(t,x,u) = f(t,x,u(t,x))+ \rho (t,x,u(t,x))\dot{W}(t,x)$ and $W$ is the multiplicative space-time white noise. We investigate the existence, uniqueness of a maximal random field solution. Moreover, we prove the stability of the solution with respect to perturbed fractional orders $\alpha , \gamma$  and the initial condition.

   237-257
7. Preface Preface Ðinh Nho Hào , Nguyễn Hoài Minh , Phan Thành Nam , Nguyễn Trọng Toán PDF

   1
8. Existence and Multiplicity Results for Nonlocal Lane-Emden Systems Rakesh Arora , Phuoc-Tai Nguyen PDF

   ▸ Abstract

   In this work, we show the existence and multiplicity for the nonlocal Lane-Emden system of the form

   \begin{array}{@{}rcl@{}} \left\{ \begin{aligned} \mathbb L u &= v^{p} + \rho \nu \quad &&\text{in } {\varOmega}, \\ \mathbb L v &= u^{q} + \sigma \tau \quad &&\text{in } {\varOmega},\\ u&=v = 0 \quad &&\text{on } \partial {\varOmega} \text{ or in } {\varOmega}^{c} \text{ if applicable}, \end{aligned} \right. \end{array} where ${\varOmega } \subset \mathbb{R}^{N}$ is a $C^2$ bounded domain, $\mathbb L$ is a nonlocal operator, ν, τ are Radon measures on Ω, p, q are positive exponents, and ρ, σ > 0 are positive parameters. Based on a fine analysis of the interaction between the Green kernel associated with $\mathbb L$, the source terms $u^q$, $v^p$ and the measure data, we prove the existence of a positive minimal solution. Furthermore, by analyzing the geometry of Palais-Smale sequences in finite dimensional spaces given by the Galerkin type approximations and their appropriate uniform estimates, we establish the existence of a second positive solution, under a smallness condition on the positive parameters ρ, σ and superlinear growth conditions on source terms. The contribution of the paper lies on our unifying technique that is applicable to various types of local and nonlocal operators.

   3-28
9. Iterative Methods with Nonconforming Time Grids for Nonlinear Flow Problems in Porous Media Thi-Thao-Phuong Hoang , Iuliu Sorin Pop PDF

   ▸ Abstract

   Partially saturated flow in a porous medium is typically modeled by the Richards equation, which is nonlinear, parabolic and possibly degenerated. This paper presents domain decomposition-based numerical schemes for the Richards equation, in which different time steps can be used in different subdomains. Two global-in-time domain decomposition methods are derived in mixed formulations: the first method is based on the physical transmission conditions and the second method is based on equivalent Robin transmission conditions. For each method, we use substructuring techniques to rewrite the original problem as a nonlinear problem defined on the space-time interfaces between the subdomains. Such a space-time interface problem is linearized using Newton’s method and then solved iteratively by GMRES; each GMRES iteration involves parallel solution of time-dependent problems in the subdomains. Numerical experiments in two dimensions are carried out to verify and compare the convergence and accuracy of the proposed methods with local time stepping.

   29-49
10. Coercivity of the Dirichlet-to-Neumann Operator and Applications to the Muskat Problem Huy Q. Nguyen PDF

    ▸ Abstract

    We consider the Dirichlet-to-Neumann operator in strip-like and half-space domains with Lipschitz boundary. It is shown that the quadratic form generated by the Dirichlet-to-Neumann operator controls some sharp homogeneous fractional Sobolev norms. As an application, we prove that the global Lipschitz solutions constructed in Dong et al. (2021) for the one-phase Muskat problem decays exponentially in time in any Hölder norm $C^\alpha, \alpha \in (0,1)$.

    51-62
11. The Spherical Maximal Function on Choquet Spaces Nguyen Cong Phuc PDF

    ▸ Abstract

    Endpoint weak-type bounds and non-endpoint strong type bounds are obtained for the spherical maximal function in the setting of Choquet spaces with respect to certain Hausdorff contents or Sobolev capacities.

    63-74
12. A Proof of the Lieb–Thirring Inequality via the Besicovitch Covering Lemma Phan Thành Nam PDF

    ▸ Abstract

    We give a proof of the Lieb–Thirring inequality on the kinetic energy of orthonormal functions by using a microlocal technique, in which the uncertainty and exclusion principles are combined through the use of the Besicovitch covering lemma.

    75-81
13. A General Estimate for the $\bar \partial$-Neumann Problem Tran Vu Khanh PDF

    ▸ Abstract

    This paper especially focuses on a general estimate, called  $(f-\mathcal M)^k$, for the $\bar \partial$-Neumann problem

    $${(f-\mathcal M)^{k}} \qquad \| f({\varLambda })\mathcal M u\|^{2}\le c(\|\bar \partial u\|^{2}+\|\bar \partial ^{*}u\|^{2}+\|u\|^{2})+C_{\mathcal M}\|u\|^{2}_{-1} $$

    for any $u\in C^{\infty }_{c}(U\cap \bar {\Omega })^{k}\cap \text {Dom}(\bar {\partial }^{*}) $, where f(Λ) is the tangential pseudodifferential operator with symbol $f((1+|\xi|^2)^{\frac 1 2})$,  $\mathcal M$ is a multiplier, and U is a neighborhood of a given boundary point $z_0$. Here the domain Ω is q-pseudoconvex or q-pseudoconcave at $z_0$. We want to point out that under a suitable choice of f and $\mathcal M$, $(f{-}\mathcal M)^{k} $ is the subelliptic, superlogarithmic, compactness and so on. Generalizing the Property (P) by Catlin (1984), we define Property $(f-\mathcal M-P)^{k} $. The result we obtain in here is: Property $(f-\mathcal M-P)^{k} $ yields the $(f-\mathcal M)^{k} $ estimate. The paper also aims at exhibiting some relevant classes of domains which enjoy Property $(f-\mathcal M-P)^{k} $.

    83-115