Communications in Mathematical Analysis and Applications
Communications in Mathematical Analysis and Applications (CMAA) publishes research papers relevant to mathematical analysis and its applications. CMAA publishes original research articles of high quality on a topic in Real and harmonic analysis, Complex analysis, Numerical analysis, Partial differential equations, Dynamical systems, Control and optimization, Probability, Mathematical biology, Statistics etc.
Reflected BSDEs Driven by RCLL Martingales with Stochastic Lipschitz Coefficient in a General Filtration: Analysis and Applications
In this paper, we investigate reflected backward stochastic differential equations with a single, discontinuous barrier, driven by a right-continuous, left-limited martingale within a general filtration. We establish the existence and uniqueness of solutions under a stochastic Lipschitz condition on the generator and a reflection process that is right-continuous with left limits. As an application, we use these results to determine fair pricing for American contingent claim options in a financial market driven by Azéma’s martingale, incorporating elements of asymmetric information.
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Classical Solutions to a Model for Heat Generation During Acoustic Wave Propagation in a Standard Linear Solid
In an open bounded real interval, this manuscript studies the evolution system \begin{cases} u_{ttt}+\alpha u_{tt} = \big(\gamma(\Theta)u_{xt}\big)_x + \big(\hat{\gamma}(\Theta)u_x\big)_x, \\ \Theta_t = D\Theta_{xx} + \Gamma(\Theta)u_{xt}^2, \end{cases} which arises as a model for the generation of heat during propagation of acoustic waves in a standard linear solid. A statement in local existence and uniqueness of classical solutions is derived for arbitrary $D>0$ and $\alpha\ge0$, for sufficiently smooth $\gamma,\hat{\gamma}$ and $\Gamma$ with $\gamma>0,\hat{\gamma}>0$ and $\Gamma\ge0$ on $[0,\infty)$, and for all suitably regular initial data of arbitrary size.
Copyright (c) 2025 Communications in Mathematical Analysis and Applications
$L^2$-Contraction and Asymptotic Stability of Large Shock for Scalar Viscous Conservation Laws
We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the flux $f(u) = u^p (2 ≤ p ≤ 4)$ in the regime of its strict convexity, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using a-contraction method with time-dependent shift and suitable weight function, which answers a question in [Blochas and Cheng, arXiv2501.01537, 2025]. Additionally, if the initial perturbation belongs to $L^1$ , then $L^2$ time-asymptotic decay rate $t^{−1/4}$ can be obtained.
Normalized Ground State Solution to a Mixed Schrödinger System in a Plane
In this paper, we establish the existence of positive ground state solutions for a class of mixed Schrödinger systems with concave-convex nonlinearities in $\mathbb{R}^2$, that is \begin{cases} -\partial_{xx}u + (-\Delta)_y^s u + \lambda_1 u = \mu_1 u^{p-1} + \beta r_1 u^{r_1-1}v^{r_2}, \\ -\partial_{xx}v + (-\Delta)_y^s v + \lambda_2 v = \mu_2 v^{q-1} + \beta r_2 u^{r_1}v^{r_2-1} \end{cases} subject to the $L^2$-norm constraints \begin{equation} \int_{\mathbb{R}^{2}} u^{2}dxdy = a, \qquad \int_{\mathbb{R}^{2}} v^{2}dxdy = b, \end{equation} where $(x,y)\in\mathbb{R}^2$, $u,v\geq0$, $s\in(1/2,1)$, $\mu_1,\mu_2,\beta>0$, $r_1,r_2>1$, the prescribed masses $a,b>0$, and the parameters $\lambda_1,\lambda_2$ appear as Lagrange multipliers. Moreover, the exponents $p,q,r_1+r_2$ satisfy $2(1+3s)/(1+s) < p,q,r_1+r_2 < 2_s$, where $2_s = 2(1+s)/(1-s)$. To obtain our main existence results, we employ variational techniques such as the mountain pass theorem, the Pohozaev manifold, Steiner rearrangement, and others, consolidating the works [Jeanjean et al., Nonlinear Differ. Equ. Appl. 31 (2024)].
Corrigendum to: "On the Eigenvalue Problem for a Bulk/Surface Elliptic System"
This note corrects a mistake in the paper ”On the eigenvalue problem for a bulk/surface elliptic system” by the author, appeared in [Vitillaro, Commun. Math. Anal. Appl. 4 (2025)].
Exponential Ergodicity for Time-Periodic McKean-Vlasov SDEs
As extensions to the corresponding results derived for time homogeneous McKean-Vlasov SDEs, the exponential ergodicity is proved for time-periodic distribution dependent SDEs in three different situations: 1) in the quadratic Wasserstein distance and relative entropy for the dissipative case; 2) in the Wasserstein distance induced by a cost function for the partially dissipative case; and 3) in the weighted Wasserstein distance induced by a cost function and a Lyapunov function for the fully non-dissipative case. The main results are illustrated by time inhomogeneous granular media equations, and are extended to reflecting McKean-Vlasov SDEs in a convex domain.
Recent Progress on Stochastic Fractional Diffusion Equations with Space-Time White Noise
We review the recent development on stochastic fractional diffusion equations with space-time white noise. We explicate the Fox $H$-function, fundamental solutions, mild Skorohod solution and its properties including existence and uniqueness, intermittent property and Hölder continuity.
Stochastic partial differential equation
Strong Convergence of Functional Stochastic Differential Equations via the Functional Itô Calculus
In general, Malliavin calculus is exploited to construct the Milstein scheme for functional stochastic differential equations due to the appearance of anticipative stochastic integrals. In this paper, by passing the Malliavin calculus, Milstein-type schemes are constructed via the functional Itô calculus for a range of functional stochastic differential equations. As regards the Milstein schemes designed, the strong convergence with the rate 1 is established for functional stochastic differential equations, where the delay measure need not be absolutely continuous with respect to the Lebesgue measure.
functional stochastic differential equation
A Stochastic Maximum Principle for Relaxed Control with General Risk Measure and Its Application in Finance
This paper deals with optimal relaxed control problem where the cost functional is a general risk measure instead of an expectation. We develop a stochastic maximum principle for this kind of optimal control problems using a variational method. Then, under the expectation optimization objective, a dynamic programming principle is studied and its connections to the adjoint process is shown. At last, the result is applied to two examples. One is a linear quadratic problem and the other is an optimal investment problem.
Long Time Behavior of the Stochastic 2D Navier-Stokes Equations
We review some basic results on existence and uniqueness of the invariant measure for the two-dimensional stochastic Navier-Stokes equations. A large part of the literature concerns the additive noise case; after revising these models, we consider our recent result [Ferrario and Zanella, Discrete Contin. Dyn. Syst. 44(1), 2024] with a multiplicative noise.
Two dimensional stochastic Navier-Stokes equations
Smooth Densities of Stochastic Differential Equations Forced by Degenerate Stable Type Noises
Malliavin calculus for stochastic differential equations (SDEs) forced by degenerate stable like noises has been intensively studied recently, see e.g. [Hao, Peng and Zhang, J. Theoret. Probab. 34 (2021); Zhang, Ann. Probab. 42(5) (2012); Zhang, Ann. Probab. 45 (2017)]. In this paper, we derive a simple inequality as a replacement of Norris type lemma and use it to show that two families of degenerate SDEs with stable like noises admit smooth density functions.
Asymptotics in Wasserstein Distance for Empirical Measures of Markov Processes
In this paper we introduce some recent progresses on the convergence rate in Wasserstein distance for empirical measures of Markov processes. For diffusion processes on compact manifolds possibly with reflecting or killing boundary conditions, the sharp convergence rate as well as renormalization limits are presented in terms of the dimension of the manifold and the spectrum of the generator. For general ergodic Markov processes, explicit estimates are presented for the convergence rate by using a nice reference diffusion process, which are illustrated by some typical examples. Finally, some techniques are introduced to estimate the Wasserstein distance between empirical measures.
Preface：Special Issue on Recent Progresses in Stochastic Differential Equations and Stochastic Partial Differential Equations
Crossing Thresholds in High Contrast Boundary Homogenization Problems: A Review
In this paper we provide a review of the state of the art in two topics of the homogenization theory: boundary homogenization for rapidly alternating boundary conditions on grill-type walls and homogenization of linear or nonlinear Robin boundary conditions in perforate media. High contrasts on the Robin boundary conditions are introduced. These contrasts are represented by new parameters and, along with the period and the sizes of “grill-crossing points”/perforations, play an important role in determining “thresholds” marking changes in the homogenized boundary conditions, either with “averaged terms” or “strange terms” or, somehow, “extreme” asymptotic behaviors for the solutions. Taking into account the advances of the last years in the topics, we also overview vector problems (cf. Winkler-Robin boundary conditions) and models in perforated media of interest in material sciences among others.
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Variational Functionals for the Characterization of BV and Sobolev Spaces
This paper examines the characterization of bounded variation (BV) and Sobolev functions by using some non-local functionals. We analyze their pointwise convergence and establish a connection with the Sobolev norm and the total variation measure. By investigating a wider class of non local functionals, we provide a deeper understanding of how these approximations capture the local properties of BV and Sobolev spaces, thereby reinforcing their applicability in the Euclidean setting.
Large Time Behavior of Solutions to Burgers Equation with a Time Delay
In this article, we consider the large time behavior of solutions $ρ$ to the one-dimensional Cauchy problem of the generalized viscous Burgers equations with a time delay. More precisely, we derive the time decay estimate of the solution to the Cauchy problem, provided by $ρ_0(0)∈L^1$ and $ρ_0∈C([−τ,0];H^1 ).$ The result is based on the combination of the weighted $L^2$ energy estimate and the $L^1$ estimate.
Multi-Level Structured Deformations: Relaxation via a Global Method Approach
We present some relaxation and integral representation results for energy functionals in the setting of structured deformations, with special emphasis given to the case of multi-level structured deformations. In particular, we present an integral representation result for an abstract class of variational functionals in this framework via a global method for relaxation and identify, under quite general assumptions, the corresponding relaxed energy densities through the study of a related local Dirichlet-type problem. Some applications to specific relaxation problems are also mentioned, showing that our global method approach recovers some previously established results.
hierarchical system of structured deformations
Homogenization of an Elliptic System Involving Non-Local and Equi-Valued Interface Conditions
In this paper, we analyze the effective behaviour of the solution of an elliptic problem in a two-phase composite material with non-standard imperfect contact conditions between its constituents. More specifically, we consider on the interface an equi-valued surface condition and a non-local flux condition involving a scaling parameter $α.$ We perform a homogenization procedure by using the periodic unfolding technique. As a result, we obtain two different effective models, depending on the scaling parameter $α.$ More precisely, in the case $α > −1,$ we are led to a standard Dirichlet problem for an elliptic equation, while in the case $α = −1,$ we get a bidomain system, consisting in the coupling of an elliptic equation with an algebraic one.
Uncertainty and Strichartz Type Inequalities for Fractional Hajłasz-Sobolev Spaces
In this paper we obtain uncertainty and Strichartz type inequalities for fractional Hajłasz-Sobolev spaces in the metric setting.
On the Eigenvalue Problem for a Bulk/Surface Elliptic System
The paper addresses the doubly elliptic eigenvalue problem where $Ω$ is a bounded open subset of $\mathbb{R}^N$ $(N ≥ 2)$ with a $C^1$ boundary $Γ = Γ_0∪Γ_1,$ $Γ_0∩Γ_1 = ∅,$ $Γ_1$ being nonempty and relatively open on $Γ.$ Moreover, $\mathcal{H}^{N-1}(\overline{\Gamma}_0 \cap \overline{\Gamma}_1)=0$ and $\mathcal{H}^{N−1} (Γ_0) > 0.$ We prove that $L^2 (Ω)×L^2 (Γ_1)$ admits a Hilbert basis constituted by eigenfunctions and we describe the behavior of the eigenvalues. Moreover, when $Γ$ is at least $C^2$ and $\overline{Γ}_0∩\overline{Γ}_1=∅,$ we give several qualitative properties of the eigenfunctions.
Preface: Proceedings of the International Conference on Elliptic and Parabolic Problems, Gaeta, Italy, 20-24 May 2024
Decay Structure of the Zener-Type Viscoelastic Plate with Type II Heat Conduction in the Whole Space
In this paper, we study the Zener-type viscoelastic plate in that heat conduction is determined by the type II Green-Naghdi theory. This model considers the finite propagation of thermal waves, which is compatible with the causality principle. We consider the Cauchy problem of this model and derive the decay estimates of solutions. Furthermore, the optimality of the decay estimate is discussed.
Suppression of Chemotactic Singularity via Poiseuille Flow in a Self-Consistent Patlak-Keller-Segel-Navier-Stokes System
In this paper, we investigate a fully parabolic Patlak-Keller-Segel-Navier-Stokes system with a self-consistent mechanism near the Poiseuille flow $A(y^2,0)$ in $\mathbb{T}×\mathbb{R},$ which is more natural than the Couette flow from a biomathematical perspective. We demonstrate that the solution to this system maintains global regularity, provided the amplitude $A$ is suitably large and the non-zero modes of the initial chemical density and vorticity are suitably small. To avoid the complex study of the spectral properties of the linear operator and its resolvent, we prove our result using a straightforward weighted energy method combined with a bootstrap argument.
fully parabolic Patlak-Keller-Segel-Navier-Stokes system.
Compact Embeddings and Pitt’s Property for Weighted Sequence Spaces of Sobolev Type
In this article we introduce a new class of weighted sequence spaces of Sobolev type and prove several compact embedding theorems for them. It is our contention that the chosen class is general enough so as to allow applications in various areas of mathematics and mathematical physics. In particular, our results constitute a generalization of those compact embeddings recently obtained in relation to the spectral analysis of a class of master equations arising in non-equilibrium statistical mechanics. As a byproduct of our considerations, we also prove a theorem of Pitt’s type asserting that under some conditions every linear bounded operator acting between such weighted spaces is compact.
Localization and Multiplicity for Stationary Stokes Systems with Variable Viscosity
In this paper we discuss the localization and the multiplicity of solutions for the stationary Stokes system with variable viscosity and a reaction force term. The results obtained apply to systems with strongly oscillating periodic viscosity and the corresponding homogenized systems.
localization and multiplicity of solutions
A Note on the Global Existence in a Fully Parabolic Patlak-Keller-Segel-Navier-Stokes System
This paper is concerned with a fully parabolic Patlak-Keller-Segel-Navier-Stokes system where $Ω ⊂\mathbb{R}^2$ is a smoothly bounded domain and the parameter $χ$ is positive. The main aim of this note is to show that if then the solution of the above system is global and bounded in time.
Local Well-Posedness and Weak-Strong Uniqueness to the Incompressible Vlasov-MHD System
In this paper, we investigate the local well-posedness of strong solutions and weak-strong uniqueness property to the incompressible Vlasov-magnetohydrodynamic (Vlasov-MHD) model in $\mathbb{R}^3_x.$ This model consists of a Vlasov equation and the incompressible MHD equations, which interact via the so-called Lorentz force. We first establish the local well-posedness of a strong solution $(f ,u,B)$ by utilizing the delicate energy method for the iteration sequence of approximate solutions, provided that the initial data $(f_0,u_0,B_0)$ are $H^2$-regular and $f_0(x,v)$ has a compact support in the velocity $v.$ We further demonstrate the weak-strong uniqueness property of solutions if $f_0(x,v)∈L^1∩L^∞(\mathbb{R}^3_x×\mathbb{R}^3_v),$ and thereby establish a rigorous connection between the strong and weak solutions to the Vlasov-MHD system. The absence of a dissipation structure in the Vlasov equation and the presence of the strong trilinear coupling term $((u−v)×B)f$ in the model pose significant challenges in deriving our results. To address these issues, we employ the method of characteristics to estimate the size of the support of $f,$ which enables us to overcome the difficulties associated with evaluating the integral $\int_{\mathbb{R}^3} ((u−v)×B)f {\rm d}v.$
A Constructive Proof of Korn’s Scaled Inequalities for Shells
One of Korn’s scaled inequalities for shells asserts that the $H^1$-norm of a displacement field of a shell with thickness $2ε$ clamped on a portion of its lateral boundary, once scaled to a domain independent of $ε,$ is bounded above by the $L^2$-norm of the corresponding scaled infinitesimal strain tensor field multiplied by a constant of order $ε^{−1}.$ We give a constructive proof to this inequality, and to other two inequalities of this type, which is thus different from the original proof of Ciarlet et al. [Arch. Rational Mech. Anal. 136 (1996), 163–190].
The Biot Stress-Right Stretch Relation for the Compressible Neo-Hooke-Ciarlet-Geymonat Model and Rivlin’s Cube Problem
The aim of the paper is to recall the importance of the study of invertibility and monotonicity of stress-strain relations for investigating the nonuniqueness and bifurcation of homogeneous solutions of the equilibrium problem of a hyperelastic cube subjected to equiaxial tensile forces. In other words, we reconsider a remarkable possibility in this nonlinear scenario: Does symmetric loading lead only to symmetric deformations or also to asymmetric deformations? If so, what can we say about monotonicity for these homogeneous solutions, a property which is less restrictive than the energetic stability criteria of homogeneous solutions for Rivlin’s cube problem. For the Neo-Hooke type materials we establish what properties the volumetric function $h$ depending on ${\rm det}F$ must have to ensure the existence of a unique radial solution (i.e. the cube must continue to remain a cube) for any magnitude of radial stress acting on the cube. The function $h$ proposed by Ciarlet and Geymonat satisfies these conditions. However, discontinuous equilibrium trajectories may occur, characterized by abruptly appearing non-symmetric deformations with increasing load, and a cube can instantaneously become a parallelepiped. Up to the load value for which the bifurcation in the radial solution is realized local monotonicity holds true. However, after exceeding this value, monotonicity no longer occurs on homogeneous deformations which, in turn, preserve the cube shape.
Preface: Asymptotic Analysis and Elasticity Theory: A Collection of Papers Dedicated to Professor Philippe G. Ciarlet on the Occasion of His 85th Birthday