More Like this

1. Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions

   https://doi.org/10.58997/ejde.2025.43  

   Bandyopadhyay, Shalmali; Lewis, Thomas; Mavinga, Nsoki (January 2025, Electronic Journal of Differential Equations)

   We establish the existence of maximal and minimal weak solutions   between ordered pairs of weak sub- and super-solutions for a coupled  system of elliptic equations with quasimonotone nonlinearities on the  boundary. We also formulate a finite difference method to approximate the  solutions and establish the existence of maximal and minimal approximations  between ordered pairs of discrete sub- and super-solutions.   Monotone iterations are formulated for constructing the maximal and minimal  solutions when the nonlinearity is monotone.  Numerical simulations are used to explore existence, nonexistence,  uniqueness and non-uniqueness properties of positive solutions.  When the nonlinearities do not satisfy the monotonicity condition, we prove the existence of weak maximal and minimal solutions using Zorn’s  lemma and a version of Kato’s inequality up to the boundary.  For more information see https://ejde.math.txstate.edu/Volumes/2025/43/abstr.html 

   more » « less
2. Solution Existence and Uniqueness for Degenerate SDEs with Application to Schrödinger-Equation Representations

   Dower, P.M.; Kaise, H.; McEneaney, W.M.; Wang, T; Zhao, R. (April 2021, Comms. communications in Information and Systems.)

   null (Ed.)

   Existence and uniqueness results for solutions of stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coeffcient is nondegenerate. Here, existence and uniqueness of strong solutions is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solutions of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coeffcient is exactly half that of the state. In addition to this degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential. The first type consists of discontinuities that may occur on a possibly high-dimensional manifold. The second consists of singularities that may occur on a smoothly parameterized curve. 

   more » « less
3. Weak solutions for a poro-elastic plate system

   https://doi.org/10.1080/00036811.2021.1953483  

   Gurvich, Elena; Webster, Justin T. (July 2021, Applicable Analysis)

   null (Ed.)

   We consider a recent plate model obtained as a scaled limit of the three-dimensional Biot system of poro-elasticity. The result is a ‘2.5’-dimensional linear system that couples traditional Euler–Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We allow the permeability function to be time dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so-called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions for implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability function. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate and exposits a variety of interesting related models and associated analytical investigations. 

   more » « less
4. Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations

   McEneaney, William M; Kaise, Hidehiro; Dower, Peter M; Zhao, Ruobing. (July 2020, 21st IFAC World Congress)

   Existence and uniqueness results for stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coefficient is nondegenerate. Here, existence and uniqueness of a strong solution is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solution of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coefficient is exactly half that of the state. In addition to the degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential and the resulting solutions to the dequantized Schrödinger equation. The first type consists of discontinuities that may occur on a possibly high-dimensional manifold (up to codimension one). The second consists of singularities that may occur on a lower-dimensional manifold (up to codimension two). 

   more » « less
5. The Littlewood–Paley decomposition for periodic functions and applications to the Boussinesq equations

   https://doi.org/10.1142/s0219530519500234  

   Dai, Yichen; Hu, Weiwei; Wu, Jiahong; Xiao, Bei (July 2020, Analysis and Applications)

   The Littlewood–Paley decomposition for functions defined on the whole space [Formula: see text] and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with [Formula: see text] as the spatial domain. This paper intends to develop parallel tools for the periodic domain [Formula: see text]. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood–Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood–Paley projections defined via the circular cutoff in [Formula: see text] with [Formula: see text] in the literature do not behave well on the Lebesgue space [Formula: see text] except for [Formula: see text]. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on [Formula: see text]. As an application of the tools developed here, we study the periodic weak solutions of the [Formula: see text]-dimensional Boussinesq equations with the fractional dissipation [Formula: see text] and without thermal diffusion. We obtain two main results. The first assesses the global existence of [Formula: see text]-weak solutions for any [Formula: see text] and the existence and uniqueness of the [Formula: see text]-weak solutions when [Formula: see text] for [Formula: see text]. The second establishes the zero thermal diffusion limit with an explicit convergence rate. 

   more » « less