Uniqueness Of Weak Solutions Research Articles

An electro-elastic contact problem with an internal state variable

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive foundation. General models for elastic materials with piezoelectirec effect, called electro-elastic materials. This paper is devoted to the study of the model involving a frictionless contact between an electro-elastic body characterized by long memory and a conductive adhesive foundation. The process is mechanically quasistatic and electrically static. We model the material with a general nonlinear electro-elastic constitutive law with internal state variable and the contact with normal compliance and unilateral constraint, where the adhesion is taken into account and a regularized electrical conductivity condition. The main feature of these models consists of the fact that their variational formulation is given by a history-dependent quasivariational inequality for the displacement field and a linear variational equation for the electric potential. A weak formulation for the model is given in the form of a coupled system for the displacement, the stress, the electric displacement, the electric potential, the bonding and the variable state internal fields. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove its unique weak solution. The proof is based on nonlinear evolution equations with monotone operators, differential equations and Banach fixed point arguments.

Complex pattern formation governed by a Cahn–Hilliard–Swift–Hohenberg system: Analysis and numerical simulations

This paper investigates a Cahn–Hilliard–Swift–Hohenberg system, focusing on a three-species chemical mixture subject to physical constraints on volume fractions. The resulting system leads to complex patterns involving a separation into phases as typical of the Cahn–Hilliard equation and small scale stripes and dots as seen in the Swift–Hohenberg equation. We introduce singular potentials of logarithmic type to enhance the model’s accuracy in adhering to essential physical constraints. The paper establishes the existence and uniqueness of weak solutions within this extended framework. The insights gained contribute to a deeper understanding of phase separation in complex systems, with potential applications in materials science and related fields. We introduce a stable finite element approximation based on an obstacle formulation. Subsequent numerical simulations demonstrate that the model allows for complex structures as seen in pigment patterns of animals and in porous polymeric materials.

An efficient and accurate mapping method for elliptic equations in irregular annular domains

In this paper, we introduce a coordinate transformation, which transforms the irregular annular domain to a unit disk. We present its basic properties. As examples, we consider Poisson type equation and Cauchy–Navier elastic equations with variable coefficients in two-dimensional irregular annular domains, and prove the existence and uniqueness of weak solutions. We also construct the mixed Fourier–Legendre spectral schemes, and derive the optimal convergence of numerical solutions under the H1-norm. The numerical results indicate that the suggested method achieves high-order accuracy.

Blow-up and global existence for a new class of parabolic p(x,⋅)-Kirchhoff equation involving double phase operator

In this paper, we are concerned with study solutions for double phase problems involving a Kirchhoff function, and nonlinearity with subcritical growth:{ut+M([u]p,q,as)Lp,q,asu=|u|r−2u,inU×[0,T),u(x,t)=0in∂U×[0,T),u(x,0)=u0(x),inU, where, Lp,q,ss is double phase operator, and M is a Kirchhoff function. Combining the Faedo-Galerking approximation with Gronwall's inequality, the existence of a unique weak solution is established. More interestingly, we study the global existence and finite time blow of solution when the initial energy is sub-critical and supercritical. Finally, the stabilization of the solution with positive initial energy is established based on Komornik's integral inequality.

Analysis of a dynamic contact problem with long memory

We examine a mathematical framework detailing a dynamic frictional contact involving thermo-viscoelastic materials characterized by long memory and damage. The contact is modeled by the normal compliance condition with Coulomb's friction law, while surface wear is also factored in. The model was in the form of a coupled system which includes an evolution equation written in inclusion form for the frictional contact, parabolic variational inequality for the damage field, nonlinear differential equation for the temperature field and an equation for the wear particles and an evolution. A variational formulation for the model was derived and the existence of the unique weak solution established, under smallness assumptions on a part of the problem data. The proof used various results from the theory of evolution inequalities and repeated fixed-point arguments.

Regularized Stein Variational Gradient Flow

AbstractThe stein variational gradient descent (SVGD) algorithm is a deterministic particle method for sampling. However, a mean-field analysis reveals that the gradient flow corresponding to the SVGD algorithm (i.e., the Stein Variational Gradient Flow) only provides a constant-order approximation to the Wasserstein gradient flow corresponding to the KL-divergence minimization. In this work, we propose the Regularized Stein Variational Gradient Flow, which interpolates between the Stein Variational Gradient Flow and the Wasserstein gradient flow. We establish various theoretical properties of the Regularized Stein Variational Gradient Flow (and its time-discretization) including convergence to equilibrium, existence and uniqueness of weak solutions, and stability of the solutions. We provide preliminary numerical evidence of the improved performance offered by the regularization.

Distributed Control for Non-Cooperative Systems Governed by Time-Fractional Hyperbolic Operators

This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system’s state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed.

Mathematical and numerical analysis for PDE systems modeling intravascular drug release from arterial stents and transport in arterial tissue.

This paper is concerned with the PDE (partial differential equation) and numerical analysis of a modified one-dimensional intravascular stent model. It is proved that the modified model has a unique weak solution by using the Galerkin method combined with a compactness argument. A semi-discrete finite-element method and a fully discrete scheme using the Euler time-stepping have been formulated for the PDE model. Optimal order error estimates in the energy norm are proved for both schemes. Numerical results are presented, along with comparisons between different decoupling strategies and time-stepping schemes. Lastly, extensions of the model and its PDE and numerical analysis results to the two-dimensional case are also briefly discussed.

On the motion of a body with a cavity filled with magnetohydrodynamic fluid

We study the dynamics of a coupled system, formed by a rigid body with a cavity entirely filled with magnetohydrodynamic compressible fluid. Our aim is to derive the global existence of the unique classical solutions and weak solutions to this system. Moreover, we show the weak-strong uniqueness principle which means that a weak solution coincides with a strong solution on the time existence of a strong solution, provided they emanate from the same initial data.

Existence of solutions to stochastic p(t,x)-Laplace equations and applications

In this article, we consider a stochastic $p(t,x)$-Laplace equation. First we use the Galerkin method to obtain a unique weak solution. Then we obtain optimal controls for the corresponding stochastic optimal control problem For more information see https://ejde.math.txstate.edu/Volumes/2024/27/abstr.html

On the contraction properties for weak solutions to linear elliptic equations with 𝐿²-drifts of negative divergence

We show the existence and uniqueness as well as boundedness of weak solutions to linear elliptic equations with L 2 L^2 -drifts of negative divergence and singular zero-order terms which are positive. Our main target is to show the L r L^r -contraction properties of the unique weak solutions. Indeed, using the Dirichlet form theory, we construct a sub-Markovian C 0 C_0 -resolvent of contractions and identify it to the weak solutions. Furthermore, we derive an L 1 L^1 -stability result through an extended version of the L 1 L^1 -contraction property.

The torsion problem of the [formula omitted]-Bilaplacian

For each bounded and open set Ω⊂RN (N≥2) with smooth boundary denoted by ∂Ω and each real number p∈(1,∞) we analyze the torsion problem of the p-Bilaplacian, namely Δ(|Δu|p−2Δu)=1 in Ω with u=Δu=0 on ∂Ω. Firstly, we show that for each p∈(1,∞) the problem has a unique weak solution up. Secondly, we prove that up converges uniformly, as p→∞, in C1(Ω¯) to a certain function, say v2, which is exactly the unique solution of the problem −Δu=1 in Ω with u=0 on ∂Ω. Moreover, for each real number q∈[1,∞), Δup converges strongly to Δv2 in Lq(Ω), as p→∞. Next, we show that each solution up is also a solution for the minimization problem T(p;Ω)≔infu∈Xp(Ω)∖{0}1|Ω|∫Ω|Δu|pdx1|Ω|∫Ωudxp where Xp(Ω)≔{u∈W2,p(Ω)∩W01,p(Ω):u(x)≥0,a.e.x∈Ω}. Further, we show that the function (1,∞)∋p↦T(p;Ω) is strictly increasing provided that Ω is a convex and bounded open set for which |Ω|−1∫Ωv2dx is small. Finally, using this monotonicity result, we give an alternative variational characterization of the constant T(p;Ω) when |Ω|−1∫Ωv2dx is small. That last variational characterization fails to hold true when |Ω|−1∫Ωv2dx>1.

Homogenization of a Multivariate Diffusion with Semipermeable Interfaces

AbstractWe study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.

Multivalued dynamics of non-autonomous reaction–diffusion equation with nonlinear advection term

In this paper, we investigate a reaction–diffusion population model with a nonlinear advection term and a time-dependent force given by the equation ut−Δu+α→⋅∇up=f(u)+h(t)in(τ,∞)×Ω,subject to the boundary condition u=0 on (τ,∞)×∂Ω. Here, Ω⊂RN with N≥1 is a bounded domain with smooth boundary, τ∈R, α→=(α1,…,αN) is a given advective direction and p>1. The presence of the nonlinear advection term α→⋅∇up introduces technical difficulties in the analysis, leading to a scenario where the uniqueness of weak solutions cannot be guaranteed. Consequently, the equation generates a multi-valued nonautonomous dynamical system. In this context, we establish the existence of minimal pullback attractors, considering universes of bounded and tempered sets. Moreover, we explore the relationships between these pullback attractors. Finally, we prove the upper semicontinuity of pullback attractors with respect to the advective vector α→.

Existence and regularity of pullback attractors for a 3D non-autonomous Navier–Stokes–Voigt model with finite delay

In this manuscript previous results [Nonlinearity 25(2012), 905–930] are extended to a non-autonomous 3D Navier–Stokes–Voigt model in which a forcing term contains memory effects. Under suitable assumptions on the function driving the delay time, the existence and uniqueness of weak solution are proved. Existence and relationships among pullback attractors in several phase-spaces are analyzed for two possible choices of the attracted universes, namely, the standard one of fixed bounded sets, and another one given by a tempered condition. Some regularity results for these attractors are also established. Compactness and attraction norms are strengthened. Since the model does not have a regularizing effect, obtaining asymptotic compactness for the associated process is a more involved task. Our proofs rely on a sharp use of the energy equality, an energy method, bootstrapping arguments and by using bi-space attractors results.

The scalar Oseen problem in two-dimensional exterior domains

In the present paper, we investigate the scalar Oseen equation in a two-dimensional exterior domain Ω, with Dirichlet boundary condition. This problem represents a linearized form of the steady Navier-Stokes equations. In order to better control the behavior at infinity of functions, the use of weighted Sobolev spaces as a functional framework turns out to be adequate. The established results are related to the existence and the uniqueness of weak solutions on Lp-theory for any real 1<p<∞. Our analysis is mainly based on the principle that linear exterior problems can be solved by combining their properties in bounded domains which are also established in this paper and their properties well known in the whole space R2.

Dynamic programming principle for backward doubly stochastic recursive optimal control problem and sobolev weak solution of the stochastic Hamilton-Jacobi-Bellman equation

In this paper, we investigate a backward doubly stochastic recursive optimal control problem wherein the cost function is expressed as the solution to a backward doubly stochastic differential equation. We present the dynamical programming principle for this type of optimal control problem and establish that the value function is the unique Sobolev weak solution to the associated stochastic Hamilton-Jacobi-Bellman equation.

Variational and numerical analysis of a dynamic elasto-viscoplastic piezoelectric frictional contact problem

Abstract This work studies a mathematical model involving a dynamic contact between two elasto-viscoplastic piezoelectric bodies with damage. The contact is modeled on a combination of a normal compliance and a normal damped response law associated with friction. We derive a variational formulation of the problem and we prove the existence and uniqueness result of the weak solution. The proof is based on the classical existence and uniqueness result for parabolic inequalities, differential equations and fixed-point arguments. This paper examines a friction problem between two piezoelectric bodies that are elastic- viscoplastic. The study uses both numerical and analytical methods. The contact surface is assumed to have a thin layer of lubricant, and a damped response contact condition is considered. The problem also takes into account electrical and frictional effects. The paper proves the existence of a unique weak solution to the problem by using variational inequalities and a fixed point argument. Additionally, a fully discrete approximation is investigated using the Euler scheme and finite element method for the spatial variable and time derivatives. Error estimates are then calculated, leading to convergence results under certain regularity conditions.

Study of a Transmission Problem with Friction Law and Increasing Continuous Terms in a Thin Layer

The aim of this paper is to establish the asymptotic analysis of nonlinear boundary value problems. The non-stationary motion is given by the elastic constructive law. The contact is described with a version of Tresca’s law of friction. A variational formulation of the model, in the form of a coupled system for the displacements and the nonlinear source terms, is derived. The existence of a unique weak solution of the model is established. We also give the problem in transpose form, and we demonstrate different estimates of the displacement and of the source term independently of the small parameter. The main corresponding convergence results are stated in the different theorems of the last section.

Solving time-fractional diffusion equations with a singular source term

This article deals with linear time-fractional diffusion equations with time-dependent singular source term. Whether the order of the time-fractional derivative is multi-term, distributed or space-dependent, we prove that the system admits a unique weak solution enjoying a Duhamel representation, provided that the time-dependence of the source term is a distribution. As an application, the square integrable space-dependent part and the distributional time-dependent part of the source term of a multi-term time-fractional diffusion equation are simultaneously recovered by partial internal observation of the solution.