Weak Solutions Research Articles

On the Pohozaev identity for the fractional p$p$‐Laplacian operator in RN$\mathbb {R}^N$

AbstractIn this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional ‐Laplacian operator and a Berestycki–Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.

Obstacle Problems with Double Boundary Condition for Least Gradient Functions in Metric Measure Spaces

AbstractIn the setting of a metric space equipped with a doubling measure supporting a (1, 1)-Poincaré inequality, we study the problem of minimizing the BV-energy in a bounded domain $$\Omega $$ Ω of functions bounded between two obstacle functions inside $$\Omega $$ Ω , and whose trace lies between two prescribed functions on the boundary. If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when $$\Omega $$ Ω is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Malý, Shanmugalingam, and Speight (2019). While such solutions are not unique in general, we show the existence of unique minimal solutions. Since candidate functions need not agree outside of the domain, standard compactness arguments fail to provide existence of weak solutions as they are defined for the problem with single boundary condition. To overcome this, we introduce a class of $$ \varepsilon $$ ε -weak solutions as an intermediate step. Our existence results generalize those of Ziemer and Zumbrun (1999), who studied this problem in the Euclidean setting with a single obstacle and single boundary condition.

On the homogeneous torsion problem for heterogeneous and orthotropic cross-sections: Theoretical and numerical aspects

For many years, torsion of arbitrary cross-sections has been a subject of numerous investigations from theoretical and numerical points of view. As it is well known, the resulting boundary value problem (BVP) governing such phenomenon happens to be a pure Neumann BVP and, therefore, its solutions are determined up to a constant. Among a large plethora of finite element method (FEM) techniques that can be used in this context, most of FEM practitioners resolve this uniqueness issue by fixing the candidate solution to a node of the domain. Although such popular and pinpointing technique is widely spread and works well for practical purposes, it does not have a continuous counterpart and therefore its justification remains a matter of debate. Hence, this self-contained work aims to address the modeling of arbitrary heterogeneous and orthotropic cross-sections as well as the theoretical and numerical aspects of their solutions. In particular, we discuss the existence of weak solutions, well-posedness, regularity of solutions, and convergence of Galerkin’s method for different variational settings (with special focus on a regularized variational approach). Moreover, we establish a connection, at a discrete level, between the convergence of solutions of well-posed variational settings and those solutions coming from the usual practice of fixing a datum at a node. Finally, we discuss some numerical aspects of all the FEM discrete formulations proposed here by performing convergence analysis in L2 and H1 norms. The section of numerical results is closed by presenting a series of study cases ranging from a square cross-section composed of two different materials to an isotropic bridge cross-section for which no analytical solution exists.

On the Existence of Global Weak Solutions to the 3D Electrically Conductive Rosensweig System and Their Convergence Towards Quasi-Equilibrium

In this article, we study an electrically conductive Rosensweig model for ferrofluids, whose Bloch–Torrey regularization was studied by Hamdache and Hamroun (Appl Math Optim 81(2):479–509, 2020). We mainly prove the global existence of weak solutions to the non-regularized model under a certain smallness condition on the electric conductivity. Hence, our result not only solves a problem that was left open by Hamdache and Hamroun, but it can also serve as a confirmation that ferrofluids are naturally poor conductors of electric current. The proof, which is interesting in itself, is quite involved and relies on the Helmohltz–Leray decomposition of the magnetic fields and the use of renormalized solutions for the magnetization. We also give a rigorous and detailed description of the convergence of the global weak solutions towards the quasi-equilibrium in the relaxation time limit regime τ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\rightarrow 0$$\\end{document}.

Existence of weak solutions to stochastic heat equations driven by truncated α-stable white noises with non-Lipschitz coefficients

We consider a class of stochastic heat equations driven by truncated α-stable white noises for α∈(1,2) with noise coefficients that are continuous but not necessarily Lipschitz continuous. We prove the existence of weak solution in probabilistic sense, taking values in two different forms under different conditions, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. More precisely, for α∈(1,2) there exists a measure-valued weak solution. However, for α∈(1,5/3) there exists a function-valued weak solution, and in this case we further show that for p∈(α,5/3) the uniform p-th moment in Lp-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in Lp sense.

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Holder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.

Weak Sharp Type Solutions for Some Variational Integral Inequalities

Weak sharp type solutions are analyzed for a variational integral inequality defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided.

On the existence of stationary threshold bilinear processes

AbstractThis article investigates some statistical and probabilistic properties of general threshold bilinear processes. Sufficient conditions for the existence of a causal, strictly and weak stationary solution for the equation defining a self-exciting threshold superdiagonal bilinear $$\left( SETBL\right) $$ S E T B L process are derived. Then it is shown that under well-specified hypotheses the higher-order moments of the SETBL process are finite. As a result, the skewness and kurtosis indexes are explicitly computed. The exact autocorrelation function is derived with an arbitrarily fixed number of regimes. Also, the covariance functions of the process and its powers are evaluated and the second (respectively, higher)-order structure is shown to be similar to that of a linear process. This implies that the considered process admits an ARMA representation. Finally, necessary and sufficient conditions for the invertibility and geometric ergodicity of a SETBL model are established. Some examples illustrate the obtained theoretical results.

On a fractional (p,q)-Laplacian equation with critical nonlinearities

In this paper, we consider the following fractional ( p , q ) -Laplacian equations with critical Hardy-Sobolev exponents { ( − Δ ) p s 1 u + ( − Δ ) q s 2 u = λ | u | r − 2 u + μ | u | p s 1 ∗ ( α ) − 2 u | x | α in Ω , u = 0 in R N ∖Ω , where 0 < s 2 < s 1 < 1 < q < r ≤ p < N s 1 , 0 $ ]]> λ , μ > 0 are two parameters, 0 ≤ α < p s 1 and p s 1 ∗ ( α ) = p ( N − α ) N − p s 1 is the fractional Hardy-Sobolev critical exponent, Ω ⊂ R N is an open bounded domain with smooth boundary. By using variational methods, we show that the problem has a nontrivial nonnegative weak solution.

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.

Analysis and Numerical Simulation of Time-Fractional Derivative Contact Problem with Friction in Thermo-Viscoelasticity

Abstract The objective of this study is to analyze a quasistatic frictional contact problem involving the interaction between a thermo-viscoelastic body and a thermally conductive foundation. The constitutive relation in our investigation is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. Additionally, the heat conduction aspect is governed by a time-fractional derivative parameter that is associated with temperature. The contact is modeled using the Signorini condition, which is a version of Coulomb’s law for dry friction. We develop a variational formulation for the problem and establish the existence of its weak solution using a combination of techniques, including the theory of monotone operators, Caputo derivative, Galerkin method, and the Banach fixed point theorem. To demonstrate the effectiveness of our approach, we include several numerical simulations that showcase the performance of the method.

On the Hm-Regularity for Second Order Elliptic Equations

Our primary objective in this paper is to discuss the Hm-Regularity for second order elliptic equations over Sobolev Spaces. We here consider the cases m = 1 and m≥ 2 separately. We revisit some elementary concepts in Functional Analysis and Abstract Harmonic Analysis before providing a proper definiton to the notion of weak solution of a Dirichlet Problem. While towards the later stages, we shall classify different types of regularity conditions, the main focus lies upon deducing appropriate ellipticity conditions in support of commenting about the existence and uniqueness of the weak solutions to a given problem. Appropriate references are provided in the bibliography section to facilitate further reading for ardent readers and researchers in this field

On the Hm-Regularity for Second Order Elliptic Equations

Our primary objective in this paper is to discuss the Hm-Regularity for second order elliptic equations over Sobolev Spaces. We here consider the cases m = 1 and m≥ 2 separately. We revisit some elementary concepts in Functional Analysis and Abstract Harmonic Analysis before providing a proper definiton to the notion of weak solution of a Dirichlet Problem. While towards the later stages, we shall classify different types of regularity conditions, the main focus lies upon deducing appropriate ellipticity conditions in support of commenting about the existence and uniqueness of the weak solutions to a given problem. Appropriate references are provided in the bibliography section to facilitate further reading for ardent readers and researchers in this field

On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces

AbstractThis paper is concerned with a parabolic evolution equation of the form , settled in a smooth bounded domain of , , and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the ‐Laplacian for suitable , the “variable‐exponent” ‐Laplacian, or even some fractional order operators. The operator is assumed to be in the form with being measurable in and maximal monotone in . The main results are devoted to proving existence of weak solutions for a wide class of functions that extends the setting considered in previous results related to the variable exponent case where . To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so‐called ‐type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators , ) to which the result can be applied.

Calderón-Zygmund estimates for nonlinear equations of differential forms with BMO coefficients

We obtain Lq-regularity estimates for weak solutions to p-Laplacian type equations of differential forms. In particular, we prove local Calderón-Zygmund type estimates for equations with discontinuous coefficients satisfying the bounded mean oscillation (BMO) condition.

A fully well-balanced hydrodynamic reconstruction

Abstract The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.

Weak solutions to the heat conducting compressible self-gravitating flows in time-dependent domains

In this paper, we consider the heat-conducting compressible self-gravitating fluids in time-dependent domains, which typically describe the motion of viscous gaseous stars. The flow is governed by the 3D Navier–Stokes–Fourier–Poisson equations where the velocity is supposed to fulfill the full-slip boundary condition and the temperature on the boundary is given by a non-homogeneous Dirichlet condition. We establish the global-in-time weak solution to the system. Our approach is based on the penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Moreover, to accommodate the non-homogeneous boundary heat flux, the concept of ballistic energy is utilized in this work.

Finite‐strain poro‐visco‐elasticity with degenerate mobility

AbstractA quasistatic nonlinear model for poro‐visco‐elastic solids at finite strains is considered in the Lagrangian frame using the concept of second‐order nonsimple materials and Kelvin–Voigt‐type viscosity. The elastic stresses satisfy static frame‐indifference, while the viscous stresses satisfy dynamic frame‐indifference. The mechanical equation is coupled to a diffusion equation for a solvent or fluid content. The latter is pulled‐back to the reference configuration. To treat the nonlinear dependence of the mobility tensor on the deformation gradient, the result by Healey and Krömer is used to show that the determinant of the deformation gradient is bounded away from zero. Moreover, the focus is on the physically relevant case of degenerate mobilities. The existence of weak solutions is shown using a staggered time‐incremental scheme and suitable energy‐dissipation inequalities.

A pseudo‐parabolic equation with logarithmic nonlinearity: Global existence and blowup of solutions

This article deals with a fourth‐order pseudo‐parabolic partial differential equation with logarithmic nonlinearity. We adopt the Faedo–Galerkin method to analyze the global existence of weak solutions and discuss the existence of a solution in subcritical and critical initial energy situations. Further, applying the concavity approach, we have shown that the solution blows up when the initial energy is subcritical, critical, and supercritical. Moreover, we have provided an upper and lower bound for blow‐up time and a decay estimate for the global solutions.