Weak Solutions Research Articles

A new approach for the regularity of weak solutions of the 3D Boussinesq system

A new approach for the regularity of weak solutions of the 3D Boussinesq system

On the Mass Transfer in the 3D Pitaevskii Model

We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in T3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}^3$$\\end{document} for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.

Regularity for critical fractional Choquard equation with singular potential and its applications

Abstract We study the following fractional Choquard equation ( − Δ ) s u + u ∣ x ∣ θ = ( I α * F ( u ) ) f ( u ) , x ∈ R N , {\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ⩾ 3 N\geqslant 3 , s ∈ 1 2 , 1 s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) , α ∈ ( 0 , N ) \alpha \in \left(0,N) , θ ∈ ( 0 , 2 s ) \theta \in \left(0,2s) , and I α {I}_{\alpha } is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of θ = 0 \theta =0 in Moroz-Van Schaftingen [Existence of groundstates for a class of nonlinear Choquardequations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state solutions for above equation with the nonlinearity involving embedding top and bottom indices, which is related to the Hardy-Littlewood-Sobolev inequality and singular term 1 ∣ x ∣ θ \frac{1}{{| x| }^{\theta }} . It is worth noting that our approach is not involving the concentration-compactness principle.

Two solutions for Dirichlet double phase problems with variable exponents

Abstract This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.

Nonlinear parabolic double phase variable exponent systems with applications in image noise removal

In this paper, a novel parabolic system involving nonlinear and nonhomogeneous partial differential operator with variable growth structure is introduced for investigating the image denoising and restoration. More precisely, our model is based on regularizing the classical models involving variable exponent operators by considering a nonlinear operator with double phase flux. We begin by investigating theoretically the solvability to the parabolic system under consideration. Under the setting of Musielak-Orlicz spaces, we build a suitable functional framework to study the proposed system. Therefore, we develop the Faedo-Galerkin approach to demonstrate the existence and uniqueness of a weak solution to our model. To illustrate our theoretical results in the context of image noise removal, we present various numerical implementations on some grayscale images. To enrich these simulations, we test the robustness efficiency of the proposed model in the so-called Magnetic Resonance Images (MRI). The obtained numerical results claim that our model is more efficient and robust against noise, in comparison (visually and quantitatively) to some existing state-of-the-art methods.

Semilinear elliptic problems via the nonlinear Rayleigh quotient with two nonlocal nonlinearities

It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: { − Δ u + V ( x ) u = μ ( I α 1 ∗ | u | q ) | u | q − 2 u − λ ( I α 2 ∗ | u | p ) | u | p − 2 u in R N , u ∈ H 1 ( R N ) , where p > q , λ , μ > 0 , α 1 ≤ α 2 ; α 1 , α 2 ∈ ( 0 , N ) , N ≥ 3 ; p ∈ ( 2 α 2 , 2 α 2 ∗ ) ; q ∈ ( 2 α 1 , 2 α 1 ∗ ) , 2 α j = ( N + α j ) / N and 2 α j ∗ = ( N + α j ) / ( N − 2 ) , j = 1 , 2 . Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp μ n > 0 and λ ∗ , λ ∗ > 0 such that our main problem admits at least two solutions for each μ > μ n where λ ∈ ( 0 , min ( λ ∗ , λ ∗ ) ) . The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each μ < μ n and λ > 0 .

On a biharmonic coupled system with non-standard nonlinearity: Existence, blow up and numerics

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples.

Reflected stochastic differential equations driven by standard and fractional Brownian motion

The reflection problem on the positive half-line with reflection at zero for a time-dependent stochastic differential equations driven by standard and fractional Brownian motion with Hurst parameter [Formula: see text] is considered. We prove the existence of weak solutions by using Euler scheme. Moreover, we show that pathwise uniqueness holds and a strong solution exists in the case of additive fractional noise and also up to a stopping time [Formula: see text] for the multiplicative case, but remains an open question beyond [Formula: see text].

The Global existence and blow-up of solutions for a class of Steklov parabolic problems

In this paper, we study weak solutions to the following Steklov parabolic problem: We prove theorems of existence of weak solutions, via Galerkin approximation. Moreover, we show the existence of solutions which blow up in a finite time.

Three weak solutions for a class of fourth order p(x)-Kirchhoff type problem with Leray-Lions operators

In this work, we study the multiplicity of a weak solution for a fourth order p(x)-Kirchhoff type problem involving the Leray-Lions type operators with no flux boundary condition. By using variational approach and critical point theory, we determine an open interval of parameters for which our problem admits at least three distinct weak solutions.

Various duality models associated with some constrained robust nonlinear optimal control problems

This paper deals with various duality results for a new class of constrained robust nonlinear control problems. Concretely, we introduce a minimisation problem associated with a vector of functionals of multiple integral type. In addition, the considered multi-objective controlled optimisation problem is governed by several mixed constraints generated by partial derivatives of second order and uncertain data. For this new class of control problems, we establish and characterise Wolfe, Mond-Weir and mixed type robust dual models. To this aim, by using the notion of convex multiple integral functional, generated by a controlled second-order Lagrangian with uncertain data, and the concept of robust weak efficient solution to the problem under study, we build a new mathematical context to state and prove the duality theorems.

Global Well-Posedness and Regularity of Weak Solutions to the Prandtl’s System

Global Well-Posedness and Regularity of Weak Solutions to the Prandtl’s System

Qualitative analysis of solutions for a degenerate partial differential equations model of epidemic spread dynamics

Compartmental models are widely used in mathematical epidemiology to describe the dynamics of infectious diseases or in mathematical models of population genetics. In this study, we study a time-dependent Susceptible-Infectious-Susceptible (SIS) Partial Differential Equation (PDE) model that is based on a diffusion-drift approximation of a probability density from a well-known discrete-time Markov chain model. This SIS-PDE model is conservative due to the degeneracy of the diffusion term at the origin. The main results of this article are the qualitative behavior of weak solutions, the dependence of the local asymptotic property of these solutions on initial data, and the existence of Dirac delta function type solutions. Moreover, we study the long-term behavior of solutions and confirm our analysis with numerical computations.

Weak solutions to coupled quadratic forward backward stochastic differential equations and Sobolev solutions to their related partial differential equations

We establish existence and uniqueness of a weak (in law sense) solution to coupled forward–backward stochastic differential equations (FBSDEs), with possibly discontinuous diffusion coefficient. We assume that the coefficient of the backward component is of quadratic growth. We first prove the existence of a Sobolev solution for the related partial differential equation (PDE) in the Sobolev space , by using compactness arguments. Next, we use Itô–Krylov's formula to get the existence of weak solution to our considered FBSDE. For the uniqueness part, we first establish the weak uniqueness of the FBSDEs then deduce the uniqueness of its related PDE by using the Feynmann–Kac formula.

Social Justice and Physical Education in the United States: The Need for New Maps

This paper is a reflection, a critique, and, hopefully, an inspiration to think about how future generations can reshape physical education in the United States. To do so, we first pay homage to our pioneers, who, we argue, were transformative leaders because they used research to respond to the sociopolitical issues of their time. In saying this, we reflect on how these ideas from that time were critically important but have also been developed for a different time, place, and demographic of people. We then trace the social justice research in the United States by highlighting the promises and pitfalls of current scholarship because it often asks “tough questions” but provides “weak solutions.” To conclude, we believe that the future of physical education needs to be about allowing those “new voices” to become the future leaders of our field. In so doing, they will change the landscape of physical education knowledge, movement, and practices.

Global existence and partial regularity for the Landau-Lifshitz-Gilbert equation with helicity

In this paper, the Landau-Lifshitz-Gilbert equation with helicity is considered. In R3 or a bounded regular domain Ω of R3, we establish the global existence of a weak solution. In Rn, a global existence criterion and uniqueness of the smooth solution are given. In R1, the local smooth solution is indeed global with large initial data. In R2, we prove the existence of a global weak solution, which is smooth with the exception of at most finite singular points.

A mathematical model for flow field with condensation discontinuity: One‐dimensional Euler equations with singular sources

Condensation induces both heat addition and mass consumption effects on the carrier flow field, leading to complex wave patterns. In this study, we aim to elucidate the structures of flow field under the influence of condensation. To achieve this, we present a mathematical model, which is formulated using Euler equations with a singular source term. Given the difficulties posed by both weak solution theory and the Dal Maso–LeFloch–Murat (DLM) theory (J. Math. Pures Appl. 483‐548 (1995)) in defining solutions to the Riemann problem, we propose a novel approach. This involves the coupling of weak solutions within two subregions, taking into consideration the discontinuity of source term. Firstly, we identify two stationary discontinuities originating from the singular source term—namely, stationary waves and composite waves. Admissibility criterions are developed for these discontinuities to facilitate the selection of physically meaningful solutions. Secondly, we employ a double classical Riemann problems (CRPs) framework to analyze the structures of Riemann solution. Our analysis reveals that the Riemann solution may exhibit seven structures with stationary wave and four structures with composite wave. Finally, the proposed model and its theoritical results are applied to validate the structure of the flow field with condensation discontinuities. For the pure heat addition problem, we successfully demonstrate all wave patterns of the flow field, perfectly matching the outcomes of numerical experiments.

In-situ poly(hexahydrotriazine) gels temporary plugging agent suitable for high-temperature reservoirs: Regulation by inorganic chloride salts

A new acid-soluble in-situ temporary plugging agent suitable for high-temperature reservoirs was developed. By regulating the aldehyde-amine condensation reaction using inorganic chlorine salts, the poly(hexahydrotriazine) gels (PHT gels) exhibit good plugging performance and high-temperature resistance and can be degraded in weak acid solutions without residue. Results show that inorganic chlorine salts have a regulating effect on the gelling time of the gelant. Without the salt or the addition of monovalent chlorine salt, the gelant gels at room temperature but does not form a gel at high temperatures. After adding divalent or trivalent inorganic salts, PHT gels are formed at high temperatures, and their gelling time increases with the increase in salt content. Mechanical compression experiments show that the compression resistance of PHT gels is almost unchanged after exposure to high temperature (140 °C) and high salt (200 g/L NaCl and 10 g/L CaCl2). Displacement experiments show that PHT gels have excellent plugging performance with a plugging rate of more than 98.50% and a maximum plugging pressure of 5.34 MPa under the high temperature and high-salt conditions mentioned above. In addition, a hydrochloric acid solution with pH ≤ 5 can achieve residue-free breakage of PHT gels. The gel-breaking fluid has a high surface activity to facilitate flowback and a core unblocking rate of more than 100% after gel-breaking, making the reservoir permeability recoverable. The plugging and acidolysis ability of PHT gels make them excellent in-situ temporary plugging agents. This study aims to provide insights into the preparation of PHT gels and their use in high-temperature reservoirs.

On the dynamic Coulomb's frictional contact problem for thermo‐piezoelectric materials

AbstractIn this work, we study, from a variational point of view, a dynamic contact problem between a thermo‐piezoelectric body and a thermally conductive foundation. The normal compliance contact condition and Coulomb's friction law are employed to model the contact. We provide existence and uniqueness results of a weak solution to the model using adequate auxiliary problems, an abstract result on nonlinear first‐order evolution inequalities and Banach fixed point argument. Finally, the continuous dependence of the solution on the surface traction force and the surface electrical charge is studied.

Weak Solutions to an Initial-Boundary Value Problem of a Consistent Three-phase-field Model

Global warming has caused the melting of glaciers, especially the Greenland sea ice which is a great threat to human beings. The phase-field method is used to describe the evolution of multi-phase microstructure of Greenland sea ice. This paper shall investigate the global existence, uniqueness and regularity of weak solutions to an initial-boundary value problem for a three-phase-field model based on criteria that lead to both physical and mathematical consistency. For global solutions, it is calculated by applying the method of continuation of local solutions. This model allows for the formulation of spurious phases, which is completely consistent with physical reality.