On the Navier-Stokes equations with temperature-dependent transport coefficients by Eduard Feireisl, Josef Málek

“…We establish long-time and large-data existence of a weak solution to the problem describing three-dimensional unsteady flows of an incompressible fluid, where the viscosity and heat-conductivity coefficients vary with the temperature. …”
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Solvability of the Brinkman-Forchheimer-Darcy Equation by Piotr Skrzypacz, Dongming Wei

“…The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with variable porosity and for small data. …”
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The Stability of the Solutions for a Porous Medium Equation with a Convection Term by Huashui Zhan, Miao Ouyang

“…The existence of the weak solution is proved by the monotone convergent method. …”
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Flow of electrorheological fluid under conditions of slip on the boundary by R. H. W. Hoppe, M. Y. Kuzmin, W. G. Litvinov, V. G. Zvyagin

“…A theorem of existence of a weak solution is proved. For this purpose the approximating-topological method is used.…”
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A New Regularity Criterion for the Three-Dimensional Incompressible Magnetohydrodynamic Equations in the Besov Spaces by TianLi LI, Wen Wang, Lei Liu

“…Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. …”
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The Partial Second Boundary Value Problem of an Anisotropic Parabolic Equation by Huashui Zhan

“…The uniqueness of weak solution can be proved without the boundary value condition on Γ2.…”
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On the surjectivity of linear transformations by M. Damlakhi, V. Anandam

“…This result is used to discuss the existence of an Lp-weak solution of Du=v where D is a differential operator with smooth coefficients and v∈Lp.…”
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Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises by Jianhua Huang, Yuhong Li, Jinqiao Duan

“…Finally, some discussions on the global weak solution of the stochastic Boussinesq equations driven by general Lévy noise are also presented.…”
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Bilinear Minimax Optimal Control Problems for a von Kárman System with Long Memory by Jin-soo Hwang

“…We show the first and twice Fréchet differentiabilities of the nonlinear solution map from a bilinear input term to the weak solution of the equation. With the Fréchet differentiabilities of the control to solution mapping, we prove the uniqueness and existence of an optimal pair and find its necessary optimality condition.…”
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The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term by Ying Wang, YunXi Guo

“…Provided that (1-∂x2)u0∈M+(R), u0∈H1(R), and u0∈L1(R), the existence and uniqueness of the global weak solution to the equation are shown to be true.…”
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The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks by P. A. Krutitskii

“…The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that weak solution of the problem does not typically exist, though the classical solution exists. …”
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WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition by Ouadie Koubaiti, Ahmed Elkhalfi, Jaouad El-mekkaoui

“…Along with the usual weak mixed formulation, we give existence and uniqueness results for weak solution. Then, we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. …”
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Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage by J. R. Fernández, M. Sofonea

“…We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. …”
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An Existence Theorem for a Nonlocal Global Pandemic Model for Insect-Borne Diseases by John R. Cannon, Daniel J. Galiffa

“…We ultimately show that a weak solution to the weak formulation of this model exists using a fixed point argument, which calls upon the construction of a weak formulation and the existence and uniqueness of an auxiliary problem.…”
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A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem by Teffera M. Asfaw

“…The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces.…”
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A Result on the Existence and Uniqueness of Stationary Solutions for a Bioconvective Flow Model by Aníbal Coronel, Luis Friz, Ian Hess, Alex Tello

“…In this note, we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem. …”
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Analysis of electroelastic frictionless contact problems with adhesion by Mircea Sofonea, Rachid Arhab, Raafat Tarraf

“…We provide variational formulations for the problems and prove the existence of a unique weak solution to each model. The proofs are based on arguments of time-dependent variational inequalities, differential equations, and fixed point. …”
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On the frictionless unilateral contact of two viscoelastic bodies by M. Barboteu, T.-V. Hoarau-Mantel, M. Sofonea

“…We derive a variational formulation of the problem and prove the existence of a unique weak solution to the model by using arguments of evolution equations with maximal monotone operators. …”
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Nonlinear eigenvalue problems in Sobolev spaces with variable exponent by Teodora-Liliana Dinu

“…In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove that if λ is large enough then there exists a nontrivial weak solution. …”
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Variational Analysis of a Dynamic Contact Problem with Wear and Damage Involving Viscoelastic Materials with Long-Term Memory by Ahmed Hamidat, Hakim Bagua, Adel Aissaoui

“…We define a variational formulation for the model and demonstrate the existence of a single weak solution to the problem. The material behaviour is explained using a viscoelastic constitutive law that includes long-term memory and damage. …”
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