Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity [PDF]
, 2020 We consider the linear transport equations driven by an incompressible flow in dimensions $$d\ge 3$$ d ≥ 3 . For divergence-free vector fields $$u \in L^1_t W^{1,q}$$ u ∈ L t 1 W 1 , q , the celebrated DiPerna-Lions theory of the renormalized solutions ...
A. Cheskidov, Xiaoyutao Luo
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On the solvability of Dirichlet problem for the weighted p-Laplacian [PDF]
Opuscula Mathematica, 2012 The paper investigates the existence and uniqueness of weak solutions for a non-linear boundary value problem involving the weighted \(p\)-Laplacian. Our approach is based on variational principles and representation properties of the associated spaces.
Ewa Szlachtowska
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Nonuniqueness of weak solutions to the Navier-Stokes equation [PDF]
Annals of Mathematics, 2017 For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations
T. Buckmaster, V. Vicol
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A variational approach for mixed elliptic problems involving the p-Laplacian with two parameters
Boundary Value Problems, 2022 By exploiting an abstract critical-point result for differentiable and parametric functionals, we show the existence of infinitely many weak solutions for nonlinear elliptic equations with nonhomogeneous boundary conditions. More accurately, we determine
Armin Hadjian, Juan J. Nieto
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Growth conditions and regularity for weak solutions to nonlinear elliptic pdes
Journal of Mathematical Analysis and Applications, 2020 We describe some aspects of the process/approach to interior regularity of weak solutions to a class of nonlinear elliptic equations in divergence form, as well as of minimizers of integrals of the calculus of variations.
Paolo Marcellini
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Weak solutions to the time-fractional g-Bénard equations
Boundary Value Problems, 2022 The Bénard problem consists in a system that couples the well-known Navier–Stokes equations and an advection-diffusion equation. In thin varying domains this leads to the g-Bénard problem, which turns out to be the classical Bénard problem when g is ...
Khadija Aayadi +3 more
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Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier–Stokes Equations in High Dimensions [PDF]
Archive for Rational Mechanics and Analysis, 2018 Consider the unforced incompressible homogeneous Navier–Stokes equations on the d-torus $${\mathbb{T}^d}$$Td where $${d \geq 4}$$d≥4 is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $${u \in L^{2} (\mathbb{T}^d)}
Xiaoyutao Luo
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Weak-very weak uniqueness to the time-dependent Ginzburg–Landau model for superconductivity in Rn
Results in Applied Mathematics, 2021 In this paper, we consider the n(n≥3)dimensional time-dependent Ginzburg–Landau model for superconductivity. First, we obtain a global existence of very weak solutions. Finally we prove a result of weak-very weak uniqueness.
Hongjun Gao, Jishan Fan, Gen Nakamura
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Some Compactness Criteria for Weak Solutions of Time Fractional PDEs [PDF]
SIAM Journal on Mathematical Analysis, 2017 The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of ...
Lei Li, Jian‐Guo Liu
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Relaxation and weak solutions of nonlocal semilinear evolution systems
Advances in Difference Equations, 2019 We give a relatively short proof of the fact that the solution set of a nonlocal semilinear differential inclusion is dense in the weak solution set of the corresponding convexified system.
N. Javaid +3 more
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