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Time—periodic weak solutions [PDF]
International Journal of Mathematics and Mathematical Sciences, 1990 In continuing from previous papers, where we studied the existence and uniqueness of the global solution and its asymptotic behavior as time t goes to infinity, we now search for a time-periodic weak solution u(t) for the equation whose weak formulation ...
Eliana Henriques de Brito
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Weak solutions with unbounded variation [PDF]
Proceedings of the American Mathematical Society, 1973 To solve a quasilinear system of hyperbolic partial differential equations with given initial data, the usual procedure is to approximate the initial data, solve the resulting problems, and show that the variation of the approximating solutions is uniformly bounded. A limiting process then can be used.
Donald P. Ballou
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Onsager's Conjecture for Admissible Weak Solutions [PDF]
Communications on Pure and Applied Mathematics, 2017 We prove that given any ...
T. Buckmaster +3 more
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Nonuniqueness of Weak Solutions to the SQG Equation [PDF]
Communications on Pure and Applied Mathematics, 2016 We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 of De Lellis and Székelyhidi in 2012. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the ...
T. Buckmaster, S. Shkoller, V. Vicol
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Existence of nontrivial weak solutions for a quasilinear Choquard equation [PDF]
Journal of Inequalities and Applications, 2018 We are concerned with the following quasilinear Choquard equation: −Δpu+V(x)|u|p−2u=λ(Iα∗F(u))f(u)in RN,F(t)=∫0tf(s)ds, $$ -\Delta_{p} u+V(x)|u|^{p-2}u=\lambda\bigl(I_{\alpha} \ast F(u)\bigr)f(u) \quad \text{in } \mathbb {R}^{N}, \qquad F(t)= \int_{0}^{t}
Jongrak Lee +3 more
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Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning [PDF]
SIAM Journal on Scientific Computing, 2020 This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via Friedrichs' seminal minimax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak ...
Fan Chen +3 more
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Uniqueness of weak solutions to a Keller-Segel-Navier-Stokes model with a logistic source
Applied Mathematics, 2021 We prove a uniqueness result of weak solutions to the nD ( n ⩾ 3) Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term.
Miaochao Chen, Sheng-Sen Lu, Qilin Liu
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Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity [PDF]
, 2020 We consider the linear transport equations driven by an incompressible flow in dimensions $$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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Weak solutions of generated Jacobian equations
Mathematics in Engineering, 2023 We prove two groups of relationships for weak solutions to generated Jacobian equations under proper assumptions on the generating functions and the domains, which are generalizations for the optimal transportation case and the standard Monge-Ampère case
Feida Jiang
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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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