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Joerg Wolf - One of the best experts on this subject based on the ideXlab platform.
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on the Liouville Type theorems for self similar solutions to the navier stokes equations
Archive for Rational Mechanics and Analysis, 2017Co-Authors: Dongho Chae, Joerg WolfAbstract:We prove Liouville Type theorems for the self-similar solutions to the Navier–Stokes equations. One of our results generalizes the previous ones by Necas–Ružicka–Sverak and Tsai. Using a Liouville Type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier–Stokes equations with the profile belonging to \({L^{p, \infty} (\mathbb{R}^3)}\) with \({p > \frac{3}{2}}\).
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on the Liouville Type theorems for self similar solutions to the navier stokes equations
arXiv: Analysis of PDEs, 2016Co-Authors: Dongho Chae, Joerg WolfAbstract:We prove Liouville Type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Ne\v{c}as-R\.{u}\v{z}i\v{c}ka-\v{S}verak and Tsai. Using the Liouville Type theorem we also remove a scenario of asymtotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $L^{p, \infty} (\Bbb R^3)$ with $p> \frac{3}{2}$.
Dongho Chae - One of the best experts on this subject based on the ideXlab platform.
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On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations
Journal of Nonlinear Science, 2020Co-Authors: Dongho Chae, Jörg WolfAbstract:In this paper, we prove a Liouville Type theorem for non-Newtonian fluid equations in $$\mathbb R^3$$ R 3 , having the diffusion term $${\varvec{A}}_p( u)=\nabla \cdot ( |{\varvec{D}}(u)|^{p-2} {\varvec{D}}(u))$$ A p ( u ) = ∇ · ( | D ( u ) | p - 2 D ( u ) ) with $$ {\varvec{D}}(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top })$$ D ( u ) = 1 2 ( ∇ u + ( ∇ u ) ⊤ ) , $$3/2
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on the Liouville Type theorems for self similar solutions to the navier stokes equations
Archive for Rational Mechanics and Analysis, 2017Co-Authors: Dongho Chae, Joerg WolfAbstract:We prove Liouville Type theorems for the self-similar solutions to the Navier–Stokes equations. One of our results generalizes the previous ones by Necas–Ružicka–Sverak and Tsai. Using a Liouville Type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier–Stokes equations with the profile belonging to \({L^{p, \infty} (\mathbb{R}^3)}\) with \({p > \frac{3}{2}}\).
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on the Liouville Type theorems for self similar solutions to the navier stokes equations
arXiv: Analysis of PDEs, 2016Co-Authors: Dongho Chae, Joerg WolfAbstract:We prove Liouville Type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Ne\v{c}as-R\.{u}\v{z}i\v{c}ka-\v{S}verak and Tsai. Using the Liouville Type theorem we also remove a scenario of asymtotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $L^{p, \infty} (\Bbb R^3)$ with $p> \frac{3}{2}$.
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Remarks on a Liouville-Type Theorem for Beltrami Flows
International Mathematics Research Notices, 2014Co-Authors: Dongho Chae, Peter ConstantinAbstract:We present a simple, short and elementary proof that if v is a Beltrami flow with a finite energy in R3 then v = 0. In the case of the Beltrami flows satisfying v ∈ Lloc(R) ∩ Lq(R3) with q ∈ [2, 3), or |v(x)| = O(1/|x|1+e) for some e > 0, we provide a different, simple proof that v = 0. AMS Subject Classification Number: 35Q31, 76B03, 76W05 keywords: Euler equations, Beltrami flows, Liouville Type theorem
G Seregin - One of the best experts on this subject based on the ideXlab platform.
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remarks on Liouville Type theorems for steady state navier stokes equations
arXiv: Analysis of PDEs, 2017Co-Authors: G SereginAbstract:Liouville Type theorems for the stationary Navier-Stokes equations are proven under certain assumptions. These assumptions are motivated by conditions that appear in Liouvile Type theorems for the heat equations with a given divergence free drift.
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a Liouville Type theorem for steady state navier stokes equations
arXiv: Analysis of PDEs, 2016Co-Authors: G SereginAbstract:A Liouville Type theorem is proven for the steady-state Navier-Stokes equations. It follows from the corresponding theorem on the Stokes equations with the drift. The drift is supposed to belong to a certain Morrey space.
Quoc Hung Phan - One of the best experts on this subject based on the ideXlab platform.
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Optimal Liouville-Type theorems for a system of parabolic inequalities
Communications in Contemporary Mathematics, 2019Co-Authors: Anh Tuan Duong, Quoc Hung PhanAbstract:We establish optimal Liouville-Type theorems for the system of parabolic inequalities ut − Δu ≥ vp,vt − Δv ≥ uq and for the scalar inequality wt − Δw ≥ wp in the whole space ℝN × ℝ and in ℝN × (0,∞...
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Optimal Liouville-Type theorems for a system of parabolic inequalities
Communications in Contemporary Mathematics, 2019Co-Authors: Anh Tuan Duong, Quoc Hung PhanAbstract:We establish optimal Liouville-Type theorems for the system of parabolic inequalities [Formula: see text] and for the scalar inequality [Formula: see text] in the whole space [Formula: see text] and in [Formula: see text]. Our optimal Liouville-Type theorems are proved for two different classes of solutions: the nontrivial nonnegative and the positive.
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A Liouville-Type theorem for cooperative parabolic systems
Discrete & Continuous Dynamical Systems - A, 2018Co-Authors: Anh Tuan Duong, Quoc Hung PhanAbstract:We prove Liouville-Type theorem for semilinear parabolic system of the form \begin{document}$u_t-\Delta u =a_{11}u^{p}+a_{12} u^rv^{s+1}$\end{document} , \begin{document}$v_t-\Delta v =a_{21} u^{r+1}v^{s}+a_{22}v^{p}$\end{document} where \begin{document}$r, s>0$\end{document} , \begin{document}$p=r+s+1$\end{document} . The real matrix \begin{document}$A=(a_{ij})$\end{document} satisfies conditions \begin{document}$ a_{12}, a_{21}\geq 0$\end{document} and \begin{document}$a_{11}, a_{22}>0$\end{document} . This paper is a continuation of Phan-Souplet (Math. Ann., 366,1561-1585,2016) where the authors considered the special case \begin{document}$s=r$\end{document} for the system of \begin{document}$m$\end{document} components. Our tool for the proof of Liouville-Type theorem is a refinement of Phan-Souplet, which is based on Gidas-Spruck (Commun. Pure Appl.Math. 34,525–598 1981) and Bidaut-Veron (Equations aux derivees partielles et applications. Elsevier, Paris, pp 189–198,1998).
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Liouville Type theorem for nonlinear elliptic system involving Grushin operator
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Anh Tuan Duong, Quoc Hung PhanAbstract:Abstract We study the degenerate elliptic system of the form { − Δ G u = v p − Δ G v = u q on R N = R N 1 × R N 2 , where Δ G : = Δ x + | x | 2 α Δ y is the Grushin operator, α ≥ 0 and p ≥ q > 1 . We establish some Liouville Type results for stable solutions of the system. In particular, we prove the comparison principle – a crucial step to establish such results. As consequences, we obtain a Liouville Type theorem for the scalar equation and provide a counterpart of the previous result in C. Cowan (2013) [7] .
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Liouville-Type theorems for nonlinear degenerate parabolic equation
Journal of Evolution Equations, 2016Co-Authors: Quoc Hung PhanAbstract:We study Liouville-Type theorems for degenerate parabolic equation of the form $${u_t-{\rm div}(|\nabla u|^{m-2}\nabla u) = u^p}$$ u t - div ( | ∇ u | m - 2 ∇ u ) = u p where $${m > 2}$$ m > 2 and $${p > m - 1}$$ p > m - 1 . We prove the optimal Liouville-Type results in dimension $${N = 1}$$ N = 1 , and for radial solutions in any dimension. We also provide some partial results for non-radial solutions in dimension $${N \geq 2}$$ N ≥ 2 . Our proofs are based on a generalized Gidas–Spruck technique, combined with the idea of Serrin and Zou (Acta Math 189(1):79–142, 2002 ) and of Bidaut-Véron (Équations aux dérivées partielles et applications. Elsevier, Paris, pp 189–198, 1998 ). Finally, we clarify and correct some of the previous results on this topic.
Elimhan N Mahmudov - One of the best experts on this subject based on the ideXlab platform.
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Optimal control of Sturm-Liouville Type evolution differential inclusions with endpoint constraints
Journal of Industrial & Management Optimization, 2020Co-Authors: Elimhan N MahmudovAbstract:The present paper studies a new class of problems of optimal control theory with linear second order self-adjoint Sturm-Liouville Type differential operators and with functional and non-functional endpoint constraints. Sufficient conditions of optimality, containing both the second order Euler-Lagrange and Hamiltonian Type inclusions are derived. The presence of functional constraints generates a special second order transversality inclusions and complementary slackness conditions peculiar to inequality constraints; this approach and results make a bridge between optimal control problem with Sturm-Liouville Type differential differential inclusions and constrained mathematical programming problems in finite-dimensional spaces.The idea for obtaining optimality conditions is based on applying locally-adjoint mappings to Sturm-Liouville Type set-valued mappings. The result generalizes to the problem with a second order non-self-adjoint differential operator. Furthermore, practical applications of these results are demonstrated by optimization of some semilinear optimal control problems for which the Pontryagin maximum condition is obtained. A numerical example is given to illustrate the feasibility and effectiveness of the theoretic results obtained.
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Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions
Journal of Optimization Theory and Applications, 2018Co-Authors: Elimhan N MahmudovAbstract:The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-Type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-Type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-Type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-Type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.
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optimization of mayer problem with sturm Liouville Type differential inclusions
Journal of Optimization Theory and Applications, 2018Co-Authors: Elimhan N MahmudovAbstract:Abstract The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-Type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-Type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-Type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-Type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.