On problem of nonexistence of dissipative estimate for discrete kinetic equations by Evgenii Vladimirovich Radkevich

“…The existence of a global solution to the discrete kinetic equations in Sobolev spaces is proved, its decomposition by summability is obtained, the influence of its oscillations generated by the interaction operator is explored. …”
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An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain by Yubing Sui, Donghao Zhang, Junying Cao, Jun Zhang

“…We further introduce some suitable Sobolev spaces and derive the weak form and an efficient discrete scheme. …”
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Constructive Sobolev gradient preconditioning for semilinear elliptic systems by Janos Karatson

“…We present a Sobolev gradient type preconditioning for iterative methods used in solving second order semilinear elliptic systems; the n-tuple of independent Laplacians acts as a preconditioning operator in Sobolev spaces. The theoretical iteration is done at the continuous level, providing a linearization approach that reduces the original problem to a system of linear Poisson equations. …”
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Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds by Chen Yuxuan

“…Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.…”
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Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum by Coutand, D, Shkoller, S

“…We establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary. The proof is founded on a new higher-order Hardy-type inequality in conjunction with an approximation of the Euler equations consisting of a particular degenerate parabolic regularization. …”

The Burgers-KdV limit in one-dimensional plasma with viscous dissipation: A study of dispersion and dissipation effects by Rong Rong, Hui Liu

“…To analyze the remaining system, we employ the energy method in Sobolev spaces to estimate its behavior. As a result, we are able to capture the Burgers-KdV dynamics over a time interval of order $ O(\varepsilon^{-1}) $, where $ \varepsilon $ represents a small parameter.…”
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Optimal Control for a Nonlocal Model of Non-Newtonian Fluid Flows by Evgenii S. Baranovskii, Mikhail A. Artemov

“…Using one result on the solvability of nonlinear operator equations with weak-to-weak and weak-to-strong continuous mappings in Sobolev spaces, we construct a weak solution that minimizes a given cost functional subject to natural conditions on the model data. …”
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A Legendre spectral method based on a hybrid format and its error estimation for fourth-order eigenvalue problems by Yuanqiang Chen, Jihui Zheng, Jing An

“…By integrating approximation results of some orthogonal projection operators in weighted Sobolev spaces, we further gave the error estimation for the approximating eigenvalues and eigenfunctions. …”
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Stability and boundedness of regular solutions for a Sisko flow in an infinite annular porous space by José Luis Díaz Palencia, Saeed ur Rahman, Abraham Otero, Pablo Salgado Sánchez

“…For this purpose, we will consider energy estimates in Sobolev spaces and develop the boundedness criteria for the resulting unsteady parabolic nonlinear equation.…”
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Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence by Dong, Zhaonan, Ern, Alexandre, Guermond, Jean-Luc

“…We estimate best-approximation errors using vector-valued finite elements for fields with low regularity in the scale of the fractional-order Sobolev spaces. By assuming that the target field enjoys an additional integrability property on its curl or its divergence, we establish upper bounds on these errors that can be localized to the mesh cells. …”
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Weak and renormalized solutions for anisotropic Neumann problems with degenerate coercivity by Mohamed Badr Benboubker, Hayat Benkhalou, Hassane Hjiaj

“…The functional setting involves anisotropic Sobolev spaces with constants exponents. …”
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Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth by Yony Raúl Santaria Leuyacc

“…Our approach is based on Trudinger-Moser type inequalities for weighted Sobolev spaces and variational methods.…”
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Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains by Tang, Tao, Wang, Li-Lian, Yuan, Huifang, Zhou, Tao

“…We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces and analyze the optimal convergence of the proposed Galerkin scheme. …”
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On the interaction between quasilinear elastodynamics and the Navier-Stokes equations by Coutand, D, Shkoller, S

“…We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. …”

A class of Schrödinger elliptic equations involving supercritical exponential growth by Yony Raúl Santaria Leuyacc

“…Our approach is based on a new Trudinger–Moser-type inequality for weighted Sobolev spaces and variational methods.…”
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فضاء الطاقة لمؤثر هرميت في R^n وفضاءات سوبوليڤ موافقة by إبراهيم إبراهيم, طالب غريبة, آصف المحمد

“…We will see that   has similar properties as    for real numbers s > o, therefore we can construct new Hilbert spaces        which are the energy spaces of powers of . They are Sobolev spaces. We  can  also  generalize  those  spaces  to   for  . …”
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Existence of solutions for $p(x)$-Laplacian equations by Rabil Ayazoglu (Mashiyev), B. Cekic, O. M. Buhrii

“…Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem.…”
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On a mathematical model of non-isothermal creeping flows of a fluid through a given domain by Anastasia Aleksandrovna Domnich, Evgenii Sergeevich Baranovskii, Mikhail Anatolievich Artemov

“…The main result of the work is a theorem on the existence of weak solutions in a subspace of the Cartesian product of two Sobolev's spaces. To prove this theorem, we give an operator interpretation of the boundary value problem, derive a priori estimates of solutions, and apply the Leray-Schauder fixed point theorem. …”
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Oscillatory solutions and smoothing of a higher-order p-Laplacian operator by José Luis Díaz Palencia, Abraham Otero

“…This variational principle is supported by the definition of generalized norms under Hilbert-Sobolev spaces, enabling focus on the oscillating properties of solutions. …”
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Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation by Bertrand Lods, Giovanni Pistone

“…This requires us to generalize our approach to Orlicz–Sobolev spaces to include derivatives.…”
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