• Title/Summary/Keyword: weak solutions

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LOCAL REGULARITY OF THE STEADY STATE NAVIER-STOKES EQUATIONS NEAR BOUNDARY IN FIVE DIMENSIONS

  • Kim, Jaewoo;Kim, Myeonghyeon
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.3
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    • pp.557-569
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    • 2009
  • We present a new regularity criterion for suitable weak solutions of the steady-state Navier-Stokes equations near boundary in dimension five. We show that suitable weak solutions are regular up to the boundary if the scaled $L^{\frac{5}{2}}$-norm of the solution is small near the boundary. Our result is also valid in the interior.

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GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC VLASOV-KLEIN-GORDON SYSTEM IN TWO DIMENSIONS

  • Xiao, Meixia;Zhang, Xianwen
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.591-598
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    • 2018
  • This paper is concerned with global existence of weak solutions to the relativistic Vlasov-Klein-Gordon system. The energy of this system is conserved, but the interaction term ${\int}_{{\mathbb{R}}^n}\;{\rho}{\varphi}dx$ in it need not be positive. So far existence of global weak solutions has been established only for small initial data [9, 14]. In two dimensions, this paper shows that the interaction term can be estimated by the kinetic energy to the power of ${\frac{4q-4}{3q-2}}$ for 1 < q < 2. As a consequence, global existence of weak solutions for general initial data is obtained.

NOTE ON LOCAL BOUNDEDNESS FOR WEAK SOLUTIONS OF NEUMANN PROBLEM FOR SECOND-ORDER ELLIPTIC EQUATIONS

  • KIM, SEICK
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.19 no.2
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    • pp.189-195
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    • 2015
  • The goal of this note is to provide a detailed proof for local boundedness estimate near the boundary for weak solutions for second order elliptic equations with bounded measurable coefficients subject to Neumann boundary condition.

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR A KELLER-SEGEL-FLUID MODEL WITH NONLINEAR DIFFUSION

  • Chung, Yun-Sung;Kang, Kyungkeun;Kim, Jaewoo
    • Journal of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.635-654
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    • 2014
  • We consider the Cauchy problem for a Keller-Segel-fluid model with degenerate diffusion for cell density, which is mathematically formulated as a porus medium type of Keller-Segel equations coupled to viscous incompressible fluid equations. We establish the global-in-time existence of weak solutions and bounded weak solutions depending on some conditions of parameters such as chemotactic sensitivity and consumption rate of oxygen for certain range of diffusive exponents of cell density in two and three dimensions.

EXISTENCE AND MULTIPLICITY OF WEAK SOLUTIONS FOR SOME p(x)-LAPLACIAN-LIKE PROBLEMS VIA VARIATIONAL METHODS

  • AFROUZI, G.A.;SHOKOOH, S.;CHUNG, N.T.
    • Journal of applied mathematics & informatics
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    • v.35 no.1_2
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    • pp.11-24
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    • 2017
  • Using variational methods, we study the existence and multiplicity of weak solutions for some p(x)-Laplacian-like problems. First, without assuming any asymptotic condition neither at zero nor at infinity, we prove the existence of a non-zero solution for our problem. Next, we obtain the existence of two solutions, assuming only the classical Ambrosetti-Rabinowitz condition. Finally, we present a three solutions existence result under appropriate condition on the potential F.

REGULARITY AND SINGULARITY OF WEAK SOLUTIONS TO OSTWALD-DE WAELE FLOWS

  • Bae, Hyeong-Ohk;Choe, Hi-Jun;Kim, Do-Wan
    • Journal of the Korean Mathematical Society
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    • v.37 no.6
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    • pp.957-975
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    • 2000
  • We find a regularity criterion for the Ostwald-de Waele models like Serrin's condition to the Navier-Stokes equations. Moreover, we show short time existence and estimate the Hausdorff dimension of the set of singular times for the weak solutions.

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WEAK SOLUTIONS OF THE EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY

  • Hwang, Jin-Soo;Nakagiri, Shin-Ichi
    • Journal of the Korean Mathematical Society
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    • v.44 no.2
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    • pp.443-453
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    • 2007
  • We study the equation of a membrane with strong viscosity. Based on the variational formulation corresponding to the suitable function space setting, we have proved the fundamental results on existence, uniqueness and continuous dependence on data of weak solutions.