Hi Jun Choe

Strong solutions of the Navier–Stokes equations for isentropic compressible fluids

We study strong solutions of the isentropic compressible Navier–Stokes equations in a domain Ω⊂R3. We first prove the local existence of unique strong solutions provided that the initial data ρ0 and u0 satisfy a natural compatibility condition. The important point in this paper is that we allow the initial vacuum: the initial density may vanish in an open subset of Ω. We then prove a new uniqueness result and stability result. Our results are valid for unbounded domains as well as bounded ones.

Strong Solutions of the Navier–Stokes Equations for Nonhomogeneous Incompressible Fluids

Abstract We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ⊂ R 3. Deriving higher a priori estimates independent of the lower bounds of the density, we prove the existence and uniqueness of local strong solutions to the initial value problem (for Ω =R 3) or the initial boundary value problem (for Ω ⊂ ⊂ R 3) even though the initial density vanishes in an open subset of Ω, i.e., an initial vacuum exists. As an immediate consequence of the a priori estimates, we obtain a continuation theorem for the local strong solutions.

Interior behaviour of minimizers for certain functionals with nonstandard growth

for all t E [0, 00). For example we see that f(t2) = atP + btq, a>O,b>O,l , P> 1, satisfy the ellipticity condition (2). Recently a number of authors have considered regularity questions of minimizers for Z under nonstandard growth conditions onf(see [l-7]). Indeed in the scalar cases Choe [3] has already obtained C’** regularity result for minimizers of Z when f(t2) = tP log(1 + t), p > 1, using Lewis’ method [8]. In this paper we prove that a locally bounded minimizer u is C,:F under the assumption that l

A Regularity Criterion for the Navier–Stokes Equations

We prove that a weak solution u = (u 1, u 2, u 3) to the Navier–Stokes equations is strong, if any two components of u satisfy Prodi–Ohyama–Serrins criterion. As a local regularity criterion, we prove u is bounded locally if any two components of the velocity lie in L 6, ∞.

On the obstacle problem for quasilinear elliptic equations of p Laplacian type

This article considers an obstacle problem for quasilinear elliptic equations of p Laplacian type. It is shown, under certain smoothness assumptions on the obstacle, that the solution to the corresponding obstacle problem has interior Holder continuous derivatives.

REGULARITY OF WEAK SOLUTIONS OF THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

In this paper, we assume a density with integrability on the space (0, T; ) for some and T > 0. Under the assumption on the density, we obtain a regularity result for the weak solutions to the compressible Navier-Stokes equations. That is, the supremum of the density is finite and the infimum of the density is positive in the domain (0, T). Moreover, Moser type iteration scheme is developed for norm estimate for the velocity.

Dirichlet Problem for the Stationary Navier-Stokes System on Lipschitz Domains

We consider the stationary Navier-Stokes system on a bounded Lipschitz domain Ω in R 3 with connected boundary ∂Ω. The main concern is the solvability of the Dirichlet problem with external force and boundary data having minimal regularity, i.e., and . Here denotes the standard Sobolev space with the pair (s, q) being admissible for the unique solvability in of the Stokes system. We show that if 1 + s ≥ 2/q in addition, then for any and satisfying the necessary compatibility condition, there exists at least one solution in of the Dirichlet problem and this solution has a complete regularity property. The uniqueness of solutions is also shown under the smallness condition on the corresponding norms of the data.

Isolated Singularity for the Stationary Navier—Stokes System

Abstract. In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in

Regularity for Certain Nonlinear Parabolic Systems

B_R \setminus \{0\}

A long time asymptotic behavior of the free boundary for an American put

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