Research Field (Fluid Dynamics)

1. The CNSS: Existence and regularities on bounded domains with corners

The first regularity result for the compressible viscous flows on a non-convex polygon is the following paper, worked with
R. B. Kellogg:

Here non-zero velocity value is assigned on a portion of the boundary, points into the region and the pressure function is obtained by integrating the continuity equation along the streamline in the direction that the non-zero velocity vector points into the region. By extracting the corner singularity of the Laplacian type, the regularity for the remainder of the solution is obtained. Later we extended this to the barotropic CNSS and the whole CNSS on the non-convex polygonal domains:

Next I myself investigated the corner singularity and regularity for the compressible viscous flows by regularizing the continuity equation, which is given in the following paper:

In next paper I studied the compressible Stokes problem with zero boundary value on convex polygonal domains by subtracting the corner singularity function by the Stokes equation. When the domain has a non-convex corner and the velocity boundary datum vanishes and the inflow boundary is empty, it is still an open problem to show existence of the nonlinear problem and to determine the possible interior layers.

In next paper I study the compressible Stokes problem with inflow boundary condition on a non-convex polyhedral cylinder in R^3 and show the regularity result by extracting the (edge) corner singularity where the stress intensity factor is not constant any more and a function of the edge variable:

Next papers investigate the issues concentrating on the jump discontinuity behavior of solutions caused by the corner singularity functions of the viscous terms and the transport phenomena of the continuity equation:

Recently, in next paper, we have resolved a very interesting problem occurred in this research direction. The problem may be thought of as a free boundary problem to resolve the curve(depending on the solution itself) emanating from the non-convex corner. A solution for the compressible viscous steady state Navier Stokes system is given for which the density has a jump discontinuity across a curve inside the domain of the problem. There are corresponding jumps in derivatives of the velocity. The solution comes from a well-posed boundary value problem on a polygonal domain with a concave corner. A formula for the decay of the jump is given. The decay formula suggests that density jumps may be noticeable in a high speed flow occurring in a very viscous, very compressible fluid.

  • J. R. Kweon. Jump discontinuities of compressible viscous flows grazing a non-convex corner, J. Math. Pures Appl. (9) 100 (2013), no. 3, 410-432.
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2. The incompressible viscous Navier-Stokes flows

Next paper demonstrates the regularity of solution for the Navier-Stokes system on any polygonal domains. It is thought that it is a critical contribution for the incompressible flows on domains with corners and will have many applications, for instance, in resolving the convergence rates of the numerical solutions, etc.

3. The time-dependent compressible viscous flows

In next papers I study the corner singularity and regularity of the solution for the time-dependent compressible viscous flows on domains with corners. The stress intensity factor that is the coefficient of the singularity function is a function of time variable and shown to have certain regularity of the fractional order:

Next paper shows the regularity result for the solution of the time-dependent whole CNSS on non-convex polygonal domains. It is a complete analysis for existence and regularity for the solution of the nonlinear problem on bounded domains with corners. The method used is different from the ones employed in above two papers.

  • J. R. Kweon. Corner singularity dynamics and regularity of compressible viscous Navier-Stokes flows,  SIAM J. Math. Anal. 44 (2012) 3127-3161.
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4. The stationary compressible viscous flows on smooth domains:

Next paper is an interesting regularity result of the CNSS in the (nearly) unit ball in R^3. The geometry of the boundary blocks further regularity for the solution to have as permitted by the data.

5. Discontinuous solutions for the CNSS on bounded domains

6. The Heat equation on a polyhedral cylinder in R^3

  • J. R. Kweon. Edge singular behavior for the Heat equation on polyhefral cylinders in R^3, Potential Anal. 38 (2013), no. 2, 589-610.
  • J. R. Kweon. For the Parabolic Lame System on Polygonal Domains: The transort equation, J. Differential Equations 255 (2013), no. 6, 1109-1131.
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