Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the $L^1-$estimate between the entropy solution and the geometric optics expansion function is bounded by $O(\varepsilon^2)$, {\it independent of} the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.Comment: 30 pages, 2 figure

Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls

We are concerned with the two-dimensional steady supersonic reacting Euler flow past Lipschitz bending walls that are small perturbations of a convex one, and establish the existence of global entropy solutions when the total variation of both the initial data and the slope of the boundary is sufficiently small. The flow is governed by an ideal polytropic gas and undergoes a one-step exothermic chemical reaction under the reaction rate function that is Lipschtiz and has a positive lower bound. The heat released by the reaction may cause the total variation of the solution to increase along the flow direction. We employ the modified wave-front tracking scheme to construct approximate solutions and develop a Glimm-type functional by incorporating the approximate strong rarefaction waves and Lipschitz bending walls to obtain the uniform bound on the total variation of the approximate solutions. Then we employ this bound to prove the convergence of the approximate solutions to a global entropy solution that contains a strong rarefaction wave generated by the Lipschitz bending wall. In addition, the asymptotic behavior of the entropy solution in the flow direction is also analyzed.Comment: 58 pages, 16 figures; SIAM J. Math. Anal. (accepted on November 1, 2016

Cylindrical Symplectic Representation and Global Regular Solution of Incompressible Navier-Stokes Equations in R3\mathbb{R}^3

The existence and uniqueness of global regular solution of incompressible Navier-Stokes equations in $\mathbb{R}^3$ are derived provided the initial velocity vector field holds a special structure.Comment: 28 pages. arXiv admin note: text overlap with arXiv:2305.1373

Symplectic Symmetry and Radial Symmetry Either Persistence or Breaking of Incompressible Fluid

The incompressible Navier-Stokes equations are considered. We find that these equations have symplectic symmetry structures. Two linearly independent symplectic symmetries form moving frame. The velocity vector possesses symplectic representation in a moving frame. The symplectic representation of two-dimensional Navier-Stokes equations holds radial symmetry persistence. On the other hand, we establish some results of radial symmetry either persistence or breaking for the symplectic representations of three-dimensional Navier-Stokes equations. Thanks radial symmetry persistence, we construct infinite non-trivial solutions of static Euler equations with given boundary condition.Comment: 60 page