arXiv subjects

Gaowei Cao

Publications and source records attributed to Gaowei Cao.

4 recordsLinked to original sources

arXiv2026

Global Dynamic Patterns of Entropy Solutions for One-dimensional Pressureless Euler System

In this paper, we are concerned with the fine properties of entropy solutions of the Cauchy problem for the one-dimensional pressureless Euler system, wherein the initial density $\rho_0$ is a locally finite Radon measure and the initial velocity $u_0\in L^\infty_{\rho_0}$. We employ the solution formula introduced by [F.M. Huang and Z. Wang, Comm. Math. Phys. 222(1) (2001), 117--146.] for this Cauchy problem to analyze the entropy solutions and obtain various new fine properties of entropy solutions; these can be summarized in four aspects: (i) Characteristics and initial waves for the Cauchy problem; (ii) Fine local structures of entropy solutions; (iii) Divides and global structures of entropy solutions; (iv) Invariants and asymptotic behaviors of entropy solutions including the asymptotic profile and the corresponding decay rates. Through these results (i)-(iv), we establish the global dynamic patterns of entropy solutions of the Cauchy problem for the $1$-D pressureless Euler system with general initial data $\rho_0$ being locally finite Radon measures and $u_0\in L^\infty_{\rho_0}$.

math.AP↗

arXiv2025

Well-Posedness of the Cauchy Problem for First-order Quasilinear Equations with Non-Lipschitz Source Terms and Its Applications

We are concerned with the well-posedness of the Cauchy problem for the first-order quasilinear equations with non-Lipschitz source terms and the global structures of the multi-dimensional Riemann solutions. For such quasilinear equations with initial data in $L^\infty$, when the source term $g(t, x, u)$ is only right-Lipschitz (not necessarily left-Lipschitz) in $u$, we first prove that the Kruzkov entropy condition is sufficient to guarantee the well-posdeness of entropy solutions. Next, we analyze the structures of global multi-dimensional Riemann solutions for scalar conservation laws with non-Lipschitz source terms, where the Riemann-type initial data consist of two different constant states separated by a smooth hypersurface. More precisely, we construct the global multidimensional Riemann solutions with non-selfsimilar structures, including nonselfsimilar shock waves and nonselfsimilar rarefaction waves, and prove that these two kinds of basic waves can be expressed via the implicit functions of functional equation determined by the initial discontinuity, the flux functions, and the non-Lipschitz source term. Moreover, we discover two new phenomena: (i) such kinds of basic waves can disappear in a finite time and (ii) the new-type rarefaction wave can contain some weak discontinuities in its interior. These behaviors are in stark contrast to the case of the Lipschitz source term, where these two kinds of basic waves persist globally and the rarefaction waves remain smooth in the interior. Finally, we provide two examples to respectively demonstrate the uniqueness of Riemann solutions in the case of the source term being non-Lipschitz from left (necessarily right-Lipschitz), and the non-uniqueness of Riemann solutions in the case of the source term being non-Lipschitz from right.

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arXiv2024

New Formula for Entropy Solutions for Scalar Hyperbolic Conservation Laws with Flux Functions of Convexity Degeneracy and Global Dynamic Patterns of Solutions

We are concerned with a new solution formula and its applications to the analysis of properties of entropy solutions of the Cauchy problem for one-dimensional scalar hyperbolic conservation laws, wherein the flux functions exhibit convexity degeneracy and the initial data are in $L^\infty$. We first introduce/validate the novel formula for entropy solutions for the Cauchy problem, which generalizes the Lax-Oleinik formula. Then, by employing this formula, we obtain a series of fine properties of entropy solutions and discover several new structures and phenomena, which include: (i) Series of results on the fine structures of entropy solutions, especially including the new criteria for all six types of initial waves for the Cauchy problem, the new structures of entropy solutions inside the backward characteristic triangle, and the new features of the formation and development of shocks such as all five types of continuous shock generation points, along with their criteria and the optimal regularities of the corresponding resulting shocks; (ii) Series of results on the global structures of entropy solutions, including the four new invariants of entropy solutions, the new criteria for the locations and speeds of divides, and the exact determination of the global structures of entropy solutions; (iii) Series of new results on the asymptotic behaviors of entropy solutions, including the asymptotic profiles and decay rates of entropy solutions for initial data in $L^\infty$, respectively in the $L^\infty$--norm and the $L^p_{{\rm loc}}$--norm. Through these results above, we obtain the global dynamic patterns of entropy solutions for scalar hyperbolic conservation laws with the flux functions satisfying (1.3) and general initial data in $L^\infty$. Moreover, the new solution formula is also extended to more general scalar hyperbolic conservation laws.

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arXiv2022

Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes

We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair $(\eta(u),q(u))$ with $\eta(u)$ of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in $L^\infty_{\rm loc}$ that satisfy the inequality: $\eta(u)_t+q(u)_x\leq \mu$ in the distributional sense for some non-negative Radon measure $\mu$. Furthermore, we extend this result to the class of weak solutions in $L^p_{\rm loc}$, based on the asymptotic behavior of the flux function $f(u)$ and the entropy function $\eta(u)$ at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness.

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