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Yanbo Hu

Publications and source records attributed to Yanbo Hu.

10 recordsLinked to original sources
arXiv2026

Loss of regularity for solutions to 1D degenerate quasilinear wave equations

In this paper, we study the loss of regularity for solutions to one-dimensional degenerate wave equations. We first consider the linear equation \( u_{tt}=\bigl(c(t,x)^2u_x\bigr)_x \) with \(c(t,x)\sim x^p\), and then the quasilinear equation \( u_{tt}=(u^{2a}u_x)_x. \) The initial data are assumed to satisfy \( u_0(x)\sim x^\alpha \) and \( u_1(x)\sim x^\beta \) near the degenerate point \(x=0\). In the linear problem, \(p\) describes the strength of the degeneracy, while \(\alpha\) and \(\beta\) describe the regularity of the initial data. In the quasilinear problem, \(\alpha\) also determines the initial degeneracy through \(u_0(x)^a\sim x^{a\alpha}\). Our main concern is the actual occurrence of loss of regularity, namely, the phenomenon in which the regularity of \(u(t,\cdot)\) near \(x=0\) becomes lower than that of \(u_0\) for \(t>0\). Previously, such a loss had mainly been established for special linear equations with coefficients depending only on time. In our previous work on the quasilinear equation, we proved local well-posedness without loss of regularity when \(\beta\geq\alpha\). In the present paper, we show that this condition is also necessary. More precisely, if \(\beta<\alpha\), then \( C_1tx^\beta\leq u(t,x)\leq C_2x^\beta \) near \(x=0\) for sufficiently small positive time. Thus the order of the solution changes from that of \(u_0\) to that of \(u_1\), and an actual loss of regularity occurs for both linear equations with time-space dependent coefficients and degenerate quasilinear equations.

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arXiv2026

Existence and singularity formation for the supersonic expanding wave of radially symmetric non-isentropic compressible Euler equations

This paper studies the existence and singularity formation of supersonic expanding waves for the radially symmetric non-isentropic compressible Euler equations of polytropic gases. We introduce a suitable pair of gradient variables to characterize the rarefaction and compression properties of the solutions. Based on their Riccati equations, we construct several useful invariant domains to establish a series of priori estimates of solutions under some assumptions on the initial data. We show that the solution is smooth in the characteristic triangle or quadrangle domain if both of these two gradient variables are non-negative at the initial time. On the other hand, when one of these two variables is very negative at some initial point, the solution forms a singularity in finite time.

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arXiv2024

Global solution and singularity formation for the supersonic expanding wave of compressible Euler equations with radial symmetry

In this paper, we define the rarefaction and compression characters for the supersonic expanding wave of the compressible Euler equations with radial symmetry. Under this new definition, we show that solutions with rarefaction initial data will not form shock in finite time, i.e. exist global-in-time as classical solutions. On the other hand, singularity forms in finite time when the initial data include strong compression somewhere. Several useful invariant domains will be also given.

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arXiv2023

Initial-boundary value problems for Poiseuille flow of nematic liquid crystal via full Ericksen-Leslie model

In this paper, we study the initial-boundary value problem for the Poiseuille flow of hyperbolic-parabolic Ericksen-Leslie model of nematic liquid crystals in one space dimension. Due to the quasilinearity, the solution of this model in general forms cusp singularity. We prove the global existence of H\"older continuous solution, which may include cusp singularity, for initial-boundary value problems with different types of boundary conditions.

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arXiv2021

On a supersonic-sonic patch arising from the Frankl problem in transonic flows

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an airfoil. Based on the methodology of characteristic decompositions, we establish the global existence and regularity of solutions in a partial hodograph coordinate system in terms of angle variables. The original problem is solved by transforming the solution in the partial hodograph plane back to that in the physical plane. Moreover, the uniform regularity of the solution and the regularity of an associated sonic curve are also verified.

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arXiv2019

On the degenerate Cauchy problem for a nonlinear variational wave system, Part I: The same wave speed case

We investigate a one-dimensional nonlinear wave system which arises from a variational principle modeling a type of cholesteric liquid crystals. The problem treated here is the Cauchy problem for the same wave speed case with initial data on the parabolic degenerating line. By introducing a partial hodograph transformation, we establish the local existence of smooth solutions in a weighted metric space based on the iteration method. A classical solution of the primary problem is constructed by converting the solution in the partial hodograph variables to that in the original variables.

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arXiv2019

Singularity for a multidimensional variational wave equation arising from nematic liquid crystals

This article is focused on a multidimensional nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle arising form the theory of nematic liquid crystals. By using the method of characteristics, we show that the smooth solutions for the spherically-symmetric variational wave equation breakdown in finite time, even for the arbitrarily small initial energy.

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arXiv2019

Sonic-supersonic solutions for the two-dimensional steady full Euler equations

This paper focuses on the structure of classical sonic-supersonic solutions near sonic curves for the two-dimensional full Euler equations in gas dynamics. In order to deal with the parabolic degeneracy near the sonic curve, a novel set of dependent and independent variables are introduced to transform the Euler equations into a new system of governing equations which displays a clear regularity-singularity structure. With the help of technical characteristic decompositions, the existence of a local smooth solution for the new system is first established in a weighted metric space by using the iteration method and then expressed in terms of the original physical variables. This is the first time to construct a classical sonic-supersonic solution near a sonic curve for the full Euler equations.

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arXiv2019

On a global supersonic-sonic patch characterized by 2-D steady full Euler equations

Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and a possible sonic curve. This paper establishes the global existence of solutions in a whole supersonic-sonic patch characterized by the two-dimensional full system of steady Euler equations and studies solution behaviors near sonic curves, depending on the proper choice of boundary data extracted from the airfoil problem and related contexts. New characteristic decompositions are developed for the full system and a delicate local partial hodograph transformation is introduced for the solution estimates. It is shown that the solution is uniformly $C^{1,\frac{1}{6}}$ continuous up to the sonic curve and the sonic curve is also $C^{1,\frac{1}{6}}$ continuous.

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