arXiv subjects

Zhuoping Ruan

Publications and source records attributed to Zhuoping Ruan.

6 recordsLinked to original sources

arXiv2023

Hyperbolic problems with totally characteristic boundary

We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u + \mathcal A(t,x,y,xD_x,D_y) u = f(t,x,y), \quad (t,x,y)\in(0,T)\times\mathbb R_+\times\mathbb R^d, \] where $\mathcal A(t,x,y,xD_x,D_y)$ is a first-order differential operator with coefficients smooth up to $x=0$ and the derivative with respect to $x$ appears in the combination $xD_x$. No boundary conditions are required in such a situation and corresponding initial-boundary value problems are effectively Cauchy problems. We introduce a certain scale of Sobolev spaces with asymptotics and show that the Cauchy problem for the operator $\partial_t + \mathcal A(t,x,y,xD_x,D_y)$ is well-posed in that scale. More specifically, solutions $u$ exhibit formal asymptotic expansions of the form \[ u(t,x,y) \sim \sum_{(p,k)} \frac{(-1)^k}{k!} x^{-p} \log^k \!x \, u_{pk}(t,y) \quad \text{as $x\to+0$} \] where $(p,k)\in\mathbb C\times\mathbb N_0$ and $\Re p\to-\infty$ as $|p|\to\infty$, provided that the right-hand side $f$ and the initial data $u|_{t=0}$ admit asymptotic expansions as $x \to +0$ of a similar form, with the singular exponents $p$ and their multiplicities unchanged. In fact, the coefficient $u_{pk}$ are, in general, not regular enough to write the terms appearing in the asymptotic expansions as tensor products. This circumstance requires an additional analysis of the function spaces. In addition, we demonstrate that the coefficients $u_{pk}$ solve certain explicitly known first-order symmetrizable hyperbolic systems in the lateral boundary. Especially, it follows that the Cauchy problem for the operator $\partial_t+\mathcal A(t,x,y,xD_x,D_y)$ is well-posed in the scale of standard Sobolev spaces $H^s((0,T)\times\mathbb R_+^{1+d})$.

math.AP↗

arXiv2023

A scheme for solving hyperbolic problems with symbolic structure

Hyperbolic problems can at times be solved employing symbolic arguments. This is especially true for the construction of forward (and backward) fundamental solutions. We formulate a corresponding abstract scheme and illustrate its practicality by a number of instructive examples.

math.AP↗

arXiv2016

Minimal regularity solutions of semilinear generalized Tricomi equations

We prove the local existence and uniqueness of minimal regularity solutions $u$ of the semilinear generalized Tricomi equation $\partial_t^2 u-t^m Δu =F(u)$ with initial data $(u(0,\cdot), \partial_t u(0,\cdot)) \in \dot{H^γ}(\mathbb R^n) \times \dot{H}^{γ-\frac2{m+2}}(\mathbb R^n)$ under the assumption that $|F(u)|\lesssim |u|^κ$ and $|F'(u)| \lesssim |u|^{κ-1}$ for some $κ>1$. Our results improve previous results of M. Beals [2] and of ourselves [15-17]. We establish Strichartz-type estimates for the linear generalized Tricomi operator $\partial_t^2 -t^m Δ$ from which the semilinear results are derived.

math.AP↗

arXiv2014

On the existence of low regularity solutions to semilinear generalized Tricomi equations in mixed type domains

In [19-20], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regularity solution problem for the semilinear mixed type equation $\p_t^2u-t^{2l-1}Δu= f(t,x,u)$ with an initial data $u(0,x)=φ(x)\in H^{s}(\Bbb R^n)$ ($0\le s<\f{n}{2}$), where $(t,x)\in\Bbb R \times\Bbb R^n$, $n\ge 2$, $l\in\Bbb N$, $f(t,x,u)$ is $C^1$ smooth in its arguments and has compact support with respect to the variable $x$. Under the assumption of the subcritical growth of $f(t,x,u)$ on $u$, we will show the existence and regularity of the considered solution in the mixed type domain $[-T_0, T_0] \times \R^n $ for some fixed constant $T_0>0$.

math.AP↗

arXiv2013

The existence and singularity structure of low regularity solutions of higher-order degenerate hyperbolic equations

This paper is a continuation of our previous work [21], where we have established that, for the second-order degenerate hyperbolic equation (\p_t^2-t^mΔ_x)u=f(t,x,u), locally bounded, piecewise smooth solutions u(t,x) exist when the initial data (u,\p_t u)(0,x) belongs to suitable conormal classes. In the present paper, we will study low regularity solutions of higher-order degenerate hyperbolic equations in the category of discontinuous and even unbounded functions. More specifically, we are concerned with the local existence and singularity structure of low regularity solutions of the higher-order degenerate hyperbolic equations \p_t(\p_t^2-t^mΔ_x)u=f(t,x,u) and (\p_t^2-t^{m_1}Δ_x)(\p_t^2-t^{m_2}Δ_x)v=f(t,x,v) in \R_+\times\R^n with discontinuous initial data \p_t^iu(0,x)=ϕ_i(x) (0\le i\le 2) and \p_t^jv(0,x)=ψ_j(x) (0\le j\le 3), respectively; here m, m_1, m_2\in\N, m_1\neq m_2, x\in\R^n, n\ge 2, and f is C^\infty smooth in its arguments. When the ϕ_i and ψ_j are piecewise smooth with respect to the hyperplane \{x_1=0\} at t=0, we show that local solutions u(t,x), v(t,x)\in L^{\infty}((0,T)\times\R^n) exist which are C^\infty away from \G_0\cup \G_m^\pm and \G_{m_1}^\pm\cup\G_{m_2}^\pm in [0,T]\times\R^n, respectively; here \G_0=\{(t,x): t\ge 0, x_1=0\} and the Γ_k^\pm = \{(t,x): t\ge 0, x_1=\pm \f{2t^{(k+2)/2}}{k+2}\} are two characteristic surfaces forming a cusp. When the ϕ_i and ψ_j belong to C_0^\infty(\R^n\setminus\{0\}) and are homogeneous of degree zero close to x=0, then there exist local solutions u(t,x), v(t,x)\in L_{loc}^\infty((0,T]\times\R^n) which are C^\infty away from \G_m\cup l_0 and \G_{m_1}\cup\G_{m_2} in [0,T]\times\R^n, respectively; here Γ_k=\{(t,x): t\ge 0, |x|^2=\f{4t^{k+2}}{(k+2)^2}\} (k=m, m_1, m_2) is a cuspidal conic surface and l_0=\{(t,x): t\ge 0, |x|=0\} is a ray.

math.AP↗

arXiv2012

On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

In this paper, we are concerned with the local existence and singularity structure of low regularity solutions to the semilinear generalized Tricomi equation $\p_t^2u-t^mΔu=f(t,x,u)$ with typical discontinuous initial data $(u(0,x), \p_tu(0,x))=(0, \vp(x))$; here $m\in\Bbb N$, $x=(x_1, ..., x_n)$, $n\ge 2$, and $f(t,x,u)$ is $C^{\infty}$ smooth in its arguments. When the initial data $\vp(x)$ is a homogeneous function of degree zero or a piecewise smooth function singular along the hyperplane ${t=x_1=0}$, it is shown that the local solution $u(t,x)\in L^{\infty}([0,T]\times\Bbb R^n)$ exists and is $C^{\infty}$ away from the forward cuspidal cone $Γ_0=\bigl{(t,x)\colon t>0, |x|^2=\ds\f{4t^{m+2}}{(m+2)^2}\bigr}$ and the characteristic cuspidal wedge $\G_1^{\pm}=\bigl{(t,x)\colon t>0, x_1=\pm \ds\f{2t^{\f{m}{2}+1}}{m+2}\bigr}$, respectively. On the other hand, for $n=2$ and piecewise smooth initial data $\vp(x)$ singular along the two straight lines ${t=x_1=0}$ and ${t=x_2=0}$, we establish the local existence of a solution $u(t,x)\in L^{\infty}([0,T]\times\Bbb R^2)\cap C([0, T], H^{\f{m+6}{2(m+2)}-}(\Bbb R^2))$ and show further that $u(t,x)\not\in C^2((0,T]\times\Bbb R^2\setminus(\G_0\cup\G_1^{\pm}\cup\G_2^{\pm}))$ in general due to the degenerate character of the equation under study; here $\G_2^{\pm}=\bigl{(t,x)\colon t>0, x_2=\pm\ds\f{2t^{\f{m}{2}+1}}{m+2}\bigr}$. This is an essential difference to the well-known result for solutions $v(t,x)\in C^{\infty}(\Bbb R^+\times\Bbb R^2\setminus (Σ_0\cupΣ_1^{\pm}\cup Σ_2^{\pm}))$ to the 2-D semilinear wave equation $\p_t^2v-Δv=f(t,x,v)$ with $(v(0,x), \p_tv(0,x))=(0, \vp(x))$, where $Σ_0={t=|x|}$, $Σ_1^{\pm}={t=\pm x_1}$, and $Σ_2^{\pm}={t=\pm x_2}$.

math.AP↗