arXiv subjects

Huicheng Yin

Publications and source records attributed to Huicheng Yin.

At least 19 recordsLinked to original sources

arXiv2026

Global small data radial symmetric solutions of 3D semilinear Euler-Poisson-Darboux equations

For the 3D semilinear Euler-Poisson-Darboux equation $\square u+\fracμ{t}\partial_tu=|u|^p$, where $t\geq1$, $μ>0$ and $p>1$, it is conjectured that there is a critical exponent $p_{crit}(3,μ)=\max\{p_s(3+μ), p_f(3)\}$ with the Strauss exponent $p_s(3+μ)=\frac{μ+4+\sqrt{μ^2+16μ+32}}{2(μ+2)}$ and the Fujita exponent $p_f(3)=\frac{5}{3}$ such that when $p>p_{crit}(3,μ)$, the small data solution $u$ exists globally, otherwise, when $1 p_{crit}(3,μ)$. Note that $p_{crit}(3,μ)=p_s(3+μ)$ for $0<μ<\frac{14}{5}$ and $p_{crit}(3,μ)=p_f(3)$ for $μ\geq\frac{14}{5}$. In the recent paper [16], the authors have obtained the global small solution $u$ for $p>\max\{\frac{5}{3},1+\frac{2}μ\}$ and $μ\geq\frac{14}{5}$. In this paper, by utilizing the hypergeometric Riemann representation and establishing some delicate pointwise spacetime weighted estimates, we prove the global existence of small data radial solution $u$ in the remaining range of $\frac{5}{3} p_{crit}(3,μ)=\frac{5}{3}$.

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arXiv2026

Shock formation for 3D steady supersonic flows with general short pulse data

This paper concerns the shock formation problem for the 3D steady supersonic potential equation of polytropic gases. The potential equation is described by a second order quasilinear wave equation $\displaystyle\sum_{i=1}^{3}\big[(\partial_iΦ)^2 - c^2(ρ)\big]\partial_i^2Φ+ 2\displaystyle\sum_{1\le i 1$), and $\partial_3Φ> c(ρ)$. For the short pulse boundary data $Φ|_{x^3=0} = δ^νΦ_0\big(\frac{r-1}δ,ω\big)$ and $\partial_3Φ|_{x^3=0}=q_0+δ^{ν-1}Φ_1\big(\frac{r-1}δ,ω\big)$ with $r=\sqrt{(x^1)^2+(x^2)^2}$, $ω=\big(\frac{x^1}{r},\frac{x^2}{r}\big)\in\mathbb{S}$, $1<ν<2$ and small $δ>0$, it is shown that a shock will be formed in a finite $x^3$-distance as long as the boundary data are supersonic and satisfy $(Φ_0,Φ_1)\not\equiv 0$. This coincides with physical phenomenon that strong compression of supersonic polytropic gases yields shocks. One of our main ingredients is to find a good unknown so that the previously imposed compatibility conditions on the short pulse initial data are removed as well as the required weighted energy estimates in the existing literatures are derived. It is expected that the method here will be applied to study the shock formation problem with general short pulse initial data for the 3D steady supersonic Euler equations of polytropic gases.

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arXiv2026

Global classical solutions to 3D irrotational compressible Euler equations of Chaplygin gases with weakly decaying initial data

We are concerned with the global classical solution problem of 3D compressible isentropic Euler equations of Chaplygin gases \[ \begin{cases} \partial_tρ+ \mathrm{div}(ρv) = 0,\\ \partial_t(ρv) + \mathrm{div}(ρv \otimes v) + \nabla p = 0,\\ ρ(0,x) = \barρ+ \varepsilonρ_0(x),\ v(0,x) = \varepsilon v_0(x). \end{cases} \] where $\barρ>0$ is a constant, $\varepsilon>0$ is small, the state equation is $p=p(ρ)=P_0-\frac{D}ρ$ with $P_0$ and $D$ being some positive constants. For the 3D compressible Euler equations of Chaplygin gases, which are a prototype of multidimensional nonlinear symmetric hyperbolic systems with totally linearly degenerate eigenvalues, there is a basic conjecture imposed by A. Majda: it typically has a global classical solution $(ρ, v)$ with $(ρ-\barρ, v)\in C([0,\infty), H^s(\Bbb R^3)) \cap C^1([0,\infty), H^{s-1}(\Bbb R^3))$ when $(ρ_0, v_0)\in H^s(\Bbb R^3)$ with $s>\frac52$ unless $(ρ, v)$ itself blows up in finite time. In this paper, under the assumptions that for any fixed constant $μ$ with $0<μ<1/2$, integer $N\geq 15$, $\mathrm{rot}\,v_0(x) \equiv 0$ and \[ \|(ρ_0, v_0)\|_{H^{N}(\mathbb{R}^3)}+\sum_{|a|\leq 13} \|\langle x\rangle^{1+μ} \nabla^a(ρ_0,v_0)\|_{L^2(\mathbb{R}^3)} \leq 1, \] we show that the classical solution $(ρ, v)$ exists globally. Our main ingredients include: establishing a series of new decay estimates of energy bounds, weighted pointwise space-time $L^\infty$-$L^2$ estimates and weighted Strichartz-type estimates for the 3D potential flow equation of Chaplygin gases.

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arXiv2026

Global existence of small data solutions to 3-D semilinear Euler-Poisson-Darboux equations

There is an interesting open question: for $n$-D ($n\ge 1$) semilinear Euler-Poisson-Darboux equation $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $μ>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,μ)=\max\{p_s(n+μ), p_f(n)\}$ with the Strauss exponent $p_{s}(n+μ)=\frac{n+μ+1+\sqrt{(n+μ)^2+10(n+μ)-7}}{2(n+μ-1)}$ and the Fujita exponent $p_f(n)=1+\frac{2}{n}$. The blowup of weak solution $u$ has been shown when $1 \max\{\frac53, 1+\frac{2}μ\}$.

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arXiv2026

The sharp lifespan of small data smooth solutions to 2-D quadratic quasilinear wave equations in exterior domains

In the paper [M. Keel, H. Smith, C.D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions. J. Amer. Math. Soc. 17 (2004), no. 1, 109-153], the authors prove that for the 3-D quadratic quasilinear wave equation in exterior domains with homogenous Dirichlet boundary value and small initial data of size $\varepsilon$, the lifespan ${\bar T}_{\varepsilon}$ of the smooth solution fulfills ${\bar T}_{\varepsilon}\ge e^{C/\varepsilon}$. However, for the corresponding 2-D quadratic quasilinear wave equation in exterior domains with homogenous Dirichlet or Neumann boundary value, so far it is still open whether the expected sharp lifespan $T_{\varepsilon}\ge\frac{C}{\varepsilon^2}$ holds or not. In this paper, we will solve this open question. Our main ingredients include: introducing the suitable Friedlander radiation field for the 2-D linear wave equation in exterior domains with homogenous Dirichlet or Neumann boundary value, constructing the delicate approximate solution, and establishing some crucial space-time decay estimates for the solutions of 2-D quasilinear wave equation in exterior domains. On the other hand, for the radial symmetric solutions to a class of 2-D quadratic quasilinear wave equation in exterior domains, the upper bound of the lifespan $T_{\varepsilon}\le\frac{C}{\varepsilon^2}$ is derived and the sharp constant $C$ is also determined explicitly.

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arXiv2026

Long time smooth solutions of 3D cubic quasilinear wave systems with small weakly decaying initial data

For the 3D cubic quasilinear wave system $\square_{c_i} u^i=G^i(u,\partial u,\partial^2u)=\displaystyle\sum_{\substack{0\le|α|,|β|,|γ|\le1 \\ 1\le j,k,l \le m}}g_{αβγ}^{ijkl}\partial^αu^j\partial^βu^k\partial^γu^l$, it is well known that global solution $u$ exists when the small smooth initial data $(u,\partial_tu)|_{t=0}$ $=(u_0(x), u_1(x))$ are compactly supported or decay rapidly at spatial infinity. However, when $(u_0, u_1)\in (H^{s+1}, H^s)$ with $s>\frac{5}{2}$ are small, it remains unknown whether $u$ exists globally or not. In this paper, we show that if $\|u_{0}\|_{H^{N+1}}+\|u_{1}\|_{H^N}\le\varepsilon$ ($N\ge 6$) is small, then the almost global solution $u$ exists in $[0, T_{\varepsilon}]$ with $T_{\varepsilon}\ge e^{C\varepsilon^{-1}}$ for the general $G(u,\partial u,\partial^2u)$ depending on $u$ and $T_{\varepsilon}\ge e^{C\varepsilon^{-2}}$ for the nonlinearity $G(\partial u,\partial^2u)$ independent of $u$, respectively. In addition, if $\displaystyle\sum_{|a|\le 5}\|\langle x\rangle^μ\partial^a_x(u _0,u_1)\|_{L^2}\le\varepsilon$ holds for any fixed constant $μ\in (0,1)$, then the solution $u$ exists globally and meanwhile the scattering property of $u$ is derived. Our main ingredients consist in establishing a series of new weighted $L^\infty-L^2$ estimates and Strichartz estimates based on the strong Huygens' principle for 3D linear wave equations.

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arXiv2026

Long time smooth solutions of 2-D quadratic quasilinear wave equations in exterior domains with Neumann boundary conditions

For the 3-D quadratic quasilinear wave equations in exterior domains with Dirichlet or Neumann boundary conditions, the global existence or the maximal existence time of small data smooth solutions have been established in the past. However, so far it is still open for the corresponding 2-D Neumann boundary value problem. In this paper, we investigate the long time existence of small data solutions to 2-D quadratic quasilinear wave equations with homogeneous Neumann boundary values. Our main ingredients include: establishing some new pointwise spacetime decay estimates for the 2-D initial boundary value problem of the divergence form wave equations, and introducing a series of good unknowns to derive the required energy estimates. The obtained results can be directly applied to the initial boundary value problem of 2-D isentropic and irrotational compressible Euler equations for both the polytropic gases and the Chaplygin gases in exterior domains with impermeable conditions, the 2-D relativistic membrane equations and 2-D membrane equations with homogeneous Neumann boundary values.

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arXiv2026

Global smooth solutions of 2-D quadratic quasilinear wave equations with null conditions in exterior domains

For 3-D quadratic quasilinear wave equations with or without null conditions in exterior domains, when the compatible initial data and Dirichlet boundary values are given, the global existence or the maximal existence time of small data smooth solutions have been established in early references. For the Cauchy problem of 2-D quadratic quasilinear wave equations with null conditions, it has been shown that the small data smooth solutions exist globally. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. In the present paper, we solve this open problem through proving the global existence of small solutions in exterior domains. Our main ingredients include: deriving new precise pointwise estimates for the initial boundary value problem of 2-D linear wave equations in exterior domains; finding appropriate divergence structures of quasilinear wave equations under null conditions; introducing a good unknown to eliminate the resulting $Q_0$ type nonlinearity, and establishing some crucial pointwise spacetime decay estimates of solutions and their derivatives.

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arXiv2026

Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, II

In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy problem of 2-D quadratically quasilinear wave equations with null conditions. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. When the 2-D quadratic nonlinearity admits a special $Q_0$ type null form, the global small solution is shown in our previous article [Hou Fei, Yin Huicheng, Yuan Meng, Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, arXiv:2411.06984]. In the present paper, we now solve this open problem through proving the global existence of small solutions to 2-D general quasilinear wave equations with null conditions in exterior domains. Our proof procedure is based on finding appropriate divergence structures of quasilinear wave equations under null conditions, introducing a good unknown to eliminate the resulting $Q_0$ type nonlinearity and deriving some new precise pointwise spacetime decay estimates of solutions and their derivatives.

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arXiv2025

Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, III

For the $2$-D semilinear wave equation with scale-invariant damping $\square u+\fracμ{t}\partial_tu=|u|^p$, where $t\geq 1$, $μ>0$ and $p>1$, it is conjectured that the global small data weak solution $u$ exists when $p>p_{s}(2+μ) =\frac{μ+3+\sqrt{μ^2+14μ+17}}{2(μ+1)}$ for $0<μ\leq 2$ and $p>p_f(2)=2$ for $μ\geq 2$. In our previous papers, the global small solution $u$ has been obtained for $p>p_{s}(2+μ)$ and $0<μ<2$ but $μ\not=1$. In the present paper, by the vector field method together with the delicate analysis on the Bessel functions, we will show the global existence of small solution $u$ for $p>2$ and $μ>2$. In forthcoming paper, for $μ=1$ and $p>p_{s}(2+μ)=p_{s}(3)=1+\sqrt 2$, the global solution $u$ is also obtained. Therefore, collecting our series of conclusions together with partial results from others, this open question has been solved completely.

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arXiv2025

Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, II

For the $2$-D semilinear wave equation with scale-invariant damping $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$ and $p>1$, in the paper [T. Imai, M. Kato, H. Takamura, K. Wakasa, The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions, J. Differential Equations 269 (2020), no. 10, 8387-8424], it is conjectured that the global small data weak solution $u$ exists when $p>p_{s}(2+μ) =\frac{μ+3+\sqrt{μ^2+14μ+17}}{2(μ+1)}$ for $μ\in (0, 2)$ and $p>p_f(2)=2$ for $μ\geq 2$. In our previous paper, the global small solution $u$ has been obtained for $p_{s}(2+μ) 2, p>2$ or $μ=1, p>p_s(μ+2)=1+\sqrt 2$.

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arXiv2025

Formation and construction of large variational shock waves for 1-D $n\times n$ quasilinear hyperbolic conservation systems

In the paper [Li Jun, Xu Gang, Yin Huicheng, On the blowup mechanism of smooth solutions to 1D quasilinear strictly hyperbolic systems with large variational initial data, Nonlinearity 38 (2025), No.2, 025016], for the 1-D $n\times n$ ($n\geqslant 3$) strictly hyperbolic system $\partial_tv+F(v)\partial_xv=0$ with some classes of large variational initial data $v(x, 0)$, the geometric blowup mechanism and the detailed singularity behaviours of $\partial_{x,t}v$ near the blowup point are studied when the $n\times n$ matrix $F(v)$ admits at least one genuinely nonlinear eigenvalue. In this paper, we focus on the formation and construction of a large variational shock wave from the blowup point for 1-D $n\times n$ quasilinear hyperbolic conservation law system $\partial_tu+\partial_xf(u)=0$ when some smooth simple wave solution is generic non-degenerate before the formation of singularity and the corresponding eigenvalue is genuinely nonlinear.

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arXiv2025

Global smooth solutions to 4D quasilinear wave equations with short pulse initial data

In this paper, we establish the global existence of smooth solutions to general 4D quasilinear wave equations satisfying the first null condition with the short pulse initial data. Although the global existence of small data solutions to 4D quasilinear wave equations holds true without any requirement of null conditions, yet for short pulse data, in general, it is sufficient and necessary to require the fulfillment of the first null condition to have global smooth solutions. It is noted that short pulse data are extensions of a class of spherically symmetric data, for which the smallness restrictions are imposed on angular directions and along the outgoing directional derivative $\partial_t+\partial_r$, but the largeness is kept for the incoming directional derivative $\partial_t-\partial_r$. We expect that here methods can be applied to study the global smooth solution or blowup problem with short pulse initial data for the general 2D and 3D quasilinear wave equations when the corresponding null conditions hold or not. On the other hand, as some direct applications of our main results, one can show that for the short pulse initial data, the smooth solutions to the 4D irrotational compressible Euler equations for Chaplygin gases, 4D nonlinear membrane equations and 4D relativistic membrane equations exist globally since their nonlinearities satisfy the first null condition; while the smooth solutions to the 4D irrotational compressible Euler equations for polytropic gases generally blow up in finite time since the corresponding first null condition does not hold.

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arXiv2025

The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein-Gordon equations

It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\square u + u = F(u,{\partial u})$ with cubic nonlinearities. However, for the short pulse initial data $(u, \partial_tu)(0, x)=({δ^{ν+1}}{u_0}({\frac{x}δ}),{δ^ν}{u_1}({\frac{x}δ}))$ with $ν\in\Bbb R$ and $(u_0, u_1)\in C_0^{\infty}(\Bbb R)$, which are a class of large initial data, we establish that when $ν\le -\frac{1}{2}$, the solution $u$ can blow up in finite time for some suitable choices of $(u_0, u_1)$ and cubic nonlinearity $F(u,{\partial u})$; when $ν>-\frac{1}{2}$, the smooth solution $u$ exists globally. Therefore, $ν=-\frac{1}{2}$ is just the critical power corresponding to the global existence or blowup of smooth short pulse solutions for the cubic semilinear Klein-Gordon equations.

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arXiv2025

Formation and construction of a shock wave for one dimensional $n\times n$ strictly hyperbolic conservation laws with small smooth initial data

Under the genuinely nonlinear assumption for 1-D $n\times n$ strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic nondegenerate condition. At first, near the unique blowup point we give a precise description on the space-time blowup rate of the smooth solution and meanwhile derive the cusp singularity structure of characteristic envelope. These results are established through extending the smooth solution of the completely nonlinear blowup system across the blowup time. Subsequently, by utilizing a new form on the resulting 1-D strictly hyperbolic system with $(n-1)$ good components and one bad component, together with the choice of an efficient iterative scheme and some involved analyses, a weak entropy shock wave starting from the blowup point is constructed. As a byproduct, our result can be applied to the shock formation and construction for the 2-D supersonic steady compressible full Euler equations ($4\times 4$ system), 1-D MHD equations ($5\times 5$ system), 1-D elastic wave equations ($6\times 6$ system) and 1-D full ideal compressible MHD equations ($7\times 7$ system).

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arXiv2025

Global existence of small data weak solutions to the semilinear wave equations with time-dependent scale-invariant damping

In this paper, we are concerned with the global existence of small data weak solutions to the $n-$dimensional semilinear wave equation $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$ with time-dependent scale-invariant damping, where $n\geq 2$, $t\geq 1$, $μ\in(0,1)\cup(1,2]$ and $p>1$. This equation can be changed into the semilinear generalized Tricomi equation $\partial_t^2u-t^mΔu=t^{α(m)}|u|^p$, where $m=m(μ)>0$ and $α(m)\in\Bbb R$ are two suitable constants. At first, for the more general semilinear Tricomi equation $\partial_t^2v-t^mΔv=t^α|v|^p$ with any fixed constant $m>0$ and arbitrary parameter $α\in\Bbb R$, we shall show that in the case of $α\leq -2$, $n\geq 3$ and $p>1$, the small data weak solution $v$ exists globally; in the case of $α>-2$, through determining the conformal exponent $p_{conf}(n,m,α)>1$, the global small data weak solution $v$ exists when some extra restrictions of $p\geq p_{conf}(n,m,α)$ are given. Returning to the original equation $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, the corresponding global existence results on the small data solution $u$ can be obtained.

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arXiv2025

Global existence for 2-D wave maps equation in exterior domains

In the paper [H. Kubo, Global existence for exterior problems of semilinear wave equations with the null condition in 2D, Evol. Equ. Control Theory 2 (2013), no. 2, 319-335], for the 2-D semilinear wave equation system $(\partial_t^2-Δ)v^I=Q^I(\partial_tv, \nabla_xv)$ ($1\le I\le M$) in the exterior domain with Dirichlet boundary condition, it is shown that the small data smooth solution $v=(v^1, \cdot\cdot\cdot, v^M)$ exists globally when the cubic nonlinearities $Q^I(\partial_tv, \nabla_xv)=O(|\partial_tv|^3+|\nabla_xv|^3)$ satisfy the null condition. We now focus on the global Dirichelt boundary value problem of 2-D wave maps equation with the form $\Box u^I=\sum_{J,K,L=1}^MC_{IJKL}u^JQ_0(u^K,u^L)$ $(1\le I\le M)$ and $Q_0(f,g)=\partial_tf\partial_tg-\sum_{j=1}^2\partial_jf\partial_jg$ in exterior domain. By establishing some crucial classes of pointwise spacetime decay estimates for the small data solution $u=(u^1, \cdot\cdot\cdot, u^M)$ and its derivatives, the global existence of $u$ is shown.

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arXiv2024

Global existence and scattering of small data smooth solutions to quasilinear wave systems on $\mathbb{R}^2\times\mathbb{T}$, II

In our previous paper [Fei Hou, Fei Tao, Huicheng Yin, Global existence and scattering of small data smooth solutions to a class of quasilinear wave systems on $\mathbb{R}^2\times\mathbb{T}$, Preprint (2024), arXiv:2405.03242], for the $Q_0$-type quadratic nonlinearities, we have shown the global well-posedness and scattering properties of small data smooth solutions to the quasilinear wave systems on $\mathbb{R}^2\times\mathbb{T}$. In this paper, we start to solve the global existence problem for the remaining $Q_{αβ}$-type nonlinearities. By combining these results, we have established the global well-posedness of small solutions on $\mathbb{R}^2\times\mathbb{T}$ for the general 3-D quadratically quasilinear wave systems when the related 2-D null conditions are fulfilled.

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