arXiv subjects

Bingbing Ding

Publications and source records attributed to Bingbing Ding.

10 recordsLinked to original sources

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 $\displaystyle\sum_{i=1}^{3}\big[(\partial_i\Phi)^2 - c^2(\rho)\big]\partial_i^2\Phi + 2\displaystyle\sum_{1\le i 1$), and $\partial_3\Phi > c(\rho)$. For the short pulse boundary data $\Phi|_{x^3=0} = \delta^\nu\Phi_0\big(\frac{r-1}{\delta},\omega\big)$ and $\partial_3\Phi|_{x^3=0}=q_0+\delta^{\nu-1}\Phi_1\big(\frac{r-1}{\delta},\omega\big)$ with $r=\sqrt{(x^1)^2+(x^2)^2}$, $\omega=\big(\frac{x^1}{r},\frac{x^2}{r}\big)\in\mathbb{S}$, $1<\nu<2$ and small $\delta>0$, 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 $(\Phi_0,\Phi_1)\not\equiv 0$. 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

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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arXiv2024

Global smooth solutions of 2D quasilinear wave equations with higher order null conditions and short pulse initial data

For the short pulse initial data with a first order outgoing constraint condition and optimal orders of smallness, we establish the global existence of smooth solutions to 2D quasilinear wave equations with higher order null conditions. Such kinds of wave equations include 2D relativistic membrane equations, 2D membrane equations, and some 2D quasilinear equations which come from the nonlinear Maxwell equations in electromagnetic theory or from the corresponding Lagrangian functionals as perturbations of the Lagrangian densities of linear wave operators. The main ingredients of the analysis here include looking for a new good unknown, finding some key identities based on the higher order null conditions and the resulting null frames, as well as overcoming the difficulties due to the slow decay of solutions to the 2-D wave equation, so that the solutions can be estimated precisely.

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arXiv2024

On global smooth solutions to the 2D isentropic and irrotational Chaplygin gases with short pulse data

This paper establishes the global existence of smooth solutions to the 2D isentropic and irrotational Euler equations for Chaplygin gases with a general class of short pulse initial data, which, in particular, resolves in this special case, the Majda's conjecture on the non-formation of shock waves of solutions from smooth initial data for multi-dimensional nonlinear symmetric systems which are totally linearly degenerate. Comparing to the 4D case, the major difficulties in this paper are caused by the slower time decay and the largeness of the solutions to the 2D quasilinear wave equation, some new auxiliary energies and multipliers are introduced to overcome these difficulties.

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arXiv2023

Global smooth solutions to 2D semilinear wave equations with large data

We are interested in coupled semi-linear wave equations satisfying the null condition in two space dimensions, a basic model in nonlinear wave equations. Our aim is to establish global existence of smooth solutions to this system with large initial data of short pulse type. Major difficulties arise due to the largeness of initial data and the slow decay nature of 2D wave equations. To overcome the difficulties, by careful examination of the local solutions, we adapt various vector-field methods to different spacetime regions with several novel weighted energy estimates.

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arXiv2023

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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arXiv2022

On the critical exponent $p_c$ of the 3D quasilinear wave equation $-\big(1+(\partial_t\phi)^p\big)\partial_t^2\phi+\Delta\phi=0$ with short pulse initial data. I, global existence

For the 3D quasilinear wave equation $-\big(1+(\partial_t\phi)^p\big)\partial_t^2\phi+\Delta\phi=0$ with the short pulse initial data $(\phi,\partial_t\phi)(1,x)=\big(\delta^{2-\varepsilon_0}\phi_0(\frac{r-1}{\delta},\omega), \delta^{1-\varepsilon_0}\phi_1(\frac{r-1}{\delta},\omega)\big)$, where $p\in\mathbb N$, $p\geq 2$, $0<\varepsilon_0<1$, $r=|x|$, $\omega= \frac{x}{r}\in\mathbb S^2$, and $\delta>0$ is sufficiently small, under the outgoing constraint condition $(\partial_t+\partial_r)^k\phi(1,x)=O(\delta^{2-\varepsilon_0})$ for $k=1,2$, we will establish the global existence of smooth large data solution $\phi$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$ being the critical exponent. In the forthcoming paper, when $1\leq p\leq p_c$, we show the formation of the outgoing shock before the time $t=2$ under the suitable assumptions of $(\phi_0,\phi_1)$.

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arXiv2013

Blowup of smooth solutions for general 2-D quasilinear wave equations with small initial data

For the 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(\nabla u)\partial_{ij}u=0$ with coefficients independent of the solution $u$, a blowup result for small data solutions has been established in [1,2] provided that the null condition does not hold and a generic nondegeneracy condition on the initial data is fulfilled. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(u, \nabla u)\partial_{ij}u=0$ with coefficients that depend simultaneously on $u$ and $\nabla u$. When the null condition does not hold and a suitable nondegeneracy condition on the initial data is satisfied, we show that smooth small data solutions blow up in finite time. Furthermore, we derive an explicit expression for the lifespan and establish the blowup mechanism.

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arXiv2013

Blowup of classical solutions for a class of 3-D quasilinear wave equations with small initial data

This paper is concerned with the small smooth data problem for the 3-D nonlinear wave equation $\partial_t^2u-\left (1+u+\p_t u\right)Δu=0$. This equation is prototypical of the more general equation $\dsize\sum_{i,j=0}^3g_{ij}(u, \nabla u)\partial_{ij}u=0$, where $x_0=t$ and $g_{ij}(u, \nabla u)=c_{ij}+d_{ij}u+\dsize\sum_{k=0}^3e_{ij}^k\partial_ku+O(|u|^2+|\nabla u|^2)$ are smooth functions of their arguments, with $c_{ij}, d_{ij}$ and $e_{ij}^k$ being constants, and $d_{ij}\neq0$ for some $(i,j)$; moreover, $\dsize\sum_{i,j,k=0}^3e_{ij}^k(\partial_ku)\p_{ij} u$ does not fulfill the null condition. For the 3-D nonlinear wave equations $\partial_t^2u-\left (1+u\right)Δu=0$ and $\partial_t^2u-\left (1+\partial_t u\right)Δu=0$, H. Lindblad, S. Alinhac, and F. John proved and disproved, respectively, the global existence of small smooth data solutions. For radial initial data, we show that the small smooth data solution of $\partial_t^2u-\left(1+u+\partial_t u\right)Δu=0$ blows up in finite time. The explicit expression of the asymptotic lifespan $T_{\varepsilon}$ as $\varepsilon\to0^+$ is also given.

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arXiv2012

On the lifespan of and the blowup mechanism for smooth solutions to a class of 2-D nonlinear wave equations with small initial data

This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu)$ $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and $(c_1'(0))^2+(c_2'(0))^2\neq 0$. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $(u(0,x), \p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$ with $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$, and $\ve>0$ is small, we will show that the classical solution $u(t,x)$ stops to be smooth at some finite time $T_{\ve}$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives $\na_{t,x}u(t,x)$, while $u(t,x)$ itself is continuous up to the blowup time $T_{\ve}$.

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