Zhu Lu | Searcharxiv

arXiv2021

Formation and construction of a multidimensional shock wave for the first order hyperbolic conservation law with smooth initial data

In this paper, the problem on formation and construction of a multidimensional shock wave is studied for the first order conservation law $\partial_t u+\partial_x F(u)+\partial_y G(u)=0$ with smooth initial data $u_0(x,y)$. It is well-known that the smooth solution $u$ will blow up on the time $T^*=-\frac{1}{\min{H(ξ,η)}}$ when $\min{H(ξ,η})<0$ holds for $H(ξ,η)=\partial_ξ(F'(u_0(ξ,η)))+\partial_η(G'(u_0(ξ,η)))$, more precisely, only the first order derivatives $\nabla_{t,x,y}u$ blow up on $t=T^*$ meanwhile $u$ itself is still continuous until $t=T^*$. Under the generic nondegenerate condition of $H(ξ,η)$, we construct a local weak entropy solution $u$ for $t\ge T^*$ which is not uniformly Lipschitz continuous on two sides of a shock surface $Σ$. The strength of the constructed shock is zero on the initial blowup curve $Γ$ and then gradually increases for $t>T^*$. Additionally, in the neighbourhood of $Γ$, some detailed and precise descriptions on the singularities of solution $u$ are given.

math.AP↗

arXiv2021

The shock formation and optimal regularities of the resulting shock curves for 1-D scalar conservation laws

The study on the shock formation and the regularities of the resulting shock surfaces for hyperbolic conservation laws is a basic problem in the nonlinear partial differential equations. In this paper, we are concerned with the shock formation and the optimal regularities of the resulting shock curves for the 1-D conservation law $\partial_tu+\partial_xf(u)=0$ with the smooth initial data $u(0,x)=u_0(x)$. If $u_0(x)\in C^{1}(\Bbb R)$ and $f(u)\in C^2(\Bbb R)$, it is well-known that the solution $u$ will blow up on the time $T^*=-\frac{1}{\min{g'(x)}}$ when $\min{g'(x)}<0$ holds for $g(x)=f'(u_0(x))$. Let $x_0$ be a local minimum point of $g'(x)$ such that $g'(x_0)=\min{g'(x)}<0$ and $g''(x_0)=0$, $g^{(3)}(x_0)>0$ (which is called the generic nondegenerate condition), then by Theorem 2 of \cite{Le94}, a weak entropy solution $u$ together with the shock curve $x=φ(t)\in C^2[T^*, T^*+\varepsilon)$ starting from the blowup point $(T^*, x^*=x_0+g(x_0)T^*)$ can be locally constructed. When the generic nondegenerate condition is violated, namely, when $x_0$ is a local minimum point of $g'(x)$ such that $g''(x_0)=g^{(3)}(x_0)=...=g^{(2k_0)}(x_0)=0$ but $g^{(2k_0+1)}(x_0)>0$ for some $k_0\in\Bbb N$ with $k_0\ge 2$; or $g^{(k)}(x_0)=0$ for any $k\in\Bbb N$ and $k\ge 2$, we will study the shock formation and the optimal regularity of the shock curve $x=φ(t)$, meanwhile, some precise descriptions on the behaviors of $u$ near the blowup point $(T^*, x^*)$ are given. Our main aims are to show that: around the blowup point, the shock really appears whether the initial data are degenerate with finite orders or with infinite orders; the optimal regularities of the shock solution and the resulting shock curve have the explicit relations with the degenerate degrees of the initial data.

math.AP↗

arXiv2014

Large time asymptotic behavior of the compressible Navier-Stokes Equations in partial Space-Periodic Domains

In this paper, we study the large time behavior of the 3-D isentropic compressible Navier-Stokes equation in the partial space-periodic domains, and simultaneously show that the related profile systems can be described by like Navier-Stokes equations with suitable "pressure" functions in lower dimensions. Our proofs are based on the energy methods together with some delicate analysis on the corresponding linearized problems.

math.AP↗