28 problems

Approximation-independence conjecture. The set of admissible boundary values, and hence the boundary layer, should be independent of the approximation method for nonlinear systems…

Carbuncle conjecture. Carbuncles are a special class of non-physical entropy solutions.

Assume that all characteristic fields are genuinely nonlinear, and consider the semidiscrete scheme referred to in [Bi]. The paper discusses uniformly Lipschitz semigroups of appro…

Vanishing viscosity conjecture. The entropic solutions of the hyperbolic system (1.1) actually coincide with the limits of solutions $u^\varepsilon$ to the parabolic system as the…

Interface stability conjecture. Aside from certain metastable phase transitions, an interface satisfying the scattering laws will remain stable under small perturbations.

Infinite-sign-change extension conjecture. The conclusion of Theorem 3 should hold when the left-hand side of $(\mathcal F5)$ changes sign infinitely many times, including the case…

Universal Jouguet applicability conjecture. The newly formulated condition derived from comparing the second derivatives of the $\mathcal U$-characteristics and the chemical shock…

Consider the nonlinear hyperbolic system, with shocks parameterized along the positive part of their shock curves. The authors refer to shocks satisfying a moderate shock size cond…

Let a system of hyperbolic conservation laws have totally linearly degenerate characteristics, and let its initial data belong to $H^s(\mathbb R^n)$ with $s\geq 1+\frac{n}{2}$. Maj…

Consider the shallow water equations with discontinuous topography and a Riemann problem whose two states have constant Riemann invariants, so that the stationary contact wave is a…

Let $Q_{\mathrm{jam}}$ be a jamiton with sonic density $\rho_s$. Let $\{Q_{\mathrm{jam}}(\rho_s^{ \mathrm{jam}})\}$ be a corresponding non-empty set of compatible jamitons parametr…

Let two compatible jamitons be given, with their parameters describing their densities, velocities, and lengths. Unique post-collision jamiton conjecture. The collision of two comp…

Algebraic stability conjecture. The operator $\Omega(k)$ has negative eigenvalues if and only if $\lambda_{\ell,1}<0$ for every $\ell=1,\ldots,N$.

Stability conjecture. If the eigenvalues of all four stability operators are negative for every wavenumber $k$, then the system (1.2) is stable.

Intermediate-domain well-posedness conjecture. The Cauchy problem for a genuinely nonlinear $2\times 2$ hyperbolic system with ${\bf L}^\infty$ initial data is well posed within an…

Square-root logarithmic convergence-rate conjecture. In the genuinely nonlinear case, one may conjecture that

The one-dimensional Burgers equation admits finite-codimension stable shock formation under suitable initial-data conditions, as established by the preceding proposition. Finite-co…

Let $H(u(\cdot,t_0),\tau)$ denote an exact solution operator mapping an initial condition $u(\cdot,t_0)$ to the solution at time $t_0+\tau$. Let $V$ be the finite-dimensional appro…

Uniqueness and classification conjecture. The solution of the Riemann problem consisting of the governing equations and initial condition is unique for any given $\gamma$ and…

Consider the modified c-Riemann solver $\mathcal{RS}_{\rm h}$ and the solver $\mathcal{RS}_{\rm v}$, and let $\Delta t$ denote a time step in a regime where chattering occurs. Vani…

Periodic nonsingularity conjecture. In a periodic domain, there exist nontrivial solutions in which no singularity forms.

Weak-discontinuity removability conjecture. The weak discontinuity could be removed by some further development of the authors' construction method.

Let $(\rho_*,v_*)(r)$ be a steady shock for the relativistic Euler equations on the Schwarzschild exterior $r>2M$, and let … where $(\delta_\rho,\delta_v)$ is a compactly supported…

Let $(\rho_*,v_*)=(\rho_*,v_*)(r)$ be a smooth steady state for the relativistic Euler equations on the Schwarzschild exterior $r>2M$. Let … where $(\delta_\rho,\delta_v)$ has comp…

Consider the one-dimensional Chaplygin gas system discussed in the paper, with the smooth solution developing a first blowup at the point where the characteristic fields lose stric…