Blow-up for compressible Euler system with space-dependent damping in 1-D

A note on characteristic decomposition for two-dimensional Euler system in van der Waals fluids

In this note, we consider a thin-film equation including a diffusion term, a fourth order term and a nonlocal source term under the periodic boundary conditions. In particular, a finite time blow-up result is established for the case of positive initial energy provided that \[ \frac{\pi^2}{a^2}\leq \frac{2}{p-1},\] where $a$ is the length of the interval and $p>1$ is the power of nonlinear force term. Also upper and lower blow-up times are estimated.

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Holder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.

This paper studies spatially periodic complex-valued solutions of the fifth-order Korteweg–de Vries (KdV)-type equations. The aim is at several fundamental issues including the existence, uniqueness and finite-time blowup problems. Special attention is paid to the Kawahara equation, a fifth-order KdV-type equation. When a Burgers dissipation is attached to the Kawahara equation, we establish the existence and uniqueness of the Fourier series solution with the Fourier modes decaying algebraically in terms of the wave numbers. We also examine a special series solution to the Kawahara equation and prove the convergence and global regularity of such solutions associated with a single mode initial data. In addition, finite-time blowup results are discussed for the special series solution of the Kawahara equation.

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.

Analysis of Blowup for a Nonlinear Degenerate Parabolic Equation

In this paper, a singular non-Newton polytropic filtration equation under the initial-boundary value condition is revisited. The finite time blow-up results were discussed when the initial energy E(u0) was subcritical (E(u0)<d), critical (E(u0)=d), and supercritical (E(u0)>d), with d being the potential depth by using the potential well method and some differential inequalities. The goal of this paper is to give a finite time blow-up result if E(u0) is independent of d. Moreover, the explicit upper bound of the blow-up time is obtained by the classical Levine’s concavity method, and the precise lower bound of the blow-up time is derived by applying an interpolation inequality.

Singularity formation for the 1D compressible Euler equations with variable damping coefficient

In this paper, we consider the long-time behavior of the initial boundary value 问题 of the compressible Euler system in the exterior domain in high dimensions $(n=2,~3)$. Assuming the initial density and velocity admit small compactly supported perturbations near constant states, we show finite-time blow-up of the initial boundary value 问题 with the impermeable boundary condition. The upper bound of the lifespan estimate with respect to the small parameter of the initial perturbations is given. To this end, a test function method is introduced, which seems quite simple. The energy equality is also involved.

We consider the Cauchy problem for the barotropic Euler system coupled to Poisson equation, in the whole space. Our main aim is to exhibit a simple functional framework that allows to handle solutions with density going to zero at infinity, but that need not be compactly supported. We have in mind in particular the 3D static solution, when the polytropic index $$\gamma $$ of the gas is equal to 6/5. Our first result is the local existence of classical solutions in a simple functional framework that does not require the velocity to tend to 0 at infinity and the density to be compactly supported. Next, following the work by Grassin and Serre dedicated to the compressible Euler system Grassin and Serre (C R Acad Sci Paris Ser I 325:721–726, 1997, Grassin (Indiana Univ Math J, 47:1397–1432, 1998), we show that if the initial density is small enough, and the initial velocity is close to some reference vector field $$u_0$$ such that the spectrum of $$Du_0$$ is positive and bounded away from zero, then the corresponding classical solution is global, and satisfies algebraic time decay estimates. Compared to our recent paper (Blanc et al. in The global existence issue for the compressible Euler system with Poisson or Helmholtz coupling), we are able to handle the 3D static solution that was mentioned above, and to show its instability, within our functional framework.

Variational principle for three-dimensional steady-state flows of an ideal fluid

7R4. Systems of Conservation Laws 2: Geometric Structures, Oscillations and Mixed Problems. - D Serre (Ecole Normale Superieure, Lyon, France). Cambridge UP, Cambridge, UK. 2000. 269 pp. ISBN 0-521-63330-3. $74.95. Reviewed by Y Horie (Los Alamos Natl Lab, Group X-7, MS D413, Los Alamos NM 87545).Conservation laws arise in many areas of continuum physics or mechanics and are described by certain types of nonlinear partial differential equations. Solving these PDEs for non-dissipative systems with appropriate initial and boundary values is known as the Cauchy problem or the mixed problem. This two-volume set of books is concerned with the modern mathematical treatment of the mixed problem. According to the author, the coverage of the book is comparable to that of Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Springer, Berlin, 1984), but the subject matters are dealt with “with more detail, but less animation.” Though important, the phenomenon of relaxation and kinetic formulation are not included. Volume 1 dealt with the fundamentals, including such topics as hyperbolicity, Lax shock, the Glimm scheme, and viscosity solution. The second volume, consisting of Chapters 8 through 15, is concerned with questions such as: the convergence of the viscosity method, stability for the Cauchy problem, admissible boundary conditions, and the multi-dimensional mixed problem. But the author cautions that “the materials in the second volume represent a significant part of the active research in the last fifteen years in the field, and are intended more for research workers and less for students.” Whether dealing with fundamentals or the state-of-the art research, this book is definitely written for post-graduate students and active researchers in mathematics. All the basic ideas are discussed fully, but for nonspecialists guiding from well-versed instructors may be helpful to wade through the materials. The readers are expected to have a good mathematical background in contemporary functional analysis. Keywords such as hyperbolic systems, shock waves, viscosity, and initial and boundary value problems are familiar to physical scientists and engineers, but the questions treated in the book are primarily mathematical: existence, uniqueness, convergence, stability, and consistency, etc. The book, however, has something illuminating to offer even to those who may not have the appetite or requisite mathematics to follow theorems and lemmas in depth. It shows what it means to think with the equations from the mathematical point of view. Half-baked knowledge is always dangerous, but this reviewer, who is not a mathematician, was still able to gain new appreciation of and fresh insight into the kinds of questions he does not normally worry about. Good examples are the treatment of stability types and admissible boundary conditions on a finite domain in mathematical terms. His formal training ends at the level of classic books by such people as Courant and Friedrichs, Whitham, and Courant and Hilbert, but he could not help wonder at the mathematical transformation of the questions one used to think intuitively about on a physical basis. At the same time, as marked by Courant (Methods of Mathematical Physics, Volume 1), one fears an increasing gap between contemporary mathematics and the interest of many scientists and engineers. Systems of Conservation Laws 2 is primarily for researchers in mathematics who are interested in the current state of knowledge and students in advanced graduate courses in PDEs. The book is also recommended to non-mathematicians as a means of getting a broad brush view of the conservation equations, nonlinear wave propagation, and shock waves from the mathematical point of view. The book is definitely recommended for university libraries.

We construct analytical solutions for the problem of nonlinear supersonic flow past slender bodies of revolution due to small amplitude oscillations. The method employed is based on the splitting of the time dependent small perturbation equation to a nonlinear time independent partial differential equation (P.D.E.) concerning the steady flow, and a linear time dependent one, concerning the unsteady flow. Solutions in the form of three parameters family of surfaces for the first equation are constructed, while solutions including one arbitrary function for the second equation are extracted. As an application the evaluation of the small perturbation velocity resultants for a flow past a right circular cone is obtained making use of convenient boundary and initial conditions in accordance with the physical problem.

Numerical simulations of shear flows of a tumbling-type nematic liquid crystal (4-Octyl-4'-cyanobiphenyl) between concentric cylinders have been performed using the Leslie-Ericksen continuum theory. The shear flow is imposed on the liquid crystal by rotating the inner cylinder. In the Leslie-Ericksen theory, the local averaged orientation of the rodlike liquid crystalline molecules is represented with a unit vector called the director, and small perturbations are added on the initial director field to obtain the director escape from the shear plane. Depending on the curvature of the flow geometry and the shear rate and also on the perturbations, four kinds of director behaviors emerge, which are the in-plane, non-twisted out-of-plane, 2π-twisted out-of-plane, and periodic director behaviors. Finally, we discuss the probability of the emergence of the director behaviors.

We study supersonic flow past a convex corner which is surrounded by quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a strong rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to separate the supersonic flow behind the rarefaction wave from the static gas. In this paper, we employ a wave front tracking method to establish structural stability of such a flow pattern under non-smooth perturbations of the upcoming supersonic flow. It is an initial-value/free-boundary problem for the two-dimensional steady non-isentropic compressible Euler system. The main ingredients are careful analysis of wave interactions and construction of suitable Glimm functional, to overcome the difficulty that the strong rarefaction wave has a large total variation.