Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the $L^1-$estimate between the entropy solution and the geometric optics expansion function is bounded by $O(\varepsilon^2)$, {\it independent of} the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.Comment: 30 pages, 2 figure

Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls

We are concerned with the two-dimensional steady supersonic reacting Euler flow past Lipschitz bending walls that are small perturbations of a convex one, and establish the existence of global entropy solutions when the total variation of both the initial data and the slope of the boundary is sufficiently small. The flow is governed by an ideal polytropic gas and undergoes a one-step exothermic chemical reaction under the reaction rate function that is Lipschtiz and has a positive lower bound. The heat released by the reaction may cause the total variation of the solution to increase along the flow direction. We employ the modified wave-front tracking scheme to construct approximate solutions and develop a Glimm-type functional by incorporating the approximate strong rarefaction waves and Lipschitz bending walls to obtain the uniform bound on the total variation of the approximate solutions. Then we employ this bound to prove the convergence of the approximate solutions to a global entropy solution that contains a strong rarefaction wave generated by the Lipschitz bending wall. In addition, the asymptotic behavior of the entropy solution in the flow direction is also analyzed.Comment: 58 pages, 16 figures; SIAM J. Math. Anal. (accepted on November 1, 2016

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows past Lipschitz Perturbed Cones

We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial-boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and asymptotic behavior of global entropy solutions in $BV$ of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure the decreasing of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interaction estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity.Comment: 50 pages; 7 figue

Stability of Inverse Problems for Steady Supersonic Flows Past Lipschitz Perturbed Cones

We are concerned with inverse problems for supersonic potential flows past infinite axisymmetric Lipschitz cones. The supersonic flows under consideration are governed by the steady isentropic Euler equations for axisymmetric potential flows, which involve a singular geometric source term. We first study the inverse problem for the stability of an oblique conical shock as an initial-boundary value problem with both the generating curve of the cone surface and the leading conical shock front as free boundaries. We then establish the existence and asymptotic behavior of global entropy solutions with bounded BV norm of this problem, when the Mach number of the incoming flow is sufficiently large and the total variation of the pressure distribution on the cone is sufficiently small. To this end, we first develop a modified Glimm-type scheme to construct approximate solutions by self-similar solutions as building blocks to balance the influence of the geometric source term. Then we define a Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, along with the construction of the approximate generating curves of the cone surface. Next, when the Mach number of the incoming flow is sufficiently large, by asymptotic analysis of the reflection coefficients in those interaction estimates, we prove that appropriate weights can be chosen so that the corresponding Glimm-type functional decreases in the flow direction. Finally, we determine the generating curves of the cone surface and establish the existence of global entropy solutions containing a strong leading conical shock, besides weak waves. Moreover, the entropy solution is proved to approach asymptotically the self-similar solution determined by the incoming flow and the asymptotic pressure on the cone surface at infinity.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with arXiv:2008.0240

Xiao-Ai-Ping, a TCM Injection, Enhances the Antigrowth Effects of Cisplatin on Lewis Lung Cancer Cells through Promoting the Infiltration and Function of CD8 +

Objectives. To investigate how Xiao-Ai-Ping injection, a traditional Chinese medicine and an ancillary drug in tumor treatment, enhances the antitumor effects of cisplatin on Lewis lung cancer (LLC) cells. Methods. LLC-bearing mice were daily intraperitoneally injected with various doses of cisplatin, Xiao-Ai-Ping, or cisplatin plus Xiao-Ai-Ping, respectively. Body weight and tumor volumes were measured every three days. Results. Combination of Xiao-Ai-Ping and cisplatin yielded significantly better antigrowth and proapoptotic effects on LLC xenografts than sole drug treatment did. In addition, we found that Xiao-Ai-Ping triggered the infiltration of CD8+ T cells, a group of cytotoxic T cells, to LLC xenografts. Furthermore, the mRNA levels of interferon-γ (ifn-γ), perforin-1 (prf-1), and granzyme B (gzmb) in CD8+ T cells were significantly increased after combination treatment of Xiao-Ai-Ping and cisplatin. In vitro studies showed that Xiao-Ai-Ping markedly upregulated the mRNA levels of ifn-γ, prf-1, and gzmb in CD8+ T cells in a concentration-dependent manner, suggesting that Xiao-Ai-Ping augments the function of CD8+ T cells. Conclusions. Xiao-Ai-Ping promotes the infiltration and function of CD8+ T cells and thus enhances the antigrowth effects of cisplatin on LLC xenografts, which provides new evidence for the combination of Xiao-Ai-Ping and cisplatin in clinic in China

Potential Regulatory Roles of MicroRNAs and Long Noncoding RNAs in Anticancer Therapies

MicroRNAs and long noncoding RNAs have long been investigated due to their roles as diagnostic and prognostic biomarkers of cancers and regulators of tumorigenesis, and the potential regulatory roles of these molecules in anticancer therapies are attracting increasing interest as more in-depth studies are performed. The major clinical therapies for cancer include chemotherapy, immunotherapy, and targeted molecular therapy. MicroRNAs and long noncoding RNAs function through various mechanisms in these approaches, and the mechanisms involve direct targeting of immune checkpoints, cooperation with exosomes in the tumor microenvironment, and alteration of drug resistance through regulation of different signaling pathways. Herein we review the regulatory functions and significance of microRNAs and long noncoding RNAs in three anticancer therapies, especially in targeted molecular therapy, and their mechanisms. Keywords: microRNAs, long noncoding RNAs, targeted therapy, chemoresistance, immune checkpoin

The emerging roles of SUMOylation in the tumor microenvironment and therapeutic implications

Abstract Tumor initiation, progression, and response to therapies depend to a great extent on interactions between malignant cells and the tumor microenvironment (TME), which denotes the cancerous/non-cancerous cells, cytokines, chemokines, and various other factors around tumors. Cancer cells as well as stroma cells can not only obtain adaption to the TME but also sculpt their microenvironment through a series of signaling pathways. The post-translational modification (PTM) of eukaryotic cells by small ubiquitin-related modifier (SUMO) proteins is now recognized as a key flexible pathway. Proteins involved in tumorigenesis guiding several biological processes including chromatin organization, DNA repair, transcription, protein trafficking, and signal conduction rely on SUMOylation. The purpose of this review is to explore the role that SUMOylation plays in the TME formation and reprogramming, emphasize the importance of targeting SUMOylation to intervene in the TME and discuss the potential of SUMOylation inhibitors (SUMOi) in ameliorating tumor prognosis