Two-Dimensional Riemann Problems:Transonic Shock Waves and Free Boundary Problems

We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.In particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

Assessment of Calf Skeletal Muscle Stiffness in Diabetic Nephropathy Patients with Medial Tibial Stress Syndrome by Two-Dimensional Shear Wave Elastography

Objective:To explore the feasibility of two-dimensional shear wave elastography in evaluating calf skeletal muscle stiffness in diabetic nephropathy patients with medial tibial stress syndrome.Methods:A total of 48 diabetic nephropathy patients with medial tibial stress syndrome from January 2020 to December 2022 were included as the study group,and 48 patients with diabetic nephropathy during the same period were included as the control group.Both groups were detected by two-dimensional shear wave elastography with ultrasonic equipment,and Young‘s modulus of the tibialis anterior muscle,tibialis posterior muscle,and gastrocnemius muscle were observed and analyzed in the two groups.Results:The Young‘s modulus values of tibialis anterior muscle,tibialis posterior muscle,and gastrocnemius muscle in the study group were significantly lower than those in the control group(P<0.05).Conclusion:Two-dimensional shear wave elastography is feasible for the evaluation of calf skeletal muscle stiffness in diabetic nephropathy patients with medial tibial stress syndrome,and has high accuracy and repeatability.This technique can be used to diagnose,treat and monitor muscle lesions in patients with diabetic nephropathy,and can also be used to assess muscle fatigue and exercise capacity,which has broad application prospects.

THE WAVE EQUATION APPROACH TO THE TWO-DIMENSIONAL INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN WITH PIECEWISE SMOOTH MIXED BOUNDARY CONDITIONS

The spectral distribution exp( ), where {} are the eigenvalues of the negative Laplacian -△=- in the (x^1,x^2)-plane, is studied for a variety of domains, where -∞< t <∞ and i=(1/2)(-1) . The dependence of (t)on the connectivity of a domain and the boundary conditions are analyzed. Particular attention is given to a general bounded domain Ω in R^2 with a smooth boundary Ω, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth parts Γj(j = 1,……,n) of Ω are considered such that Some geometrical properties of Ω(e.g., the area of Ω, the total lengths of the boundary, the curvature of its boundary, etc.) are determined, from the asymptotic expansions of (t) for |t| → 0.

Diagnostic problems in two-dimensional shear wave elastography of the liver

Two-dimensional shear wave elastography(2D-SWE)is used in the clinical setting for observation of the liver.Unfortunately,a wide spectrum of artifactual images are frequently encountered in 2D-SWE,the precise mechanisms of which remain incompletely understood.This review was designed to present many of the artifactual images seen in 2D-SWE of the liver and to analyze them by computer simulation models that support clinical observations.Our computer simulations yielded the following suggestions:(1)When performing 2D-SWE in patients with chronic hepatic disease,especially liver cirrhosis,it is recommended to measure shear wave values through the least irregular hepatic surface;(2)The most useful 2D-SWE in patients with focal lesion will detect lesions that are poorly visible on B-mode ultrasound and will differentiate true tumors from pseudo-tumors(e.g.,irregular fatty change);and(3)Measurement of shear wave values in the area posterior to a focal lesion must be avoided.

Contact Problem in Decagonal Two-Dimensional Quasicrystal

As a new structure of solid matter quasicrystal brings profound new ideas to the traditional condensed matter physics, its elastic equations are more complicated than that of traditional crystal. A contact problem of decagonal two? dimensional quasicrystal material under the action of a rigid flat die is solved satisfactorily by introducing displacement function and using Fourier analysis and dual integral equations theory, and the analytical expressions of stress and displacement fields of the contact problem are achieved. The results show that if the contact displacement is a constant in the contact zone, the vertical contact stress has order -1/2 singularity on the edge of contact zone, which provides the important mechanics parameter for contact deformation of the quasicrystal.

THE RIEMANN PROBLEM FOR A TWO-DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS WITH NON-CLASSICAL SHOCK WAVES

The Riemann problem for a two-dimensional 2 x 2 nonstrictly hyperbolic system of nonlinear conservation laws has been solved thoroughly for any given initial data which are constant in each quadrant. The non-classical shockwaves, which are labelled as delta-shock waves, appear in some solutions. The solutions have been obtained are not unique. Due to the specific property of the system considered, there are no rarefaction waves in solution. This paper is divided into three parts. The first part constructs Riemann solutions for initial data involving two contact discontinuities while the second considers the case for other initial data. The last part briefly discusses the non-uniqueness of the solutions.

TWO-DIMENSIONAL RIEMANN PROBLEMS:FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS

In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections： 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.

Differential diagnosis of different types of solid focal liver lesions using two-dimensional shear wave elastography

BACKGROUND The clinical management and prognosis differ between benign and malignant solid focal liver lesions(FLLs),as well as among different pathological types of malignant FLLs.Accurate diagnosis of the possible types of solid FLLs is important.Our previous study confirmed the value of shear wave elastography(SWE)using maximal elasticity(Emax)as the parameter in the differential diagnosis between benign and malignant FLLs.However,the value of SWE in the differential diagnosis among different pathological types of malignant FLLs has not been proved.AIM To explore the value of two-dimensional SWE(2D-SWE)using Emax in the differential diagnosis of FLLs,especially among different pathological types of malignant FLLs.METHODS All the patients enrolled in this study were diagnosed as benign,malignant or undetermined FLLs by conventional ultrasound.Emax of FLLs and the periphery of FLLs was measured using 2D-SWE and compared between benign and malignant FLLs or among different pathological types of malignant FLLs.RESULTS The study included 32 benign FLLs in 31 patients and 100 malignant FLLs in 96 patients,including 16 cholangiocellular carcinomas(CCCs),72 hepatocellular carcinomas(HCCs)and 12 liver metastases.Thirty-five FLLs were diagnosed as undetermined by conventional ultrasound.There were significant differences between Emax of malignant(2.21±0.57 m/s)and benign(1.59±0.37 m/s)FLLs(P=0.000),and between Emax of the periphery of malignant(1.52±0.39 m/s)and benign(1.36±0.44 m/s)FLLs(P=0.040).Emax of liver metastases(2.73±0.99 m/s)was significantly higher than that of CCCs(2.14±0.34 m/s)and HCCs(2.14±0.46 m/s)(P=0.002).The sensitivity,specificity and accuracy were 71.00%,84.38%and 74.24%respectively,using Emax>1.905 m/s(AUC 0.843)to diagnose as malignant and 23 of 35(65.74%)FLLs with undetermined diagnosis by conventional ultrasound were diagnosed correctly.CONCLUSION Malignant FLLs were stiffer than benign ones and liver metastases were stiffer than primary liver carcinomas.2D-SWE with Emax was a useful complement to conventional ultrasound for the differential diagnosis of FLLs.

ASYMPTOTIC STABILITY OF SHOCK WAVES FOR THE OUTFLOW PROBLEM OF A HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY

This paper is concerned with an ideal polytropic model of non-viscous and heatconductive gas in a one-dimensional half space. We focus our attention on the outflow problem when the flow velocity on the boundary is negative and we prove the stability of the viscous shock wave and its superposition with the boundary layer under some smallness conditions.Our waves occur in the subsonic area. The intrinsic properties of our system are more challenging in mathematical analysis, however, in the subsonic area, the lack of a boundary condition on the density provides us with a special manner for defining the shift for the viscous shock wave, and helps us to construct the asymptotic profiles successfully. New weighted energy estimates are introduced and the perturbations on the boundary are handled by some subtle estimates.

Rational and Periodic Wave Solutions of Two-Dimensional Boussinesq Equation

Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.

Model for liver hardness using two-dimensional shear wave elastography,durometer,and preoperative biomarkers

BACKGROUND Post-hepatectomy liver failure(PHLF)increases morbidity and mortality after liver resection for patients with advanced liver fibrosis and cirrhosis.Preoperative liver stiffness using two-dimensional shear wave elastography(2D-SWE)is widely used to evaluate the degree of fibrosis.However,the 2D-SWE results were not accurate.A durometer measures hardness by quantifying the ability of a material to locally resist the intrusion of hard objects into its surface.However,the durometer score can only be obtained during surgery.To measure correlations among 2D-SWE,palpation by surgeons,and durometermeasured objective liver hardness and to construct a liver hardness regression model.METHODS We enrolled 74 hepatectomy patients with liver hardness in a derivation cohort.Tactile-based liver hardness scores(0-100)were determined through palpation of the liver tissue by surgeons.Additionally,liver hardness was measured using a durometer.Correlation coefficients for durometer-measured hardness and preoperative parameters were calculated.Multiple linear regression models were constructed to select the best predictive durometer scale.Receiver operating characteristic(ROC)curves and univariate and multivariate analyses were used to calculate the best model’s prediction of PHLF and risk factors for PHLF,respectively.A separate validation cohort(n=162)was used to evaluate the model.RESULTS The stiffness measured using 2D-SWE and palpation scale had good linear correlation with durometer-measured hardness(Pearson rank correlation coefficient 0.704 and 0.729,respectively,P<0.001).The best model for the durometer scale(hardness scale model)was based on stiffness,hepatitis B virus surface antigen,and albumin level and had an R2 value of 0.580.The area under the ROC for the durometer and hardness scale for PHLF prediction were 0.807(P=0.002)and 0.785(P=0.005),respectively.The optimal cutoff value of the durometer and hardness scale was 27.38(sensitivity=0.900,specificity=0.660)and 27.87(sensitivity=0.700,specificity=0.787),respectively.Patients with a hardness scale score of>27.87 were at a significantly higher risk of PHLF with hazard ratios of 7.835(P=0.015).The model’s PHLF predictive ability was confirmed in the validation cohort.CONCLUSION Liver stiffness assessed by 2D-SWE and palpation correlated well with durometer hardness values.The multiple linear regression model predicted durometer hardness values and PHLF.

Eigenfunction expansion method and its application to two-dimensional elasticity problems based on stress formulation

This paper proposes an eigenfunction expansion method to solve twodimensional （2D） elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial differential equations of the above 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence and the completeness of two normed orthogonal eigenfunction systems in some space are obtained, which belong to the two block operators arising in the operator matrix. Moreover, the general solution to the above 2D problem is given by the eigenfunction expansion method.

Comparison of point and two-dimensional shear wave elastography of the spleen in healthy subjects

BACKGROUND Few systematic comparative studies of the different methods of physical elastography of the spleen are currently available.AIM To compare point shear wave and two-dimensional elastography of the spleen considering the anatomical location(upper,hilar,and lower pole).METHODS As part of a prospective clinical study,healthy volunteers were examined for splenic elasticity using four different ultrasound devices between May 2015 and April 2017.The devices used for point shear wave elastography were from Siemens(S 3000)and Philips(Epiq 7),and those used for two-dimensional shear wave elastography were from GE(Logiq E9)and Toshiba(Aplio 500).In addition,two different software versions(5.0 and 6.0)were evaluated for the Toshiba ultrasound device(Aplio 500).The study consisted of three arms:A,B,and C.RESULTS In study arm A,200 subjects were evaluated(78 males and 122 females,mean age 27.9±8.1 years).In study arm B,113 subjects were evaluated(38 men and 75 women,mean age 26.0±6.3 years).In study arm C,44 subjects were enrolled.A significant correlation of the shear wave velocities at the upper third of the spleen(r=0.33088,P<0.0001)was demonstrated only for the Philips Epiq 7 device compared to the Siemens Acuson S 3000.In comparisons of the other ultrasound devices(GE,Siemens,Toshiba),no comparable results could be obtained for any anatomical position of the spleen.The influencing factors age,gender,and body mass index did not show a clear correlation with the measured shear wave velocities.CONCLUSION The absolute values of the shear wave elastography measurements of the spleen and the two different elastography methods are not comparable between different manufacturers or models.

TWO-DIMENSIONAL STRESS WAVE ANALYSIS IN INCOMPRESSIBLE ELASTIC SOLIDS

Two-dimensional stress wares in n general incompressible elastic solid are investigated. First, baxic equations for simple wares and shock waves are presented for a general strain energy junction. Then the characteristic ware speeds and the associated characteristic vectors are deduced. It is shown that there usually exist two simple waves and two shock wares. Finally, two examples are given for the case of plane strain deformation and antiplane strain deformation, respectively. It is proved that, in the case of plane strain deformation, the oblique reflection problem of a plane shock is not solvable in general.

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

THE RIEMANN PROBLEM FOR A TWO-DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS WITH NON-CLASSICAL WEAK SOLUTIONS:SOME ONE-J AND NON-R INITIAL DATA

The Riemann problem for a two-dimensional 2 x 2 nonstrictly hyperbolic system of nonlinear conservation laws has been considered for constant initial data having discontinuities on three rays with vertex at the origin. The solutions are constructed for some one-J and non-R initial data. One kind of new discontinuity, which is labelled as the delta-shock wave, appears in some solutions. The delta-shock wave is a discontinuity plane that is the suport of a generalized function.

EXACT SOLUTION FOR A TWO-DIMENSIONAL LAMB'S PROBLEM DUE TO A STRIP IMPULSE LOADING

By applying the integral transform method and the inverse transformation technique based upon the two types of integration, the present paper has successfully obtained an exact algebraic solution for a two-dimensional Lamb＇s problem due to a strip impulse loading for the first time. With the algebraic result, the excitation and propagation processes of stress waves, including the longitudinal wave, the transverse wave, and Rayleigh-wave, are discussed in detail. A few new conclusions have been drawn from currently available integral results or computational results.

Centered simple waves for the two-dimensional pseudo-steady isothermal ?ow around a convex corner

The two-dimensional(2D) pseudo-steady isothermal flow, which is isentropic and irrotational, around a convex corner is studied. The self-similar solutions for the supersonic flow around the convex corner are constructed, where the properties of the centered simple wave are used for the 2D isentropic irrotational pseudo-steady Euler equations. The geometric procedures of the center simple waves are given. It is proven that the supersonic flow turns the convex corner by an incomplete centered expansion wave or an incomplete centered compression wave, depending on the conditions of the downstream state.

The interaction of nonlinear waves in two-dimensional lattice

This paper investigates the collision between two nonlinear waves with arbitrary angle in two-dimensional nonlinear lattice. By using the extended Poincarge-Lighthill-Kuo perturbation method, it obtains two Korteweg-de Vries equations for nonlinear waves in both the ζ and η directions, respectively, and derives the analytical phase shifts after the collision of two nonlinear waves. Finally, the solution of u（υ） up to O（ε^3） order is given.

The interaction of nonlinear waves in two-dimensional dust crystals

This paper investigates the collision between two nonlinear waves with different propagation directions in two- dimensional dust crystals. Using the extended Poincare-Lighthill-Kuo perturbation method, two Korteweg-de Vries equations for nonlinear waves in both the ξ and η directions are obtained, respectively, and the analytical phase shifts and trajectories after the collision of two nonlinear waves are derived. Finally, the effects of parameters of the lattice constant a, the arbitrary constant u0η, the forces f（r）, and the colliding angle θ on the phase shifts of both colliding nonlinear waves are examined.