Partial Differential Equations of Mixed Type—Analysis and Applications

Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of mixed type is still in its infancy. The aim of this expository paper is to show – through several longstanding fundamental problems in fluid mechanics, differential geometry, and other areas – that many nonlinear PDEs arising in these areas are no longer of standard types, but lie at the boundaries of the classification of PDEs or, indeed, go beyond the classification and are of mixed type. Some interrelated connections, historical perspectives, recent developments, and current trends in the analysis of nonlinear PDEs of mixed type are also presented.

1. Linear Partial Differential Equations of Mixed Type

Three of the basic types of PDEs are elliptic, hyperbolic, and parabolic, following the classification introduced by Jacques Salomon Hadamard in 1923 (see Figure 1).

The prototype of second-order elliptic equations is the Laplace equation:

$$\begin{equation} \Delta _\mathbf{x}u\coloneq \sum _{j=1}^n \partial _{x_jx_j}u=0 \qquad \text{ for } \mathbf{x}=(x_1, \dots , x_n)\in \mathbb{R}^{n}. {\tag{1.1}} \end{equation}$$

This equation often describes physical equilibrium states whose solutions are also called harmonic functions or potential functions, where $\partial _{x_jx_j}$ is the second-order partial derivative in the $x_j$-variable, $j=1,\dots , n$. The simplest representative of hyperbolic equations is the wave equation:

$$\begin{equation} \partial _{tt}u-\Delta _\mathbf{x}u=0 \qquad \quad \text{for }(t, \mathbf{x})\in \mathbb{R}^{n+1}, {\tag{1.2}} \end{equation}$$

which governs the propagation of linear waves (such as acoustic waves and electromagnetic waves). The prototype of second-order parabolic equations is the heat equation:

$$\begin{equation} \partial _t u-\Delta _\mathbf{x}u=0 \qquad \quad \text{for }(t, \mathbf{x})\in \mathbb{R}^{n+1}, {\tag{1.3}} \end{equation}$$

which often describes the dynamics of temperature and diffusion/stochastic processes.

Figure 1.

Jacques Salomon Hadamard (December 8, 1865–October 17, 1963) first introduced the classification of PDEs in 16.

At first glance, the forms of the Laplace/heat equations and the wave equation look quite similar. In particular, any steady solution of the wave/heat equations is a solution of the Laplace equation, and a solution of the Laplace equation often determines an asymptotic state of the time-dependent solutions of the wave/heat equations. However, the properties of the solutions of the Laplace/heat equations and the wave equation are significantly different. One important difference is in terms of the infinite versus finite speed of propagation of the solution, while another pertains to the gain versus loss of regularity of the solution; see 1416 and the references cited therein. Since the solutions of elliptic/parabolic PDEs share many common features, we focus mainly on PDEs of mixed elliptic-hyperbolic type from now on.

The distinction between the elliptic and hyperbolic types can be seen more clearly from the classification of two-dimensional ($2$-D) constant-coefficient second-order PDEs:

$$\begin{equation} a_{11} \partial _{x_1x_1}u+2a_{12}\partial _{x_1x_2}u +a_{22}\partial _{x_2x_2}u=f(\mathbf{x}) {\tag{1.4}} \end{equation}$$

for $\mathbf{x}=(x_1,x_2)\in \mathbb{R}^2$. Let $\lambda _1\le \lambda _2$ be the two constant eigenvalues of the $2\times 2$ symmetric coefficient matrix $(a_{ij})_{2\times 2}$. Then Equation 1.4 is classified as elliptic if

$$\begin{equation} \det (a_{ij})>0 \iff \lambda _1\lambda _2>0 \iff a_{12}^2-a_{11}a_{22}<0, {\tag{1.5}} \end{equation}$$

while it is classified as hyperbolic if

$$\begin{equation} \det (a_{ij})<0 \iff \lambda _1\lambda _2<0 \iff a_{12}^2-a_{11}a_{22}>0. {\tag{1.6}} \end{equation}$$

Notice that the left-hand side of Equation 1.4 is analogous to the quadratic (homogeneous) form:

$$a_{11} \xi _1^2+ 2a_{12}\xi _1\xi _2+a_{22}\xi _2^2$$

for conic sections. Thus, the classification of Equation 1.4 is consistent with the classification of conic sections and quadratic forms in algebraic geometry, based on the sign of the discriminant: $a_{12}^2-a_{11}a_{22}$. The corresponding quadratic curves are ellipses (incl. circles), hyperbolas, and parabolas (see Figure 2).

Figure 2.

Types of conic sections: parabolas, ellipses, and hyperbolas.

This classification can also be seen by taking the Fourier transform on both sides of Equation 1.4:

$$\begin{equation} (a_{11}\xi _1^2+2a_{12}\xi _1\xi _2 +a_{22}\xi _2^2)\hat{u}(\boldsymbol{\xi })= -\hat{f}(\boldsymbol{\xi }) {\tag{1.7}} \end{equation}$$

for $\boldsymbol{\xi }=(\xi _1, \xi _2)\in \mathbb{R}^2$. Here $\hat{w}(\boldsymbol{\xi })=\frac{1}{2\pi }\int _{\mathbb{R}^2} w(\mathbf{x})e^{-i \mathbf{x}\cdot \boldsymbol{\xi }}\, {\mathrm{d}}\mathbf{x}$ is the Fourier transform of a function $w(\mathbf{x})$, such as $u(\mathbf{x})$ and $f(\mathbf{x})$ for 1.7. When Equation 1.4 is elliptic, the Fourier transform $\hat{u}(\boldsymbol{\xi })$ of solution $u(\mathbf{x})$ gains two orders of decay for the high Fourier frequencies (i.e., $|\boldsymbol{\xi }|\gg 1$) so that the solution gains the regularity of two orders from $f(\mathbf{x})$. When Equation 1.4 is hyperbolic, $\hat{u}(\boldsymbol{\xi })$ fails to gain two orders of decay for the high Fourier frequencies along the two characteristic directions in which $a_{11}\xi _1^2+2a_{12}\xi _1\xi _2 +a_{22}\xi _2^2=0$, even though it still gains two orders of decay for the high Fourier frequencies away from these two characteristic directions.

For the classification above, a general homogeneous constant-coefficient second-order PDE (i.e., $f(\mathbf{x})=0$) with 1.5 or 1.6 can be transformed correspondingly into the Laplace equation 1.1 with $n=2$, or the wave equation 1.2 with $n=1$, via the corresponding coordinate transformations. This reveals the beauty of the classification theory that was first introduced by Hadamard in 16.

On the other hand, for general variable-coefficient second-order PDEs:

$$\begin{equation} a_{11}(\mathbf{x})\partial _{x_1x_1}u +2 a_{12}(\mathbf{x})\partial _{x_1x_2}u +a_{22}(\mathbf{x})\partial _{x_2x_2}u=f(\mathbf{x}), {\tag{1.8}} \end{equation}$$

the situation is different. The classification depends upon the signature of the eigenvalues $\lambda _j(\mathbf{x}), j=1,2$, of the coefficient matrix $(a_{ij}(\mathbf{x}))$. In general, $\lambda _1(\mathbf{x})\lambda _2(\mathbf{x})$ may change its sign as a function of $\mathbf{x}$, which leads to the mixed elliptic-hyperbolic type of 1.8. Equation 1.8 is elliptic when $\lambda _1(\mathbf{x})\lambda _2(\mathbf{x})>0$ and hyperbolic when $\lambda _1(\mathbf{x})\lambda _2(\mathbf{x})<0$ with a transition boundary/region where $\lambda _1(\mathbf{x})\lambda _2(\mathbf{x})=0$.

Three of the classical prototypes for linear PDEs of mixed elliptic-hyperbolic type are as follows:

(i)

The Lavrentyev-Bitsadze equation:$$\partial _{x_1x_1}u +{\mathrm{sign}} (x_1)\partial _{x_2x_2} u=0.$$

This equation exhibits a jump transition at $x_1=0$. It becomes the Laplace equation 1.1 in the half-plane $x_1>0$ and the wave equation 1.2 in the half-plane $x_1<0$, and changes its type from elliptic to hyperbolic via the jump-discontinuous coefficient ${\mathrm{sign}} (x_1)$.

(ii)

The Tricomi equation: $\qquad \partial _{x_1x_1}u + x_1\partial _{x_2x_2}u =0$.

This equation is of hyperbolic degeneracy at $x_1=0$. It is elliptic in the half-plane $x_1>0$ and hyperbolic in the half-plane $x_1<0$, and changes its type from elliptic to hyperbolic through the degenerate line $x_1=0$. This equation is of hyperbolic degeneracy in the domain $x_1\le 0$, where the two characteristic families coincide perpendicularly to the line $x_1=0$. The degeneracy of the equation is determined by the classical elliptic or hyperbolic Euler-Poisson-Darboux equation:⁠¹

¹

J. Hadamard, La Théorie des Équations aux Dérivées Partielles, in French, Éditions Scientifiques, Peking; Gauthier-Villars Éditeur, Paris, 1964.

✖

$$\begin{equation} \partial _{\tau \tau }u \pm \partial _{x_2x_2}u +\frac{\beta }{\tau }\partial _\tau u =0, {\tag{1.9}} \end{equation}$$

with $\beta =\frac{1}{3}$ for $\tau =\frac{2}{3}|x_1|^{\frac{3}{2}}$, and signs “$\pm$” corresponding to the half-planes $\pm x_1>0$ for $\mathbf{x}$ to lie in.

(iii)

The Keldysh equation: $\qquad x_1 \partial _{x_1x_1}u + \partial _{x_2x_2}u =0$.

This equation is of parabolic degeneracy at $x_1=0$. It is elliptic in the half-plane $x_1>0$ and hyperbolic in the half-plane $x_1<0$, and changes its type from elliptic to hyperbolic through the degenerate line $x_1=0$. This equation is of parabolic degeneracy in the domain $x_1\le 0$, in which the two characteristic families are quadratic parabolas lying in the half-plane $x_1<0$, and tangential at contact points to the degenerate line $x_1=0$. Its degeneracy is also determined by the classical elliptic or hyperbolic Euler-Poisson-Darboux equation 1.9 with $\beta =-\frac{1}{4}$ for $\tau =\frac{1}{2}|x_1|^{\frac{1}{2}}$.

For such a linear PDE, the transition boundary (i.e., the boundary between the elliptic and hyperbolic domains) is known a priori. Thus, one traditional approach is to regard such a PDE as a degenerate elliptic or hyperbolic PDE in the corresponding domain, and then to analyze the solution behavior of these degenerate PDEs separately in the elliptic and hyperbolic domains with degeneracy on the transition boundary, determined, say, by the Euler-Poisson-Darboux type equations as 1.9. Another successful approach for dealing with such a PDE is the fundamental solution approach. With this approach, we first understand the behavior of the fundamental solution of the mixed-type PDE, especially its singularity, from which analytical/geometric properties of the solutions can then be revealed, since the fundamental solution is a generator of all of the solutions of the linear PDE. Great effort and progress have been made in the analysis of linear PDEs of mixed type by many leading mathematicians since the early 20th century (cf. 461618 and the references cited therein). Still, there are many important problems regarding linear PDEs of mixed type which require further understanding.

In the sections to come, we show, through several longstanding fundamental problems in fluid mechanics, differential geometry, and other areas, that many nonlinear PDEs arising in mathematics and science are no longer of standard type, but are in fact of mixed type. In contrast to the linear case, the transition boundary for a nonlinear PDE of mixed type is often a priori unknown, and the nonlinearity generates additional singularities in general. Thus, many classical methods and techniques for linear PDEs no longer work directly for nonlinear PDEs of mixed type. The lack of effective unified approaches is one of the main obstacles for tackling the elliptic/hyperbolic phases together for nonlinear PDEs of mixed type. Over the course of the last eight decades, the PDE research community has been largely partitioned according to the approaches taken to the analysis of different classes of PDEs (elliptic/hyperbolic/parabolic). However, advances in the analysis of nonlinear PDEs over the last several decades have made it increasingly clear that many difficult questions faced by the community lay at the boundaries of this classification or, indeed, go beyond this classification. In particular, many important nonlinear PDEs that arise in longstanding fundamental problems across diverse areas are of mixed type. As we will show in §2–§4, below, these problems include steady transonic flow problems and shock reflection/diffraction problems in gas dynamics, high-speed flow, and related areas (cf. 236121315181920), and isometric embedding problems with optimal target dimensions and assigned regularity/curvatures in elasticity, geometric analysis, materials science, and other areas (cf. 1117). The solution to these problems will advance our understanding of shock reflection/diffraction phenomena, transonic flows, properties/classifications of elastic/biological surfaces/bodies/manifolds, and other scientific issues, and lead to significant developments of these areas and related mathematics. To achieve these goals, a deep understanding of the underlying nonlinear PDEs of mixed type (for instance, the solvability, the properties of solutions, etc.) is key.

2. Nonlinear PDEs of Mixed Type and Steady Transonic Flow Problems in Fluid Mechanics

In many applications, fluid flows are often regarded as time-independent; this is the case for some longstanding fundamental problems, such as that of transonic flows past multi-dimensional (M-D) obstacles (wedges/conic bodies, airfoils, etc.), or de Laval nozzles; see Figures 3–4. Furthermore, steady-state solutions are often global attractors as time-asymptotic equilibrium states, and serve as building blocks for constructing time-dependent solutions (cf. 6121315). The underlying nonlinear PDEs governing these fluid flows are generically of mixed type.

Figure 3.

NASA’s first Schlieren photo of shock waves interacting between two aircraft (taken in March 2019).

Figure 4.

Diagram of a de Laval nozzle for the approximate flow velocity.

Our first example is steady potential fluid flows governed by the steady Euler equations of the conservation law of mass and Bernoulli’s law:

$$\begin{equation} \operatorname {div}(\rho \nabla \varphi )=0, \,\, {1\over 2}|\nabla \varphi |^2+{1\over \gamma -1}\rho ^{\gamma -1}=\frac{B_0}{\gamma -1} {\tag{2.1}} \end{equation}$$

for $\mathbf{x}\in \mathbb{R}^n$ after scaling, where $\rho$ is the density, $\varphi$ is the velocity potential (i.e., $\boldsymbol{v}=\nabla \varphi$ is the velocity), $\gamma >1$ is the adiabatic exponent for the ideal gas, $B_0/(\gamma -1)$ is the Bernoulli constant, and $\nabla$ is the gradient in $\mathbf{x}$. System 2.1, along with its time-dependent version (see 3.1 below), is one of the first PDEs to be written down by Euler (cf. Figure 5), and has been employed widely in aerodynamics and other areas in instances when the vorticity waves are weak in the fluid flow under consideration (cf. 36121315). System 2.1 for the steady velocity potential $\varphi$ can be rewritten as

$$\begin{equation} \operatorname {div}\big (\rho _B(|\nabla \varphi |)\nabla \varphi \big )=0 {\tag{2.2}} \end{equation}$$

with $\rho _B(q)=\big (B_0-(\gamma -1)q^2/2\big )^{{1}/(\gamma -1)}$. Equation 2.2 is a nonlinear conservation law of mixed elliptic-hyperbolic type:

•

strictly elliptic (subsonic) if $|\nabla \varphi |<c_*\coloneq \sqrt {2B_0/(\gamma +1)}$;

•

strictly hyperbolic (supersonic) if $|\nabla \varphi |>c_*$.

The transition boundary here is $|\nabla \varphi |=c_*$ (sonic), a degenerate set of 2.2, which is a priori unknown, since it is determined by the solution itself.

Figure 5.

Leonhard Euler (April 15, 1707–September 18, 1783) formulated the Euler equations for fluid mechanics; these are among the first PDEs to be written down.

Figure 6.

In 1936, Ludwig Prandtl (February 4, 1875–August 15, 1953) identified, via the shock polar analysis, two oblique shock configurations when a steady uniform supersonic gas flow passes a solid wedge.

Similarly, the time-independent full Euler flows are governed by the steady Euler equations:

$$\begin{equation} \operatorname {div}(\rho \boldsymbol{v})=0,\quad \operatorname {div}(\rho \boldsymbol{v}\otimes \boldsymbol{v}) +\nabla p=0, \quad \operatorname {div}\big (\rho \boldsymbol{v}(E+\frac{p}{\rho })\big )=0, {\tag{2.3}} \end{equation}$$

where $p$ is the pressure, $\boldsymbol{v}$ is the velocity, and $E=\frac{1}{2}|\boldsymbol{v}|^2+e$ is the energy with $e=\frac{p}{(\gamma -1)\rho }$ as the internal energy determined by the thermodynamic constitutive equation of state. System 2.3 is a system of conservation laws of mixed-composite hyperbolic-elliptic type:

•

strictly hyperbolic when $|\boldsymbol{v}|>c$ (supersonic);

•

mixed-composite elliptic-hyperbolic (two of these are elliptic and the others are hyperbolic) when $|\boldsymbol{v}|< c$ (subsonic),

where $c=\sqrt {\gamma p/\rho }$ is the sonic speed. The transition boundary between the supersonic/subsonic phase is $|\boldsymbol{v}|=c$, a degenerate set of the solution of System 2.3, which is a priori unknown.

Many fundamental transonic flow problems in fluid mechanics involve these nonlinear PDEs of mixed type. One of these is a classical shock problem in which an upstream steady uniform supersonic gas flow passes a symmetric straight-sided solid wedge

$$\begin{equation} W\coloneq \{\mathbf{x}=(x_1,x_2)\in \mathbb{R}^2: |x_2|<x_1\tan \theta _{\mathrm{w}}, x_1>0\}, {\tag{2.4}} \end{equation}$$

whose (half-wedge) angle $\theta _{\mathrm{w}}$ is less than the detachment angle $\theta ^{\mathrm{d}}_{\mathrm{w}}$ (cf. Figure 7).

Figure 7.

Two steady solutions with shocks around the solid wedge with an angle $\theta _{\mathrm{w}}\in (0, \theta _{\mathrm{w}}^{\mathrm{s}})$ or even $\theta _{\mathrm{w}} \in [\theta _{\mathrm{w}}^{\mathrm{s}}, \theta _{\mathrm{w}}^{\mathrm{d}})$.

Since this problem involves shocks, its global solution should be a weak solution of Equation 2.2 or System 2.3 in the distributional sense (which admits shocks)⁠² in the domain under consideration (see 7). For example, for Equation 2.2, a shock is a curve across which $\nabla \varphi$ is discontinuous. If $\Lambda ^+$ and $\Lambda ^-(\coloneq \Lambda \setminus \overline{\Lambda ^+})$ are two nonempty open subsets of a domain $\Lambda \subset \mathbb{R}^2$, and $\mathcal{S}\coloneq \partial \Lambda ^+\cap \Lambda$ is a $C^1$-curve across which $\nabla \varphi$ has a jump, then $\varphi \in C^1(\Lambda ^{\pm }\cup \mathcal{S})\cap C^2(\Lambda ^{\pm })$ is a global weak solution of 2.2 in $\Lambda$ if and only if $\varphi$ is in $W^{1,\infty }_{\mathrm{loc}}(\Lambda )$⁠³ and satisfies Equation 2.2 in $\Lambda ^\pm$ and the Rankine-Hugoniot conditions on $\mathcal{S}$:

²

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-RCSAM, No. 11, SIAM, Philadelphia, Pennsylvania, 1973.

✖

³

A $W^{k,p}$ function, for $1\le p\le \infty$ and $k\ge 1$ integer, is a real-valued function such that itself and its (weak) derivatives up to order $k$ are all $L^p$ functions.

✖

$$\begin{equation} \begin{aligned} &\varphi _{\Lambda ^+\cap \mathcal{S}}=\varphi _{\Lambda ^-\cap \mathcal{S}},\\ &\rho _B(|\nabla \varphi |^2)\nabla \varphi \cdot \boldsymbol{\nu }|_{\Lambda ^+\cap \mathcal{S}} =\rho _B(|\nabla \varphi |^2)\nabla \varphi \cdot \boldsymbol{\nu }|_{\Lambda ^-\cap \mathcal{S}}, \end{aligned} {\tag{2.5}} \end{equation}$$

where $\boldsymbol{\nu }$ is the unit normal to $\mathcal{S}$ in the flow direction; i.e., $\nabla \varphi \cdot \boldsymbol{\nu }|_{\Lambda ^\pm \cap \mathcal{S}}>0$. A piecewise smooth solution with discontinuities satisfying 2.5 is called an entropy solution of 2.2 if it satisfies the following entropy condition: The density $\rho$ increases in the flow direction of $\nabla \varphi _{\Lambda ^+\cap \mathcal{S}}$ across any discontinuity. Then such a discontinuity is called a shock (see 12); see also Figure 8.⁠⁴

⁴

Author of the picture: Konrad Jacobs. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

✖

Figure 8.

Richard Courant (January 8, 1888–January 27, 1972) and Kurt Otto Friedrichs (September 28, 1901–December 31, 1982); their monumental book 12 has had a great impact upon the development of the M-D theory of shock waves and nonlinear PDEs of hyperbolic/mixed types.

For this problem, there are two configurations: the weak oblique shock reflection with supersonic/subsonic downstream flow (determined by the sonic angle $\theta _{\mathrm{w}}^{\mathrm{s}}$), and the strong oblique shock reflection with subsonic downstream flow; both of these satisfy the entropy condition, as was discovered by Prandtl (cf. Figure 6). The weak oblique shock is transonic with subsonic downstream flow for $\theta _{\mathrm{w}}\in (\theta _{\mathrm{w}}^{\mathrm{s}}, \theta _{\mathrm{w}}^{\mathrm{d}})$, while the weak oblique shock is supersonic with supersonic downstream flow for $\theta _{\mathrm{w}}\in (0, \theta _{\mathrm{w}}^{\mathrm{s}})$. However, the strong oblique shock is always transonic with subsonic downstream flow. The question of physical admissibility of one or both of the strong/weak shock reflection configurations was hotly debated for eight decades in the wake of Courant-Friedrichs 12 and von Neumann 20, and has only recently been better understood (cf. 7 and the references cited therein). There are two natural approaches to understanding this phenomenon: One is to examine whether these configurations are stable under steady perturbations, and the other is to determine whether these configurations are attainable as large-time asymptotic states (i.e., the Prandtl-Meyer problem); both approaches involve the analysis of nonlinear PDEs 2.2 or 2.3 of mixed type.

Mathematically, the steady stability problem can be formulated as a free boundary problem with the perturbed shock-front:

$$\begin{equation} \mathcal{S}=\{\mathbf{x}: x_2 = \sigma (x_1), x_1 \ge 0\} {\tag{2.6}} \end{equation}$$

with $\sigma (0)=0$ and $\sigma (x_1)>0$ for $x_1>0$ as a free boundary (with the Rankine-Hugoniot conditions, say 2.5, as free boundary conditions) to determine the domain behind $\mathcal{S}$:

$$\begin{equation} \Omega =\{\mathbf{x}\in \mathbb{R}^2: b(x_1)<x_2<\sigma (x_1), x_1>0\}, {\tag{2.7}} \end{equation}$$

and the downstream flow in $\Omega$ for Equation 2.2 or System 2.3 of mixed elliptic-hyperbolic type, where $x_2=b(x_1)$ is the perturbation of the flat wedge boundary $x_2=x_1\tan \theta _{\mathrm{w}}$. Such a global solution of the free boundary problem provides not only the global structural stability of the steady oblique shock, but also a more detailed structure of the solution.

Figure 9.

The leading steady shock $x_2=\sigma (x_1)$ as a free boundary under the perturbation.

Supersonic (i.e., supersonic-supersonic) shocks correspond to the case when $\theta _{\mathrm{w}}\in (0, \theta _{\mathrm{w}}^{\mathrm{s}})$; these are shocks of weak strength. The local stability of such shocks was first established in the 1960s. The global stability and uniqueness of the supersonic oblique shocks for both Equation 2.2 and System 2.3 have been solved for more general perturbations of both the upstream steady flow and the wedge boundary, even in $BV$,⁠⁵ by purely hyperbolic methods and techniques (cf. 7 and the references cited therein).

⁵

A $BV$ function is a real-valued function whose total variation is bounded.

✖

For transonic (i.e., supersonic-subsonic) shocks, it has been proved that the oblique shock of weak strength is always stable under general steady perturbations. However, the oblique shock of strong strength is stable only conditionally for a certain class of steady perturbations that require the exact match of the steady perturbations near the wedge-vertex and the downstream condition at infinity, which reveals one of the reasons why the strong oblique shock solutions have not been observed experimentally. In these stability problems for transonic shocks, the PDEs (or parts of the systems) are expected to be elliptic for global solutions in the domains determined by the corresponding free boundary problems; that is, we solve an expected elliptic free boundary problem. However, the earlier methods and approaches for elliptic free boundary problems do not directly apply to these problems, such as the variational methods, the Harnack inequality approach, and other elliptic methods/approaches. The main reason for this is that the type of equations needs to be controlled before we can apply these methods, and this requires some strong a priori estimates. To overcome these difficulties, the global structure of the problems is exploited, which allows us to derive certain properties of the solution so that the type of equations and the geometry of the problem can be controlled. With this, the free boundary problem, as described above, has been solved by an iteration procedure; see Chen-Feldman 7 and the references cited therein for more details.

When a subsonic flow passes through a de Laval nozzle, the flow may form a supersonic bubble with a transonic shock (see Figure 4); full understanding of how the geometry of the nozzle helps to create/stabilize/destabilize the transonic shock requires a deep understanding of the nonlinear PDEs of mixed type. Likewise, for the Morawetz problem for a steady subsonic flow past an airfoil, experimental results show that a supersonic bubble may be formed around the airfoil (see Figures 10–11), and the flow behavior is determined by the solution of a nonlinear PDE of mixed type.

Figure 10.

Transonic flow patterns on an airfoil showing flow patterns at and above the critical Mach number.

Figure 11.

Aerodynamic condensation evidences of supersonic expansion fans around a transonic aircraft.

Some fundamental problems for transonic flow posed in the 1950s–60s (e.g., 36121519) remain unsolved, though some progress has been made in recent years (e.g., 6713 and the references cited therein).

3. Nonlinear PDEs of Mixed Type and Shock Reflection/Diffraction Problems in Fluid Mechanics and Related Areas

In general, fluid flows are time-dependent. We now describe how some longstanding M-D time-dependent fundamental shock problems in fluid mechanics can naturally be formulated as problems for nonlinear PDEs of mixed type through a prototype: the shock reflection-diffraction problem.

When a planar shock separating two constant states (0) and (1), with constant velocities and densities $\rho _0<\rho _1$ (state (0) is ahead or to the right of the shock, and state (1) is behind the shock), moves in the flow direction (i.e., $v_1>0$) and hits a symmetric wedge 2.4 with (a half-wedge) angle $\theta _{\mathrm{w}}$ head-on at time $t = 0$, a reflection-diffraction process takes place for $t > 0$. A fundamental question that arises is which types of wave patterns of shock reflection-diffraction configurations may be formed around the wedge. The complexity of these configurations was first reported by Ernst Mach (cf. Figure 12), who observed two patterns of shock reflection-diffraction configurations: Regular reflection (two-shock configuration) and Mach reflection (three-shock/one-vortex-sheet configuration); these are shown in Figure 14, below.⁠⁶ The issue remained dormant until the 1940s, when John von Neumann 1920 (also cf. Figure 13) and other mathematical/experimental scientists (cf. 261215 and the references cited therein) began extensive research into all aspects of shock reflection-diffraction phenomena. It has been found that the situation is much more complicated than that which Mach originally observed; the shock reflection can be divided into more specific subpatterns, and various other patterns of shock reflection-diffraction configurations such as supersonic regular reflection, subsonic regular reflection, attached regular reflection, double Mach reflection, von Neumann reflection, and Guderley reflection may occur; see 261215 and the references cited therein (also see Figures 14–19, below). Then the fundamental scientific issues include:

⁶

M. Van Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford, 1982.

✖

(i)

the structures of the shock reflection-diffraction configurations;

(ii)

the transition criteria between the different patterns of the configurations;

(iii)

the dependence of the patterns upon physical parameters such as the wedge angle $\theta _{\mathrm{w}}$, the incident-shock-wave Mach number (i.e., the strength of the incident shock), and the adiabatic exponent $\gamma >1$.

Figure 12.

Ernst Waldfried Josef Wenzel Mach (18 February 1838 – 19 February 1916), who first observed the complexity of shock reflection-diffraction configurations (1878).

Figure 13.

John von Neumann (December 28, 1903–February 8, 1957), who proposed the sonic conjecture and the detachment conjecture for shock reflection-diffraction configurations.

In particular, several transition criteria between the different patterns of shock reflection-diffraction configurations have been proposed; these include the sonic conjecture and the detachment conjecture, both put forward by von Neumann 19 (see also 26).

Figure 14.

Three patterns of shock reflection-diffraction configurations.

To present this more clearly, we now focus on the Euler equations for time-dependent compressible potential flow, which consist of the conservation law of mass and Bernoulli’s law:

$$\begin{equation} \partial _t\rho +\operatorname {div}(\rho \nabla \Phi )=0, \quad \partial _t \Phi +{1\over 2}|\nabla \Phi |^2+{1\over \gamma -1}\rho ^{\gamma -1} =\frac{\rho _0^{\gamma -1}}{\gamma -1} {\tag{3.1}} \end{equation}$$

for $(t, {\mathbf{x}})\in \mathbb{R}_+\times \mathbb{R}^2$ after scaling, where $\Phi$ is the time-dependent velocity potential (i.e., $\boldsymbol{v}=\nabla \Phi$ is the velocity). Equivalently, System 3.1 can be reduced to the nonlinear wave equation of second-order:

$$\begin{equation} \partial _t {\rho }(\partial _t\Phi , \nabla _\mathbf{x}\Phi )+ \nabla _\mathbf{x}\cdot \big ({\rho }(\partial _t\Phi , \nabla _\mathbf{x}\Phi )\nabla _\mathbf{x}\Phi \big )=0, {\tag{3.2}} \end{equation}$$

with $\rho (\partial _t\Phi ,\nabla _\mathbf{x}\Phi )= \big (\rho _0^{\gamma -1}-(\gamma -1)(\partial _t\Phi +\frac{1}{2}|\nabla _\mathbf{x}\Phi |^2)\big )^{\frac{1}{\gamma -1}},$ which is one of the original motivations for the extensive study of nonlinear wave equations.

Mathematically, the shock reflection-diffraction problem is a $2$-D lateral Riemann problem for 3.1 or 3.2 in domain $\mathbb{R}^2\setminus \overline{W}$ with $\rho _0, \rho _1, v_1>0$ satisfying

$$\begin{equation} \rho _1>\rho _0, \quad v_1=(\rho _1-\rho _0)\sqrt {\frac{2(\rho _1^{\gamma -1}-\rho _0^{\gamma -1})}{\rho _1^2-\rho _0^2}}. {\tag{3.3}} \end{equation}$$

Problem 3.1 (Shock Reflection-Diffraction Problem).

Piecewise constant initial data, consisting of state $(0)$ with velocity $\mathbf{v}_0=(0,0)$ and density $\rho _0>0$ on $\{x_1>0\}\setminus \overline{W}$ and state $(1)$ with velocity $\mathbf{v}_1=(v_1, 0)$ and density $\rho _1>0$ on $\{x_1 < 0\}$ connected by a shock at $x_1=0$, are prescribed at $t = 0$, satisfying 3.3. Seek a solution of the Euler system 3.1, or Equation 3.2, for $t\ge 0$, subject to the initial data and the boundary condition $\nabla \Phi \cdot \boldsymbol{\nu }_{\mathrm{w}}=0$ on $\partial W$, where $\boldsymbol{\nu }_{\mathrm{w}}$ is the unit outward normal to $\partial W$.

Problem 3.1 is invariant under scaling: $(t, {\mathbf{x}}, \Phi )\rightarrow (\alpha t, \alpha {\mathbf{x}}, \frac{\Phi }{\alpha })$ for any $\alpha \neq 0$. Thus the problem admits self-similar solutions in the form:

$$\begin{equation} \Phi (t, {\mathbf{x}})=t\phi (\boldsymbol{\xi })\qquad \quad \text{for }\boldsymbol{\xi }=\frac{{\mathbf{x}}}{t}. {\tag{3.4}} \end{equation}$$

Then the pseudo-potential function $\varphi (\boldsymbol{\xi })=\phi (\boldsymbol{\xi })-\frac{1}{2}|\boldsymbol{\xi }|^2$ satisfies the equation:

$$\begin{equation} {\mathrm{div}}\big (\rho _B(|\mathrm{D}\varphi |^2,\varphi )\mathrm{D}\varphi \big )+2\rho _B(|\mathrm{D}\varphi |^2,\varphi )=0, {\tag{3.5}} \end{equation}$$

with $\rho _B(|\mathrm{D}\varphi |,\varphi )= \bigl (\rho _0^{\gamma -1}- (\gamma -1)(\frac{1}{2}|\mathrm{D}\varphi |^2+\varphi )\bigr )^{\frac{1}{\gamma -1}},$ where the divergence ${\mathrm{div}}$ and gradient $\mathrm{D}$ are with respect to $\boldsymbol{\xi }\in \mathbb{R}^2$. Define the pseudo-sonic speed $c=c(|\mathrm{D}\varphi |,\varphi )$ by

$$\begin{equation} c^2(|\mathrm{D}\varphi |,\varphi )= \rho ^{\gamma -1}(|\mathrm{D}\varphi |^2,\varphi ) =B_0-(\gamma -1)\big (\frac{1}{2}|\mathrm{D}\varphi |^2+\varphi \big ). {\tag{3.6}} \end{equation}$$

Equation 3.5 is of mixed elliptic-hyperbolic type:

•

strictly elliptic if $|\mathrm{D}\varphi |<c(|\mathrm{D}\varphi |, \varphi )$ (pseudo-subsonic);

•

strictly hyperbolic if