Abstract
We consider the inverse problem for time-dependent semilinear transport equations. We show that time-independent coefficients of both the linear (absorption or scattering coefficients) and nonlinear terms can be uniquely determined, in a stable way, from the boundary measurements, by applying a linearization scheme and Carleman estimates for the linear transport equations. We establish results in both Euclidean and general geometry settings.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
We investigate the time-dependent transport equation with nonlinear term in this article. Let \(\Omega \subset \mathbb R^d\), \(d\geqq 2\), be an open bounded and convex domain with smooth boundary \(\partial \Omega \). We denote
for \(T>0\). We also denote the outgoing and incoming boundaries of \(S\Omega \) by \(\partial _+ S \Omega \) and \(\partial _- S \Omega \) respectively which are defined as follows:
where n(x) is the unit outer normal vector at \(x\in \partial \Omega \) and \(\langle v,w \rangle \) is the dot product in \(\mathbb R^d\). Moreover, \(\partial _\pm S\Omega _T:=(0,T)\times \partial _\pm S\Omega .\) Let the function \(f\equiv f(t,x,v)\) be the solution to the following initial boundary value problem for the nonlinear transport equation:
where T is sufficiently large, \(\sigma \equiv \sigma (x,v)\) is the absorption coefficient and the scattering operator K takes the form
with the scattering coefficient \(\mu \equiv \mu (x,v',v)\) and the normalized measure, that is, \(\int _{\mathbb {S}^{d-1}} \,\textrm{d}\omega (v')=1\), where \(d\omega (v')\) is the measure on \(\mathbb {S}^{d-1}\).
In this paper, we are interested in the inverse problem for the nonlinear transport equation in (1.1). The main objectives are to determine the nonlinearity N, absorption \(\sigma \) and the scattering coefficient \(\mu \) by the boundary data. The problem is motivated by applications in the photoacoustic tomography, in which the nonlinear excitation is observed due to two-photon absorption effect of the underlying medium, see [29, 43, 44, 47] and the references therein.
There has been extensive study in the inverse coefficient problem for the transport equation. The associated inverse problem is concerned with determining unknown properties (such as absorption and scattering coefficients, \(\sigma \) and \(\mu \)) from the albedo operator which maps from incoming to outgoing boundary. The uniqueness result was studied in [10, 12,13,14,15, 46] and stability estimates were derived in [3,4,5,6, 38, 48, 49]. See also recent references [2, 45]. Moreover, related studies in the Riemannian setting can be found in [1, 39,40,41,42]. As for the nonlinear transport equation, the unique determination for the kinetic collision kernel was derived in [30] for the stationary Boltzmann equation and in [36] for the time-dependent Boltzmann equation. In addition to the recovery of the collision kernel, the determination of the Lorentzian spacetime, i.e. the first order information, from the source-to-solution map for the Boltzmann equation was considered in [7].
The main strategy we applied here is using the Carleman estimate for the linear transport equation and the linearization technique. A Carleman estimate, established by Carleman [9], is an \(L^2\) weighted estimate for a solution to a partial differential equation with large parameters. Roughly speaking, a special weight function in the Carleman estimate is chosen to control irrelevant information and then extract the desired properties. The Carleman estimates have been successfully applied in solving inverse problems for various equations. We refer readers to the related references [8, 18, 19, 21, 22, 26, 38] for applications the inverse transport problem. As for the linearization technique, it deals with nonlinear equations in inverse problems to reduce the nonlinear equation to the linear one. In this paper, we apply the higher order linearization whose feature is that it introduces small parameters into the problem for the nonlinear equation. Then differentiating it multiple times with respect to these parameters to earn simpler and new linearized equations. For more detailed discussions and related studies, see for instance [11, 25, 35] for hyperbolic equations, [17, 20, 23, 24, 27, 28, 31,32,33,34, 37] for elliptic equations, and [29, 30, 36] for kinetic equations.
1.1 Main Results
Throughout this paper, we suppose that T is sufficiently large which depends on the domain. Suppose that \(\sigma \in L^\infty (S\Omega )\) and \(\mu \in L^\infty (S\Omega ^2)\) and there exist positive constants \(\sigma ^0\) and \(\mu ^0\) such that
Moreover, suppose that \(\mu \) satisfies
for almost every \((x,v)\in S\Omega \). The assumption (1.4) means that the absorption effect is stronger than the scattering effect in the medium.
Now we denote the measurement operator \(\mathcal {A}_{\sigma , \mu ,N}\) by
It follows from Theorem 2.6 in Section 2 that the initial boundary value problem (1.1) is well-posed for small initial and boundary data \((f_0,f_-)\). Specifically, there exists a small parameter \(\delta >0\) such that when
the initial boundary value problem (1.1) has a unique solution. Hence, the map \(\mathcal {A}_{\sigma , \mu ,N}\) is well-defined within the class of small given data.
The paper is devoted to investigating the inverse coefficient problem for the transport equation with nonlinearity. We study the reconstruction of the absorption coefficient (or scattering coefficient) as well as the nonlinear term from the measurement operator. In that follows, we illustrate the main results on \(\mathbb R^d\) (discussed in Section 3) and also results on Riemannian manifolds (discussed in Section 4) separately.
1.1.1 Inverse Problems in Euclidean Space
In the first theme of the paper, we consider the problem (1.1) with the nonlinear term \(N(x,v,f):S\Omega \times \mathbb R\rightarrow \mathbb R\) satisfying the following conditions:
This implies that N can be expanded into a power series
which converges in the \(L^\infty (S\Omega )\) topology with \(q^{(k)}(x,v):=\partial _z^kN(x,v,0)\in L^\infty (S\Omega )\).
For a fixed vector \(\gamma \in \mathbb {S}^{d-1}\), we say a function p is in the set \(\Lambda \) if p satisfies
for some fixed constant \(\gamma _0>0\). We state the first main result. The inverse problem here is to recover \(\sigma \) and N provided that \(\mu \) is given.
Theorem 1.1
Let \(\Omega \) be an open bounded and convex domain with smooth boundary. Suppose that \(\sigma _j\in L^\infty (S\Omega )\) and \(\mu \in L^\infty (S\Omega ^2)\) satisfy (1.3) and (1.4) for \(j=1,\,2\). Let \(N_j:S\Omega \times \mathbb R\rightarrow \mathbb R\) satisfy the assumption (1.7) with \(q^{(k)}\) replaced by \(q^{(k)}_j\) for \(j=1,2\), respectively. Let \(\sigma _j\), \(\mu (\cdot ,\cdot ,v)\), \(q_j^{(k)}\) for all \(k\geqq 2\) be in \(\Lambda \). If
for any \(h\in L^\infty (S\Omega )\) with \(\Vert h\Vert _{L^\infty (S\Omega )}\leqq \delta \) for sufficiently small \(\delta \), then
Remark 1.1
We would like to point out that the constant \(\gamma _0\) indeed can be chosen to be arbitrarily small as long as \(\gamma _0>0\). In this case, the condition (1.9) becomes less restrictive in the sense that the coefficients only need to vanish in a small subset of \(\mathbb {S}^{d-1}\) in order to make the above uniqueness results hold. We refer to Section 3 for detailed discussions and for more relaxed conditions, instead of (1.9), on \(\sigma _j,\, \mu \) and \(q_j^{(k)}\).
Remark 1.2
On the other hand, suppose that \(\sigma \) is given and \(\mu \) is unknown and is of the form \(\mu :=\tilde{\mu }(x,v)p(x,v',v)\). In this case, we can also recover \(\tilde{\mu }\), see Proposition 3.5 for details. Combining with the reconstruction of N(x, v, z), we obtain the determination of both the scattering coefficient and the nonlinear term provided that \(\sigma \) is known.
Moreover, we also consider the problem when the nonlinear term has the form
where \(N_0\) satisfies
and
for a positive integer \(\ell \geqq 2\) and constants \(C_1,C_2>0\), independent of f. For instance, when \(\ell =2\), \(N_0(f)\) can represent the quadratic nonlinearity such as \(N_0(f)=f^2\) or \(f\int _{\mathbb {S}^{d-1}} f\textrm{d}\omega (v')\). The latter example finds applications in photoacoustic tomography with nonlinear absorption effect and we refer the interested readers to the references [29, 43].
Theorem 1.2
Let \(\Omega \) be an open bounded and convex domain with smooth boundary. Suppose that \(\sigma _j\in L^\infty (S\Omega )\) and \(\mu \in L^\infty (S\Omega ^2)\) satisfy (1.3) and (1.4) for \(j=1,\,2\). Let \(N_j(x,v,f)=q_j(x,v)N_0(f)\), where \(q_j\in L^\infty (S\Omega )\) for \(j=1,2\) and \(N_0\) satisfies (1.10)-(1.11) with \(\partial ^2_z N_0(0)>0\). Let \(\sigma _j\), \(\mu (\cdot ,\cdot ,v)\), \(q_j\) be in \(\Lambda \). If
for any \(h\in L^\infty (S\Omega )\) with \(\Vert h\Vert _{L^\infty (S\Omega )}\leqq \delta \) for sufficiently small \(\delta \), then
Remark 1.3
Similarly, as discussed in Remark 1.2, if \(\sigma \) is now given, then we can recover \(\tilde{\mu }\) and N from the boundary data as well.
1.1.2 Inverse Problems on Manifolds
The second theme of the paper is the inverse problems for the transport equation on manifolds.
We denote M the interior of a compact non-trapping Riemannian manifold \((\overline{M}, g)\) with smooth strictly convex (with respect to the metric g) boundary \(\partial M\). Since \(\overline{M}\) is non-trapping, any maximal geodesic will exit \(\overline{M}\) in finite time, i.e. have finite length. M plays the role of \(\Omega \) in the manifold case, and thus we naturally generalize the notations for \(\Omega \) (e.g. \(S\Omega \), \(S\Omega _T\), \(\partial _\pm S\Omega _T\), etc.) to corresponding notations for M (See e.g. SM, \(SM_T\), \(\partial _\pm SM_T\), etc.). See Section 2 for more details.
We consider the following initial boundary value problem:
Here X is the geodesic vector field which generates the geodesic flow on SM, see Section 2 for more details. In particular, \(X=v\cdot \nabla _x\) in the Euclidean case. The equation (1.12) is in the absence of the scattering effect, due to our Carleman estimates on Riemannian manifolds in Section 4. The Carleman weight function chosen in this paper is naturally associated with the geodesic flow of the Riemannian manifold, which depends on both the position x and the direction v, and therefore makes it hard to control the scattering term by other terms in the estimate, see also Remark 4.1. Since the main scope of the paper is recovering the nonlinearity of the transport equation, we do not pursue further the inverse problem with the scattering term in the Riemannian case.
Let \(\mathcal A_{\sigma ,N}:=\mathcal A_{\sigma ,0,N}\) be the measurement operator associated with the problem (1.12). Analogous to the results in the Euclidean case, we have the following two main results on Riemannian manifolds:
Theorem 1.3
Let M be the interior of a compact non-trapping Riemannian manifold \(\overline{M}\) with smooth strictly convex boundary \(\partial M\). Suppose that \(\sigma _j\in L^\infty (SM)\) satisfy (1.3) for \(j=1,\,2\). Let \(N_j:SM\times \mathbb R\rightarrow \mathbb R\) satisfy the assumption (1.7) in the manifold with \(q^{(k)}\) replaced by \(q^{(k)}_j\) for \(j=1,2\), respectively. If
for any \(h\in L^\infty (SM)\) with \(\Vert h\Vert _{L^\infty (SM)}\leqq \delta \) for sufficiently small \(\delta \), then
Moreover, when the nonlinear term takes the form \(N(x,v,f)=q(x,v)N_0(f)\), we have the following result:
Theorem 1.4
Suppose that \(\sigma _j\in L^\infty (SM)\) satisfies (1.3) for \(j=1,\,2\). Let \(N_j(x,v,f)=q_j(x,v)N_0(f)\), where \(q_j\in L^\infty (SM)\) for \(j=1,2\) and \(N_0\) satisfies (1.10)-(1.11) in the manifold with \(\partial ^2_z N_0(0)>0\). If
for any \(h\in L^\infty (SM)\) with \(\Vert h\Vert _{L^\infty (SM)}\leqq \delta \) for sufficiently small \(\delta \), then
Remark 1.4
We actually only need much less data to stably determine both \(\sigma \) and N. To be more specific, fix positive \(h\in L^\infty (SM)\) with \(X^\beta h\in L^\infty (SM)\) for \(\beta =1,2\), consider the initial boundary value condition \((\varepsilon h,0)\) for \(|\varepsilon |\) sufficiently small, we establish the following stability result:
If in addition \(\sigma =\sigma _1=\sigma _2\), then
The constants C in both estimates are independent of \(\sigma _j\) and \(q_j\), \(j=1,2\). See Proposition 4.4 and Proposition 4.5 for more details. Similar results hold when the nonlinear term \(N_j\), \(j=1,2\) satisfy the assumption (1.7), see e.g. Lemma 3.6 and the proof of Lemma 3.8.
The rest of this part of the paper is organized as follows: in Section 2, we introduce the notations and function spaces, and also establish several preliminary results, including boundedness of solutions to the linear equation, Maximum principle, and the well-posedness problem for the nonlinear transport equation. We investigate the reconstruction of the unknown coefficients in the Euclidean setting and prove Theorem 1.1 in Section 3. In particular, we establish an improved version of the Carleman estimate of [38]. In Section 4, we first deduce the Carleman estimate and the energy estimate in a Riemannian manifold. With these estimates, Theorem 1.3 follows directly by applying similar arguments as in the proof of Theorem 1.1. Furthermore, in the case of \(N =q N_0(f)\), we show the unique determination of q, which immediately implies the uniqueness of N in Theorem 1.4. Finally, we note that the techniques for showing Theorem 1.4 can also be applied to prove Theorem 1.2.
2 Preliminaries
In this section, we will discuss the forward problem for the initial boundary value problem for the nonlinear transport equation. In particular, we will prove the well-posedness result on a more general setting, namely, the Riemannian manifold. All the results discussed in this section are also valid in the Euclidean space and will be utilized in Section 3.
2.1 Notations and Spaces
In order to investigate the transport equation on a Riemannian manifold, we need to introduce the related notations first. Most of the notations below are similar to the ones we saw earlier in Section 1, but with \(\Omega \) replaced by the manifold M.
Let M be the interior of a compact Riemannian manifold \((\overline{M}, g)\), of dimension \(d\geqq 2\), with a Riemannian metric g and strictly convex boundary \(\partial M\). Suppose that \(\overline{M}\) is non-trapping. Let TM be the tangent bundle of M. We denote the unit sphere bundle of the manifold (M, g) by
where \(\langle \cdot \,,\,\cdot \rangle _{g(x)}\) is the inner product on the tangent space \(T_xM\). Let \(\partial _+ S M\) and \(\partial _- SM\) be the outgoing and incoming boundaries of SM respectively and they are defined by
where n(x) is the unit outer normal vector at \(x\in \partial M\). For any point \(x\in M\), let \(S_xM:=\{v:\, (x,v)\in SM\}\). Moreover, we also denote
Let \(T>0\), we denote \(SM_T:=(0,T)\times SM\) and \(\partial _\pm SM_T:=(0,T)\times \partial _\pm SM\).
For every point \(x\in M\) and every vector \(v\in S_xM\), let \(\gamma _{x,v}(s)\) be the maximal geodesic satisfying the initial conditions
Since M is non-trapping, \(\gamma _{x,v}\) is defined on the finite interval \([-\tau _-(x,v),\tau _+(x,v)]\). Here the two travel time functions
are determined by \(\gamma (\pm \tau _\pm (x,v))\in \partial M\). In particular, they satisfy \(\tau _+(x,v)=\tau _-(x,-v)\) for all \((x,v)\in SM\) and \(\tau _-(x,v)|_{\partial _-SM}=\tau _+(x,v)|_{\partial _+SM}=0\). Denote the geodesic flow by
Let X be the generating vector field of the geodesic flow \(\phi _{t}(x,v)\), that is, for a given function f on SM, \(Xf(x,v)=\frac{\textrm{d}}{\textrm{d}t}f(\phi _t(x,v))|_{t=0}\). Notice that in the Euclidean space \(\mathbb R^d\), \(\phi _t(x,v)=(x+tv,\,v)\) and \(X=v\cdot \nabla _x\) where v is independent of x.
We define the spaces \(L^p(SM)\) and \(L^p(SM_T)\), \(1\leqq p<\infty \), with the norm
with \(d\Sigma =d\Sigma (x,v)\) the volume form of SM. Moreover, for the spaces \(L^p(\partial _\pm SM_T)\), we define its norm to be
where \(d\xi (x,v):=\langle n(x),v\rangle _{g(x)}d\tilde{\xi }(x,v)\) with \(d\tilde{\xi }\) the standard volume form of \(\partial SM\). Note that in the Euclidean setting since v is independent of x, we denote \(d\tilde{\xi }= d\lambda (x)d\omega (v)\), where \(d\lambda \) is the measure on \(\partial \Omega \) and \(d\omega (v)\) is the measure on \(\mathbb {S}^{d-1}\). We also define the spaces \(H^k(0,T;L^2(SM))\) for positive integer k with the norm
When \(p=\infty \), \(L^\infty (SM)\), \(L^\infty (SM_T)\) and \(L^\infty (\partial _\pm SM_T)\) are the standard vector spaces consisting of all functions that are essentially bounded.
We first study the forward problem for the linear transport equation in Section 2.2. Equipped with this, we apply the contraction mapping principle to deduce the unique existence of solution to the nonlinear transport equation in Section 2.3.
2.2 Forward Problem for the Linear Transport Equation
We consider the initial boundary value problem for the linear transport equation with the source \(S\equiv S(t,x,v)\):
there the scattering operator K on the manifold takes the form
We will demonstrate the existence of a solution to (2.2) by proving that the corresponding integral equation has a solution. To achieve this, we study the following simpler case first.
Proposition 2.1
Suppose that \(\sigma \in L^\infty (SM)\) and \(f_-\in L^\infty (\partial _-SM_T)\). The solution f of the problem
is
where H is the Heaviside function, that is, H satisfies \(H(s)=0\) if \(s<0\) and \(H(s)=1\) if \(s>0\).
To simplify the notation, in the formulation above we denote \(\tau _-:=\tau _-(x,v)\) for a fixed \((x,v)\in SM\).
Proof
For a fixed \((x,v)\in SM\) and \(0<t<T\), let
The equation (2.4) can be written as
whose solution is
Choosing \(s=\tau _-(x,v)\), we have
which leads to
by applying the change of variable \(\tilde{\eta }=-\eta +\tau _-(x,v)\). By taking \(F(0)=f(t-\tau _-(x,v),\phi _{-\tau _-(x,v)}(x,v))\) which vanishes if \(t\leqq \tau _-(x,v)\), we obtain the desired result.\(\square \)
Let’s study the integral formulation of the linear transport equation (2.2).
Proposition 2.2
Suppose that \(\sigma \in L^\infty (SM)\) and \(\mu \in L^\infty (SM^2)\) satisfy (1.3) and (1.4). Let \(S\in L^\infty (SM_T)\), \(f_0\in L^\infty (SM)\), and \(f_-\in L^\infty (\partial _-SM_T)\). Then the solution f to (2.2) satisfies the integral formulation of the transport equation:
In the Euclidean case, this result can be found in Proposition 4 (page 233), combining with Remark 12, in [16]. To make the paper self contained, we provide below the proof for the Riemannian case.
Proof
We first consider the homogeneous boundary condition, that is, \(f_-=0\). Multiplying
on both sides of the transport equation in (2.2), we get
where we denote \(g:=K(f)+S\). By solving the differential equation (2.7) and then multiplying \(e^{-\int ^{t}_0\sigma (\phi _{\eta +k}(x,v))\textrm{d}\eta }\) on both sides of the solution, we have
Replacing \(\phi _{t+k}(x,v)\) by \(\phi _0(x,v)=(x,v)\) (that is, taking \(k=-t\)) in (2.8) gives
Moreover, we apply the change of variables \(\tilde{\eta }=-\eta +t\) so that (2.9) becomes
We then apply another change of variables \(\tilde{s}=-s+t\) so that
From (2.10) and (2.11), taking \(f_0(\phi _{-t}(x,v))=0\) if \(\phi _{-t}(x,v)\notin \Omega \) (namely, \(t\geqq \tau _-(x,v)\)), we derive that the solution satisfies the integral equation with \(f_-\equiv 0\).
Next, in the case of a nonhomogeneous boundary condition \(f_-\ne 0\), we let \(f_1\) be the solution of (2.4) and look for the solution f of the problem (2.2) in the form \(f=f_1+w\), where w is the solution of
Since w has the homogeneous boundary condition, w satisfies the integral equation with \(f_-=0\). Therefore, combining this with (2.5), we finally deduce that \(f=f_1+w\) satisfies (2.6).\(\square \)
In the following we will see that solving the integral equation (2.6) is equivalent to solving (2.2). Hence once we show that the integral equation (2.6) has a unique solution, this is sufficient to say that the well-posedness of (2.2) holds.
Proposition 2.3
Under the hypothesis of Proposition 2.2, if f satisfies the integral equation (2.6), then f is the solution to (2.2). Moreover, there exists a unique solution to the integral equation (2.6).
Proof
Step 1: Equivalence. Below we will show that if there exists a function f satisfying (2.6), then such f is a solution to (2.2). Notice that
We apply the operator \(\partial _t+X\) to the right-hand side of the integral formula (2.6) to get
Here we used the fact that \(\tau _-(\phi _k(x,v))=\tau _-(x,v)+k\).
Now we consider \(I_1\) - \(I_3\) separately. For \(I_1\), we have
For \(I_2\),
We denote \(m=s-k\), then
Combining the above 3 terms together, we have
Finally, it’s easy to check that \(f(0,x,v)=f_0(x,v)\) if \((x,v)\in SM\), and \(f(t,x,v)=f_-(t,x,v)\) if \((x,v)\in \partial _-SM\) and \(t>0\). We thus conclude that f is a solution to (2.2). Combining with Proposition 2.2, we see that to show the forward problem of (2.2), it is sufficient to find a solution to the integral equation.
Step 2: Existence of solutions to the integral equation. We define a sequence of functions \(f^{(n)}\) in the following ways:
and for \(n\geqq 0\),
Let \(w^{(n+1)}:= f^{(n+1)}-f^{(n)}\) for \(n\geqq 0\) and then be represented as
Recall that in (1.4) for almost every \((x,v)\in SM\), \(\mu \) satisfies
From this, we can derive that
for \((t,x,v)\in SM_T\). We then denote the scalar value \(\kappa \) by
It is clear that \(0\leqq \kappa <1\) since \(0\leqq \sigma \leqq \sigma ^0\). Due to the monotonicity of \(e^{-\int ^s_0\sigma (\phi _{-r}(x,v))dr}\) with respect to s, we obtain
Next, we estimate the third term on the right-hand side of (2.13). From (1.3), we derive that
Thus (2.13) and \(\sigma \geqq 0\) lead to
Combining these estimates (2.16)-(2.17) together, we can derive that
with \(0\leqq \kappa <1\). This implies that the series \(\sum ^\infty _{n=0}w^{(n+1)}\) is convergent and thus the partial sum
converges to a limit f in \(L^\infty (SM_T)\). In particular, f satisfies the integral equation:
and, furthermore, f is also a solution of (2.2) due to Step 1.
Step 3: Unique solution for the integral equation. Finally we show the uniqueness of the solution. Let \(f_1\) and \(f_2\) in \(L^\infty (SM_T)\) be the solutions to (2.6). Let \(w:=f_1-f_2\in L^\infty (SM_T)\). Then w satisfies the integral equation:
Following the argument as in (2.15), we obtain
This implies that \(w\equiv 0\).\(\square \)
From the above discussion, we have shown that there exists a unique solution f to the integral equation. Due to the equivalence, such f is also a solution to (2.2). Hence we can now conclude the following well-posedness result for the problem (2.2).
Proposition 2.4
(Well-posedness for linear transport equation) Suppose that \(\sigma \in L^\infty (SM)\) and \(\mu \in L^\infty (SM^2)\) satisfy (1.3) and (1.4). Let \(S\in L^\infty (SM_T)\), \(f_0\in L^\infty (SM)\) and \(f_-\in L^\infty (\partial _-SM_T)\). We consider the following problem:
Then (2.19) has a unique solution f in \(L^\infty (SM_T)\) satisfying
where the constant C depends on \(\sigma \), T.
Proof
From Proposition 2.2 and Proposition 2.3, it is clear that the solution f to (2.19) uniquely exists. Moreover, using a similar argument as in (2.15), we can derive the stability estimate (2.20) from (2.6).\(\square \)
It has been proved in [16], Theorem 3, p229] that when \(f_-\equiv 0\), \(S\geqq 0\) and \(f_0\geqq 0\), the solution is nonnegative. In the next proposition, we show the maximum principle for the transport equation, namely, the solution to (2.19) is strictly positive if \(S\geqq 0\), the initial and boundary data are strictly positive.
Proposition 2.5
(Maximum principle) Suppose the hypotheses in Proposition 2.4 hold and suppose that \(S\geqq 0\). If \(f_0\geqq c> 0\) and \(f_-\geqq c>0\) for some positive constant c, then there exists a positive constant \(\tilde{c}\) such that \(f\geqq \tilde{c} >0\) in \(SM_T\).
Proof
From (2.13), \(\sigma \leqq \sigma ^0\) in (1.3), and the hypothesis \(f_0,\,f_-\geqq c> 0\), we obtain
This implies \(K(f^{(0)})\geqq 0\) due to \(\mu \geqq 0\). Hence, by induction, we can derive from (2.14) that for \(n\geqq 0\),
We therefore have an increasing sequence converging to a function f(t, x, v), which satisfies \(f(t,x,v)\geqq e^{-T\sigma ^0}c>0\). Alternatively, we can apply the proof in Proposition 2.3, which gives that \(f^{(n)}\rightarrow f\) in \(L^\infty (SM_T)\) as \(n\rightarrow \infty \). Hence this also leads to the same result, that is,
This completes the proof.\(\square \)
2.3 Forward Problem for the Nonlinear Transport Equation
Equipped with the well-posedness result for the linear equation, we will prove the unique existence of solution for the following problem:
Theorem 2.6
(Well-posedness for nonlinear transport equation) Let M be the interior of a compact non-trapping Riemannian manifold \(\overline{M}\) with strictly convex boundary \(\partial M\). Suppose that \(\sigma \) and k satisfy (1.3) and (1.4). Then there exists a small parameter \(0<\delta <1\) such that for any
the problem (2.21) has a unique small solution \(f\in L^\infty (SM_T)\) satisfying
where the positive constant C is independent of f, \(f_0\) and \(f_-\).
Proof
To show the existence, let \((f_0,f_-) \in \mathcal {X}^M_\delta \), we first consider the following problem for the linear equation:
By Proposition 2.4, there exists a unique solution \(\hat{f}\) of (2.23) that satisfies
where the constant \(C>0\) is independent of \(\hat{f}\), \(f_-\) and \(f_0\).
Now we let \(w:=f-\hat{f}\). We observe that if such function w exists, then w must satisfies the following problem:
To prove (2.25) has a solution, we apply the contraction mapping principle. We denote the set
where the parameter \(\eta >0\) will be determined later. For \(\varphi \in \mathcal {G}\), we define the function F by
Then \(F(\varphi )\in L^\infty (SM_T)\) due to (2.24) and the hypothesis of N(f). In particular, Proposition 2.4 yields that the problem
is uniquely solvable for any \(\varphi \in \mathcal {G}\). We now denote \(\mathcal {L}^{-1}:F(\varphi )\in L^\infty (SM_T)\mapsto \tilde{w}\in L^\infty (SM_T)\) the solution operator for the problem (2.26) and also define the map \(\Psi \) on the set \(\mathcal {G}\) by
In the following, we will show that \(\Psi \) is a contraction map on \(\mathcal {G}\). To this end, we first show that \(\Psi (\mathcal {G})\subset \mathcal {G}\). Taking \(\varphi \in \mathcal {G}\), from (1.7), the Taylor’s Theorem, and Proposition 2.4, we derive that
where constant \(C>0\) is independent of \(\delta \) and \(\eta \). Note that both \(\partial ^2_zN(x,v,0)\) and
are bounded in \(SM_T\). We then take \(\delta ,\, \eta \) sufficiently small with \(0<\delta<\eta <1\) such that
which implies \(\Psi \) maps \(\mathcal {G}\) into itself.
Moreover, for any \(\varphi _1,\, \varphi _2 \in \mathcal {G}\), from Proposition 2.4, we can also derive that
We estimate
Here we used the fact that \(N_r\) is Lipschitz in z with the Lipschitz constant independent of \(x,\, v\) due to the boundedness of \(\partial _z^kN\). In addition, we choose small \(\delta ,\, \eta \) so that
This yields that \(\Psi \) is a contraction map. By the contraction mapping principle, there exists a unique \(w\in \mathcal {G}\) so that \(\Psi (w)=w\), which then satisfies the problem (2.25). Also w satisfies the estimate
We further take \(\delta ,\,\eta \) small enough so that \(C \left( (\delta +\eta ) +(\delta +\eta )^{2}\right) \leqq 1/2\) and, therefore, the term containing \(\Vert w\Vert _{L^\infty (SM_T)}\) on the right-hand side can then be absorbed by the left-hand side, it follows that
Finally we conclude that \(f=w+\hat{f}\) is the solution to the problem (2.21) and it satisfies
due to (2.24). This completes the proof.\(\square \)
3 Inverse Problems in the Euclidean Space
In this section, we will discuss the inverse problem for the nonlinear transport equation in the Euclidean space. The main objective is to show that the nonlinear term as well as the absorption coefficient (or scattering coefficient) can be recovered from the boundary measurements. Notice that as mentioned previously, the well-posedness result in Section 2 also holds in the domain \(\Omega \) in \(\mathbb R^d\).
Recall the following notations in Section 1:
Suppose that the absorption coefficient \(\sigma \in L^\infty (S\Omega )\) and scattering coefficient \(\mu \in L^\infty (S\Omega ^2)\) are known and satisfy (1.3) and (1.4). We consider the nonlinear term N that satisfies (1.7) and takes the form
where \(q^{(k)}(x,v) = \partial _{z}^kN(x,v,0)\in L^\infty (S\Omega )\) and the series converges in \(L^\infty (S\Omega )\).
Let f be the solution to the initial boundary value problem:
The unique existence of small solution f follows by applying Theorem 2.6, which is also valid in the Euclidean space. Recall that we denote the measurement operator by
In Section 2, we have defined backward/forward exit time in the Riemannian manifold. We will adapt these definitions in the Euclidean setting here. For \((x,v)\in S\Omega \), the backward exit time \(\tau _-(x,v)\) is defined by
This is the time at which a particle \(x\in \Omega \) with velocity \(-v\) leaves the domain \(\Omega \). Similarly, we define the forward exit time \(\tau _+(x,v)\) for every \((x,v)\in S\Omega \) by
In particular, when \((x,v)\in \partial _\pm S\Omega \), we have \(\tau _\pm (x,v)=0\). Suppose that T is sufficiently large so that \(T>\text {diam}\,\Omega \), where the notation \(\text {diam}\,\Omega \) denotes the diameter of \(\Omega \).
This section is structured as follows. We first study the reconstruction of the linear coefficients in Section 3.1 under suitable assumptions. Standing on this result, we will show that the nonlinear term can be uniquely determined from the measurement in Section 3.2.
3.1 Recover \(\sigma \) or \(\mu \)
To recover the unknown \(\sigma \) and \(\mu \), we apply the first order linearization to reduce the nonlinear equation to a linear equation without the unknown N(x, v, f). From this, the Carleman estimate for the transport equation is applied to achieve the goal.
For small parameter \(\varepsilon \), the well-posedness result in Theorem 2.6 yields that there is a unique small solution \(f(t,x,v)\equiv f(t,x,v;\varepsilon )\) to (3.1) with initial data \(f|_{t=0}=\varepsilon h\) and boundary data \(f|_{\partial _-S\Omega _T} =\varepsilon g\). We can obtain the differentiability of the solution \(f=f(t,x,v;\varepsilon )\) with respect to \(\varepsilon \) by adapting the proof of [29], Proposition A.4], where the differentiability is discussed for a nonlinear transport equation, to our setting. Hence, we have the k-th derivative of f with respect to \(\varepsilon \) at \(\varepsilon =0\), which is defined by
for any integer \(k\geqq 1\).
Now we perform the first linearization of the problem (3.1) with respect to \(\varepsilon \) at \(\varepsilon =0\). Due to the well-posedness result, the nonlinear term is eliminated and only the linear terms are preserved. Then (3.1) becomes
Hence the problem is reduced to studying the inverse coefficient problem for the above linear transport equation. Note that the unique determination of \((\sigma , \mu )\) from the albedo operator was shown in [13,14,15] by applying the singular decomposition of the operator under suitable assumptions. One might recover both \(\sigma \) and \(\mu \) by directly applying these existing results for the linear equation. However, additional assumptions might be needed to deduce the uniqueness and stability results in our setting. Therefore, to be consistent with the assumptions we have made in this paper, we will only focus on applying the Carleman estimate to recover either \(\sigma \) or \(\mu \) by assuming that the other one is given.
Let us briefly discuss how to build the Carleman estimate for the transport equation with linear Carleman weight function \(\varphi \), see also [38]. First we note that the Carleman estimate is valid under the geometric assumption on the velocity. For a fixed vector \(\gamma \in \mathbb {S}^{d-1}\), we denote the subset V of the unit sphere by
for some positive constant \(\gamma _0\). For a fixed \(0<\beta <\gamma _0\), there exists a constant \(a>0\) so that \(\gamma \cdot v-\beta \geqq a>0\) in V. Then we define the function
Next we define the weight function \(\varphi \in C^2([0,T]\times \overline{\Omega })\) by
It follows that \((\partial _t+v\cdot \nabla _x)\varphi = B(v)>0\), which is essential in the derivation of the Carleman estimate later.
Moreover, we define the transport operator
Let \(w(t,x,v)=e^{s\varphi }f(t,x,v)\) for \(s>0\). We define the linear operator L by
We denote \(Q:=(0,T)\times \Omega \). From the identity
applying the integration by parts, one can derive the Carleman estimate in the following proposition.
Proposition 3.1
For a fixed \(\gamma _1>0\), suppose that \((\sigma ,\mu )\) satisfy
and
for some constants \(C_\sigma ,\,C_\mu > 0\). Let \(f \in H^1(0,T;L^2(S\Omega ))\) satisfy \(v\cdot \nabla _x f \in L^2(S\Omega _T)\) and \(f(T,x,v)=0\). Suppose the initial data \(f(0,x,\cdot )\) is supported in V. Then there exist positive constants \(C=C(a,\gamma _0)\) and \(s_0=s_0(d,\gamma _1,C_\sigma ,C_\mu , \Vert \sigma \Vert _{L^\infty })\) so that for all \(s\geqq s_0>0\), we have
Proof
Since \(f(T,x,v)=0\), for any vector \(v\in \mathbb {S}^{d-1}\), applying the integration by parts leads to the following estimate:
Using (3.5), we can bound the last term by the third term on the right, that is,
if s is large enough. Since \(w=e^{s\varphi }f(t,x,v)\), integrating over \(\mathbb {S}^{d-1}\) yields the Carleman estimate without the scattering:
by noting that \(B\geqq a>0\) in V and \(f(0,x,\cdot )\) is supported in V.
To derive (3.7), we observe that
Due to \(B^{-1}\mu \in L^2(\mathbb {S}^{d-1})\), applying Hölder’s inequality, we get
It leads to
which can then be absorbed by the second term on the left-hand side of (3.8) provided that s is large enough. This ends the proof.\(\square \)
We still need the following energy estimate (it can be showed by adapting the argument in [Lemma 2.1, [38]] for our case \(V=\mathbb {S}^{d-1}\) and, therefore, we omit the proof here):
Lemma 3.2
Let \(\Omega \) be an open bounded and convex domain with smooth boundary. Suppose that \(\sigma \in L^\infty (S\Omega )\) and \(\mu \in L^\infty (S\Omega ^2)\) satisfy (1.3) and (1.4). Let \(f_0\in L^\infty (S\Omega )\) satisfy \((v\cdot \nabla _x)^\beta f_0\in L^\infty (S\Omega )\), \(\beta =1,\,2\). Let f be the solution to the problem
and satisfy \(f \in H^2(0,T;L^2(S\Omega ))\) and also \((v\cdot \nabla _x) f \in H^1(0,T;L^2(S\Omega ))\). Suppose that the source term has the form
with \(\widetilde{S}\in L^\infty (S\Omega )\) and
for some constant \(c_3>0\). Then there exists a constant \(C>0\), which depends on \(c_3\), \(\Vert \sigma \Vert _{L^\infty (S\Omega )}\), and \(\Vert \mu \Vert _{L^\infty (S\Omega ^2)}\), so that
The next theorem states the main estimate which will be used to prove the inverse coefficient/source problems. It indicates that partial information of the source term can be revealed by applying the Carleman estimate on the cut-off function of \(\partial _t u\) on the time variable see [38] for a similar argument.
Theorem 3.3
Under the same conditions and hypotheses of Lemma 3.2, let \(S_0(0,x,v)\), \(f_0(x,v)\) and \(\mu (x,v',\cdot )\) be supported in V. Suppose that \(\sigma \) and \(\mu \) satisfy (3.5) and (3.6). Suppose that
for some fixed constants \(c_1,c_2>0\), and
Then there exists a positive constant C, depending on \(c_j\) (\(j=1,2,3\)), \(\Vert \sigma \Vert _{L^\infty (S\Omega )}\), and \(\Vert \mu \Vert _{L^\infty (S\Omega ^2)}\), so that
Remark 3.1
From the proof below, one can see that the condition (3.12) can be replaced by a slightly relaxed assumption:
for some fixed constant \(c_0>0\). Moreover, the conditions (3.5) (with small \(\gamma _1\)) and (3.6) are satisfied if \(\sigma \) and \(\mu (\cdot ,\cdot ,v)\) satisfy (3.12).
Proof
Let T be large enough so that
that is,
This implies
Due to the continuity of \(\varphi \), there exist constants \(\zeta >0\), \(r_0\) and \(r_1\) so that
and
We consider the function
where f is the solution to (3.10) and \(\chi \in C^\infty _c(\mathbb R)\) is a smooth cut-off function satisfying \(0\leqq \chi \leqq 1\) and
Hence z satisfies \( z(T,x,v)=0, \) \(z|_{\partial _-S\Omega _T}=0\), and the nonhomogeneous transport equation
Note that since \(S_0(0,x,v)\), \(f_0(x,v)\) and \(\mu (x,v',\cdot )\) are supported in V, from (3.10), it follows that the initial data
is also supported in V, which satisfies the hypothesis of Proposition 3.1. Now, applying Proposition 3.1 yields that
with
for some constant \(C_1>0\). Next we analyze the first term on the RHS of (3.16). To this end, since \(\partial _t S_0\) is bounded and \(\varphi (t,x)\leqq \varphi (0,x)\), we obtain
by applying the assumption (3.12), where \(C>0\) depends on \(c_3\). In addition, the second term on the RHS of (3.16) is controlled by applying (3.11) and thus we obtain
by noting that \(\partial _t f|_{\partial _-S\Omega _T}=0\), \(\partial _t\chi =0\) in \([0,T-2\zeta ]\cup [T-\zeta ,T]\) and \(\varphi \leqq r_0\) in \([T-2\zeta ,T]\). Furthermore, (3.15) yields
Combining (3.16)-(3.19) together, it follows that
Finally, using the facts that (3.12), \(S_0(0,x,v)\geqq c_1\) in V and \(\varphi (0,x)\geqq r_1>r_0\), the first two terms on the RHS will be absorbed by the LHS once s is sufficiently large. This results in
which ends the proof.\(\square \)
Remark 3.2
In the case that \(f_0\equiv 0\), the term \(K(f_0)\) in z(0, x, v) vanishes automatically. Hence the assumption on the support of \(\mu \) in Theorem 3.3 can be removed.
With Theorem 3.3, we state and prove the uniqueness and stability estimate for the linear coefficients.
Proposition 3.4
Let \(\Omega \) be an open bounded and convex domain with smooth boundary. Suppose that \(\sigma _j\in L^\infty (S\Omega )\) and \(\mu \in L^\infty (S\Omega ^2)\) satisfy (1.3) and (1.4) for \(j=1,\,2\). Let \(N_j:S\Omega \times \mathbb R\rightarrow \mathbb R\) satisfy the assumption (1.7) with \(q^{(k)}\) replaced by \(q^{(k)}_j\) for \(j=1,2\), respectively. For \(\varepsilon >0\), let \(f_j\) be the unique small solution to
and \(F^{(1)}_j=\partial _\varepsilon |_{\varepsilon =0} f_j\), \(j=1,2\). If \(\sigma _1\), \(\sigma _2\) and \(\mu (\cdot ,\cdot ,v)\) satisfy (3.12), then
for \(h\in L^\infty (S\Omega )\) with support in V satisfying \(0<c_1\leqq h\leqq c_2\) in \(\Omega \times V\) for some positive constants \(c_1,\,c_2\) and \((v\cdot \nabla _x)^\beta h\in L^\infty (S\Omega )\), \(\beta =1,\,2\).
In particular, if \(\mathcal {A}_{\sigma _1,\mu ,N_1}(f_0,0)=\mathcal {A}_{\sigma _2,\mu ,N_2}(f_0,0)\) for any \((f_0,0)\in \mathcal {X}^\Omega _\delta \), then
Proof
Let \(w^{(1)}:=F_1^{(1)}- F_2^{(1)}\), where \(F^{(1)}_j\) is the solution to (3.3) with \(\sigma \) replaced by \(\sigma _j\). Then w is the solution to
From the hypothesis, we have that \(F^{(1)}_2(0,x,v)=h\) is strictly positive in \(\Omega \times V\) and also bounded from above in \(S\Omega \). Moreover, \(F_2^{(1)}\) and \(\partial _t F_2^{(1)}\) are in \(L^\infty (S\Omega _T)\). Indeed, one can see this by taking derivative with respect to t on (3.3):
By the well-posedness result in Proposition 2.4, the solution \(\partial _t F^{(1)}_2\) for (3.21) exists in \(L^\infty (S\Omega _T)\) if \((\tilde{h},0)\in \mathcal {X}^\Omega _\delta \). Hence, combining these together, there exist positive constants \(c_1,\,c_2,\,c_3\) surely that
Then \(F^{(1)}_2\), acting as the source \(S_0\), satisfies all the conditions in Theorem 3.3.
In addition, since \(\partial _tF^{(1)}_2\in L^\infty (S\Omega _T)\) and thus is in \(L^2(S\Omega _T)\), from the equation (3.3), we can derive that \(v\cdot \nabla _x F^{(1)}_2\in L^2(S\Omega _T)\). Similarly, we can differentiate (3.21) again with respect to t to derive that \(\partial _t^2 F^{(1)}_2\in L^2(S\Omega _T)\) which leads to \((v\cdot \nabla _x) \partial _tF^{(1)}_2\in L^2(S\Omega _T)\). Applying the same argument, one can also deduce that \(\partial _t^\beta F_1^{(1)}\in L^2(S\Omega _T)\) with \(\beta =1,\,2\), then it implies \((v\cdot \nabla _x)F^{(1)}_1, \,(v\cdot \nabla _x)\partial _t F^{(1)}_1\in L^2(S\Omega _T)\). Hence we obtain that \(w^{(1)}=F_1^{(1)}- F_2^{(1)}\) satisfies the hypothesis
in Theorem 3.3. We finally get \(\Vert \sigma _1-\sigma _2\Vert _{L^2(S\Omega )}\leqq C \Vert \partial _tF^{(1)}_1-\partial _tF^{(1)}_2\Vert _{L^2(\partial _+S\Omega _T)}\), due to Theorem 3.3. Since \(\mathcal {A}_{\sigma _1,\mu ,N_1}=\mathcal {A}_{\sigma _2,\mu ,N_2}\) implies \(\partial _tF^{(1)}_1=\partial _tF^{(1)}_2\) on \(\partial _+S\Omega _T\), the uniqueness \(\sigma _1=\sigma _2\) then holds.\(\square \)
Remark 3.3
In the proposition, we impose the assumption that \((v\cdot \nabla _x)^\beta h\in L^\infty (S\Omega )\) for \(\beta =1,2\), so that the term \(w^{(1)}=F^{(1)}_1-F^{(1)}_2\) has enough regularity for applying Theorem 3.3.
On the other hand, when \(\sigma \) is given, we study below the reconstruction of \(\mu \).
Proposition 3.5
Under the same assumptions as in Proposition 3.4, suppose that \(\sigma \in L^\infty (S\Omega )\) and \(\mu _j\in L^\infty (S\Omega ^2)\) satisfy (1.3) and (1.4) for \(j=1,\,2\). Assume that \(\mu _j:=\tilde{\mu }_j(x,v)p(x,v',v)\) with \(\tilde{\mu }_j\in L^\infty (S\Omega )\) and \(p(x,v',v)\in L^\infty (S\Omega ^2)\) and \(p(x,v',v)\geqq c>0\) for some positive constant c. Let \(f_j\) be the unique small solution to
and \(F^{(1)}_j=\partial _\varepsilon |_{\varepsilon =0} f_j\), \(j=1,2\). If \(\sigma \), \(\tilde{\mu }_1\), \(\tilde{\mu }_2\) and \(p(\cdot ,\cdot ,v)\) satisfy (3.12), then
for \(h\in L^\infty (S\Omega )\) with support in V satisfying \(0<c_1\leqq h\leqq c_2\) in \(\Omega \times V\) for some positive constants \(c_1,\,c_2\) and \((v\cdot \nabla _x)^\beta h\in L^\infty (S\Omega )\), \(\beta =1,\,2\).
In particular, if \(\mathcal {A}_{\sigma ,\mu _1,N_1}(f_0,0)=\mathcal {A}_{\sigma ,\mu _2,N_2}(f_0,0)\) for any \((f_0,0)\in \mathcal {X}^\Omega _\delta \), then
Proof
Let \(w^{(1)}:=F_1^{(1)}- F_2^{(1)}\), where \(F^{(1)}_j\) is the solution to (3.3) with \((\sigma ,\mu )\) replaced by \((\sigma ,\mu _j)\). Then \(w^{(1)}\) is the solution to
The source term is \((K_1-K_2)F_2^{(1)}= (\tilde{\mu }_1-\tilde{\mu }_2)(x,v) \int p(x,v',v) F_2^{(1)}(t,x,v') \textrm{d}v'\). Following a similar argument as of that of the proof of Proposition 3.4, we can deduce that \(\tilde{\mu }_1=\tilde{\mu }_2\) by applying Theorem 3.3.\(\square \)
3.2 Recover the Nonlinear term
From Section 3.1, we have discussed how to reconstruct one unknown linear coefficient from the measurement \(\mathcal {A}_{\sigma ,\mu ,N}\) provided that the other one is given. Therefore, in this section, we suppose that \((\sigma ,\,\mu )\) are recovered and only focus on the reconstruction of the nonlinear term.
The setting is as follows: suppose that \(f_j\), \(j=1,2\) are the solutions to
where the nonlinear term \(N_j\) satisfy \(N_j(x,v,f)= \sum _{k=2}^\infty q_j^{(k)}(x,v){f^k\over k!}\).
To recover \(N_j\), it is sufficient to recover every \(q_j^{(k)}\), \(k\geqq 2\). To this end, we apply the induction argument and also rely on the higher order linearization technique to extract out the information of \(q_j^{(k)}\) from the measurement.
To simplify the notation, we denote the operator \(\mathcal {T}\) by
Recall that \(F^{(k)}_j=\partial ^k_\varepsilon |_{\varepsilon =0} f_j\). When \(k=2\), the function \(F_j^{(2)}\), \(j=1,2\), satisfies the problem
due to the well-posed result \(f_j(t,x,v;0)=0\). Notice that since both \(F^{(1)}_j\), \(j=1,2\) satisfy (3.3) with the same data, the well-posedness for the initial boundary value problem for the transport equation yields that \(F^{(1)}:=F^{(1)}_1=F^{(1)}_2\).
We are ready to recover the coefficient \(q^{(2)}_j\).
Lemma 3.6
Suppose that \(\sigma \in L^\infty (S\Omega )\) and \(\mu \in L^\infty (S\Omega ^2)\) satisfy (1.3) and (1.4). Let \(\sigma \), \(\mu (\cdot ,\cdot ,v)\), \(q_1^{(2)}\) and \(q_2^{(2)}\) satisfy (3.12). If \(h\in L^\infty (S\Omega )\) with support in V satisfying \(0<c_1\leqq h\leqq c_2\) in \(\Omega \times V\) for some positive constants \(c_1,\,c_2\) and \((v\cdot \nabla _x)^\beta h\in L^\infty (S\Omega )\), \(\beta =1,\,2\), then
for some constant \(C>0\).
Proof
Let \(w^{(2)}: = F^{(2)}_1-F^{(2)}_2\) and then \(w^{(2)} \in L^\infty (S\Omega _T)\) satisfies
Following a similiar argument as in Proposition 3.4, Proposition 2.4 yields that \(F^{(1)}\) and \(\partial _t F^{(1)}\) are in \(L^\infty (S\Omega _T)\). Therefore, we can derive that there exist positive constants \(c_1,\,c_2,\,c_3\) so that
Then Theorem 3.3 leads to the estimate (3.25) immediately.\(\square \)
Since \(\mathcal {A}_{\sigma ,\mu ,N_1}=\mathcal {A}_{\sigma ,\mu ,N_2}\) implies \(\partial _t F^{(2)}_1=\partial _t F^{(2)}_2\) on \(\partial _+S\Omega _T\), from Lemma 3.6, it suggests that
when the boundary measurements are the same.
To recover the higher order terms \(q^{(m)}_j\), \(m > 2\), notice that the function \(F^{(m)}_j\), \(j=1,2\) satisfy the problem
where
Notice that the remainder term \(R_{m,j}\) only contains the derivatives of \(f^k_j\) up to order \(m-1\), that is, \(F^{(1)}_j,\ldots , F^{(m-1)}_j\), and also \(q^{(2)}_j,\ldots ,q^{(m-1)}_j\).
Lemma 3.7
Let \(m\geqq 3\). Suppose that \(\mathcal {A}_{\sigma ,\mu ,N_1}(f_0,0)=\mathcal {A}_{\sigma ,\mu ,N_2}(f_0,0)\) for all \((f_0,0)\in \mathcal {X}^\Omega _\delta \) and also \(q^{(k)}_1= q^{(k)}_2\) for \(k=2,\ldots , m-1\). Then for any \(1\leqq k \leqq m-1\), we have
Proof
We proceed by applying the induction argument. First we consider the case \(m=3\). Since \(\mathcal {A}_{\sigma ,\mu ,N_1}=\mathcal {A}_{\sigma ,\mu ,N_2}\), we have \(q^{(2)}:=q^{(2)}_1= q^{(2)}_2\) due to Lemma 3.6. Based on this, \(F^{(2)}_1\) and \(F^{(2)}_2\) satisfy the same initial boundary value problem with the same source \(q^{(2)}(x,v) (F^{(1)})^2\). The well-posedness theorem yields that
Hence the case \(m=3\) holds.
Next by the induction, suppose that if \(q^{(k)}_1= q^{(k)}_2\) for \(k=2,\ldots , m-2\), then \(F^{(k)}_1 = F^{(k)}_2\) in \(S\Omega _T\), \(1\leqq k\leqq m-2\), holds. It is sufficient to show that \(F^{(m-1)}_1 = F^{(m-1)}_2\) when \(q^{(k)}_1= q^{(k)}_2\) for \(k=2,\ldots , m-1\). To this end, we observe that \(F^{(m-1)}_j\) satisfy
It is clear that \(q^{(m-1)}_1 (x,v)(F^{(1)})^{m-1}= q^{(m-1)}_2 (x,v)(F^{(1)})^{m-1}\) and \(R_{m-1,1}=R_{m-1,2}\) by applying \(q^{(k)}_1= q^{(k)}_2\) for \(k=2,\ldots , m-1\) and the definition of \(R_{m-1,j}\), which only contains \(F^{(1)}_j,\ldots , F^{(m-2)}_j\) and \(q^{(2)}_j,\ldots ,q^{(m-2)}_j\). Therefore, \(F^{(m-1)}_j\) satisfies the same initial boundary value problem with the same source, which then leads to \(F^{(m-1)}_1 = F^{(m-1)}_2\) due to the well-posedness theorem again. This completes the proof.\(\square \)
With Lemma 3.6 and Lemma 3.7, we can now stably and uniquely recover all the terms \(q^{(k)}_j\) for all \(k\geqq 2\).
Lemma 3.8
Suppose all conditions in Lemma 3.6 and Lemma 3.7 hold. Let \(q_1^{(m)}\) and \(q_2^{(m)}\) satisfy (3.12) for \(m\geqq 2\). If \(\mathcal {A}_{\sigma ,\mu ,N_1}(f_0,0)=\mathcal {A}_{\sigma ,\mu ,N_2}(f_0,0)\) for all \((f_0,0)\in \mathcal {X}^\Omega _\delta \), then
Proof
For any fixed positive integer \(m\geqq 2\), we will show that \(q_1^{(m)}=q_2^{(m)}\) by applying the induction argument. Recall that we have shown the case \(m=2\), that is, \(q_1^{(2)}=q_2^{(2)}\). By the induction argument, we suppose that \(q_1^{(k)}=q_2^{(k)}\) for all \(2\leqq k\leqq m-1\). The objective is to prove \(q_1^{(m)}=q_2^{(m)}\). From Lemma 3.7, we can derive that \(F^{(k)}_1 = F^{(k)}_2\) in \(S\Omega _T\) for \(k=1,\ldots , m-1\). This implies that \(R_{m,1} = R_{m,2}\) by the definition of \(R_{m,j}\). Hence, we derive that
Note that as discussed above, for sufficiently small and well-chosen data \(h>0\), there exist positive constants \(c_1,\, c_2,\,c_3\) so that
for any integer \(m\geqq 2\). With these estimates, we can apply the Carleman estimate again in Theorem 3.3 to the problem (3.29) to recover the m-th order term, namely,
for some constant \(C>0\). Thus \(q_1^{(m)}=q_2^{(m)}\) follows by the fact that \(\partial _t F^{(m)}_1=\partial _t F^{(m)}_2\) on \(\partial _+S\Omega _T\).\(\square \)
Finally, we prove Theorem 1.1
Proof of Theorem 1.1
Since \(\mathcal {A}_{\sigma _1,\mu ,N_1}=\mathcal {A}_{\sigma _2,\mu ,N_2}\) implies \(\partial _t F^{(m)}_1 = \partial _t F^{(m)}_2\) on \(\partial _+S\Omega _T\) for \(m\geqq 2\), with (1.8), Proposition 3.4 and Lemma 3.8 immediately yield the result.\(\square \)
4 Inverse Problems on Riemannian Manifolds
Let M be the interior of a compact Riemannian manifold \(\overline{M}\) with strictly convex boundary \(\partial M\), of dimension \(\geqq 2\). Let f be the solution to the problem
The objective of the section is to recover \(\sigma \) and N(x, v, f). For this purpose, we will deduce the Carleman estimate and energy estimate on the Riemannian manifolds. Since we could not find the relative results for the transport equation on manifolds in the literature, we prove them in the upcoming subsection. Once these are established, we will then turn to the determination of \(\sigma \) and N.
4.1 Carleman Estimate on Riemannian Manifolds
Let \(\sigma \in L^\infty (SM)\), we denote the operators
where X is the geodesic vector field on SM. Recall that \(\tau _+(x,v)\) is the forward exit time of the geodesic starting at \((x,v)\in SM\). Since the manifold is non-trapping, there exists \(D>0\), such that \(0<\tau _+(x,v)<D\) for all \((x,v)\in SM\). In particular, we let D be the least upper bound for \(\tau _+\) on SM.
Since
and \(\tau _+(\phi _t (x,v)) = \tau _+(x,v)-t\), we have
In particular, \(\tau _+\) is a smooth function on SM.
We define the phase function \(\varphi \) by
for some positive constant \(\beta \), so that
We first deduce the Carleman estimate for the transport equation on a Riemannian manifold.
Theorem 4.1
(Carleman estimate) Let \(\sigma \in L^\infty (SM)\). There exists \(s_0\) and \(C>0\), such that for all \(s\geqq s_0>0\)
for \(u\in H^1(0,T;L^2(SM))\) and \(Xu\in L^2(SM_T)\). Here \(\textrm{d}\Sigma =\textrm{d}\Sigma (x,v)\) the volume form of SM, \(\textrm{d}\xi (x,v)=\left<v,n(x)\right>_{g(x)}\textrm{d}\tilde{\xi }(x,v)\) with n(x) the unit outer normal vector at \(x\in \partial M\) and \(\textrm{d}\tilde{\xi }\) the volume form of \(\partial SM\).
Proof
Now let \(w(t,x,v)=e^{s\varphi (t,x,v)} u(t,x,v)\), we define
We integrate Lw over \([0,T]\times SM\) to get
Notice that \(Lw=e^{s\varphi } P_0 u\) and \(w =e^{s\varphi (t,x,v)} u\), the above calculation gives
To incorporate the absorbing coefficient \(\sigma \), observe that
which yields that
Since \(\sigma \in L^\infty (SM)\), by choosing sufficiently large s, the second term on the left-hand side of the above inequality can be absorbed by the first term on the right-hand side, it follows that there exist \(s_0\) and \(C>0\) (independent of s), for all \(s\geqq s_0\), surely that
\(\square \)
Remark 4.1
It is worth mentioning that different from Proposition 3.1 for the Euclidean case, the Carleman estimate on Riemannian manifolds does not contain the scattering term. This is due to the fact that our weight function \(\varphi =-\beta t-\tau _+\) depends on the direction v. In the proof of Proposition 3.1, see also [38], it is essential that the weight function \(\varphi \) is independent of v, so that the integral \(\int _0^T \int _{SM} e^{2s\varphi }|K(u)|^2\,\textrm{d}\Sigma \textrm{d}t\) can be absorbed by the term \(s^2\int _0^T\int _{SM}e^{2s\varphi }u^2\,\textrm{d}\Sigma \textrm{d}t\) for \(s>0\) sufficiently large.
On the other hand, if we replace \(-\tau _+\) by some globally defined function \(\psi (x)\) independent of v, then \(X\psi (x,v)=\left<v,\nabla \psi (x)\right>_{g(x)}\) can not always be positive. In this case, \((\partial _t+X)\varphi =-\beta +X\psi \) could be negative on \(SM_T\), consequently the Carleman estimate can not hold for such weight function. Therefore, one can not expect to find a globally defined Carleman weight independent of v to prove similar Carleman estimates.
Next, we derive an energy estimate, which will be used to establish uniqueness and stability results for an inverse source problem of linear transport equations on manifolds later.
Lemma 4.2
Suppose \(\sigma \in L^\infty (SM)\) satisfies (1.3). Let \(f_0\in L^\infty (SM)\) satisfy \(X f_0\in L^\infty (SM)\), and \(f_-\in L^\infty (\partial _-SM_T)\) satisfy \(\partial _t f_-\in L^\infty (\partial _-SM_T)\). Suppose that the source term has the form
with \(\widetilde{S}\in L^\infty (SM)\),
for some fixed constant \(c>0\). Let f be the solution to the problem
Then there exists a positive constant C, depending on c, T, and \(\Vert \sigma \Vert _{L^\infty (SM)}\), so that
for any \(0\leqq t\leqq T\) and
for \(f\in H^2(0,T;L^2(SM))\) and \(Xf\in H^1(0,T;L^2(SM))\).
Proof
Taking derivative of the transport equation with respect to the time t gives
Then we multiply \(2\partial _t f\) to (4.5) and integrate over SM to get
where the constant \(C>0\) depends on \(\sigma \) and c. Here we are using the fact that \(\int _{\partial _+ SM}|\partial _t f|^2\, \textrm{d}\xi (x,v)\geqq 0\). We denote \(E(t)=\int _{SM}|\partial _t f|^2(t,x,v)\,\textrm{d}\Sigma \), integrate (4.6) over the time interval (0, t), then
for \(0\leqq t\leqq T\). In the meantime, let \(t=0\) in the transport equation, we obtain \(\partial _tf(0,x,v)+Xf_0+\sigma f_0=S(0,x,v)\), which gives
Therefore
where C depends on \(\sigma ,\,c\) and T. We apply the Gronwall’s inequality to get
which proves the estimate (4.3).
To prove (4.4), we return to (4.6) and integrate it over (0, T), by (4.3), we obtain
where C depends on \(\sigma ,\,c\) and T.\(\square \)
Finally, we will apply the Carleman estimate in Theorem 4.1 and Lemma 4.2 to control the source term in the transport equation.
Theorem 4.3
Suppose \(\sigma \in L^\infty (SM)\) satisfies (1.3). Let \(f_0\in L^\infty (SM)\) satisfy \(X^\beta f_0\in L^\infty (SM)\) with \(\beta =1,\,2\), and \(f_-\) satisfy \(\partial _t f_-\in L^\infty (\partial _-SM_T)\). Suppose that the source term has the form of
with \(\widetilde{S}\in L^\infty (SM)\),
for some fixed constants \(c_1,c_2,c_3>0\). Let f be the solution to the problem
Then there exists a positive constant C, depending on \(c_1\), \(c_2\), \(c_3\) and \(\Vert \sigma \Vert _{L^\infty (SM)}\), so that
for \(f\in H^2(0,T;L^2(SM))\) and \(Xf\in H^1(0,T;L^2(SM))\).
Proof
We choose \(T>D/\beta \), where D is the least upper bounded for \(\tau _+\), then for any \((x,v)\in SM\)
Since \(\varphi \) is continuous, there exist \(\delta >0\) and \(-\beta T<\alpha _1<\alpha _2<-D\) such that
Let \(\chi \in C^\infty _0(\mathbb R)\) be a cut-off function, such that \(0\leqq \chi \leqq 1\) and
Let \(u(t,x,v)=\chi (t) \partial _t f(t,x,v)\), then
Moreover, \(u(T,x,v)=0\) and, from the transport equation,
We apply Theorem 4.1 to u and use \(\varphi (t,x,v)\leqq \varphi (0,x,v)\) for \(t\geqq 0\) in SM,
By Lemma 4.2 and \(0<c_1\leqq S_0(0,x,v)\), it follows that
Since \(\varphi (0,x,v)>\alpha _2\), for s large enough so that \({s\over 2}>CT\), we can derive that
Since \(\alpha _2>\alpha _1\), we choose s large enough such that \({s\over 2} se^{2s\alpha _2}-Ce^{2s\alpha _1}>0\) we have
This completes the proof.\(\square \)
4.2 Reconstruction of the Nonlinear Term on a Riemannian Manifold
Let \(f \equiv f(t,x,v;\varepsilon )\) be the solution to the problem
With the help of the Carleman estimate and linearization techinique, we are ready to show the following two cases of N:
4.2.1 The Case \(N=\sum q^{(k)}{f^k\over k!}\)
We show the first main result in the Riemannian case.
Proof of Theorem 1.3
Consider \(N(x,v,f)=\sum _{k=2}^\infty q^{(k)}(x,v){f^k\over k!}\), we can follow similar arguments as in Section 3.1 and Section 3.2 in the absence of the scattering term for the problem (4.9) to recover the unknown terms. In particular, we can deduce that \(\sigma _1=\sigma _2\) and also \(q_1^{(k)}=q_2^{(k)}\) in SM so that \(N_1(x,v,f)=N_2(x,v,f)\) by utilizing Theorem 4.3.\(\square \)
4.2.2 The Case \(N=qN_0(f)\)
Suppose that the nonlinear term has the form
where \(N_0\) satisfies
and
for a positive integer \(\ell \geqq 2\) and constants \(C_1,C_2>0\), independent of f. We can show as in the proof of Theorem 2.6 that the well-posedness of (4.9) holds under the assumptions (4.10)-(4.11).
We will apply Theorem 4.3 to recover \(\sigma \) and q. The strategy is to recover \(\sigma \) by applying the first linearization and Theorem 4.3. After that, we will employ the second linearization to recover q.
The first linearization of the problem (4.9) with respect to \(\varepsilon \) at \(\varepsilon =0\) is
where recall that \(F^{(1)}(t,x,v) : =\partial _\varepsilon |_{\varepsilon =0} f(t,x,v;\varepsilon )\). The problem now is reduced to studying the inverse coefficient problem for linear transport equations.
Proposition 4.4
Suppose that \(\sigma _j\in L^\infty (SM)\) satisfies (1.3) and \(q_j\in L^\infty (SM)\) for \(j=1,\,2\). Let \(h\in L^\infty (SM)\) satisfy \(0<c_1\leqq h\leqq c_2\) for some positive constants \(c_1,\,c_2\) and \(X^\beta h\in L^\infty (S\Omega )\), \(\beta =1,\,2\). Let \(f_j\) be the solution to the problem (4.9) with \(\sigma \) replaced by \(\sigma _j\) and q replaced by \(q_j\), and \(F^{(1)}_j=\partial _\varepsilon |_{\varepsilon =0} f_j\), \(j=1,\,2\). Then
for some constant \(C>0\).
Proof
To obtain (4.13), let \(w^{(1)}:=F_1^{(1)}- F_2^{(1)}\) and then w is the solution to
Note that \(F^{(1)}_2|_{t=0}=h\) is strictly positive in SM and also satisfies \(\Vert F^{(1)}_2\Vert _{L^\infty (SM_T)}, \Vert \partial _t F^{(1)}_2\Vert _{L^\infty (SM_T)}\leqq C\) for some constant \(C>0\). By Theorem 4.3, it immediately implies (4.13). The proof is complete.\(\square \)
With the establishment of Proposition 4.4, if \(\mathcal {A}_{\sigma _1,N_1}=\mathcal {A}_{\sigma _2,N_2}\), then \(\sigma _1=\sigma _2\). Hence we let \(\sigma :=\sigma _1=\sigma _2\). Now we will recover q in the nonlinear term. To do so, we apply the linearization again at \(\varepsilon =0\) to obtain
Since \(f_j|_{\varepsilon =0}=0\), by assumption (4.11), we get \(\partial _z N_0(0)=0\) and
Notice that since \(F^{(1)}_j\), \(j=1,\,2\) satisfy the same transport equation with the same initial data and boundary data on \(\partial _-SM_T\). The well-posedness theorem yields that
Therefore we denote \(M_0(F^{(1)}):=\partial _\varepsilon ^2|_{\varepsilon =0}N_0(f_j)=\partial _z^2 N_0(0)\, (F^{(1)})^2\) for \(j=1,\,2\). The function \(w^{(2)}:= F^{(2)}_1-F^{(2)}_2\) satisfies
According to Theorem 4.3, we then have the second main result in the Riemannian case.
Proposition 4.5
Suppose that \(\sigma \in L^\infty (SM)\) satisfies (1.3) and \(q_j\in L^\infty (SM)\) for \(j=1,\,2\). Let \(h\in L^\infty (SM)\) satisfy \(0<c_1\leqq h\leqq c_2\) for some positive constants \(c_1,\,c_2\) and \(X^\beta h\in L^\infty (SM)\), \(\beta =1,\,2\). Suppose that the nonlinear term satisfies \(\partial _z^2 N_0(0)>0\). Let \(f_j\) be the solution to the problem (4.9) with q replaced by \(q_j\), and \(F^{(2)}_j=\partial ^2_\varepsilon |_{\varepsilon =0} f_j\), \(j=1,\,2\). Then
for some constant \(C>0\) depending on \(\sigma ,\,h\) and \(N_0\).
Moreover, if \(\mathcal {A}_{\sigma _1,N_1}(f_0,0)=\mathcal {A}_{\sigma _2,N_2}(f_0,0)\) for any \((f_0,0)\in \mathcal {X}^M_\delta \), then
Proof
Since \(0<c_1 \leqq F^{(1)}(0,x,v)=h\leqq c_2\) and \(\partial _z^2 N_0(0)>0\), we have that \(0\leqq \tilde{c}_1\leqq M_0(F^{(1)})\leqq \tilde{c}_2\) for some positive constants \(\tilde{c}_1, \tilde{c}_2\). One the other hand, by applying Proposition 2.4 to the problem (4.12), we obtain that \(\Vert F^{(1)}\Vert _{L^\infty (SM_T)}\leqq C\Vert h\Vert _{L^\infty (SM)}\), and \(\Vert \partial _t F^{(1)}\Vert _{L^\infty (SM_T)}\leqq C\Vert Xh+\sigma h\Vert _{L^\infty (SM)}\leqq C(\Vert Xh\Vert _{L^\infty (SM)}+\Vert h\Vert _{L^\infty (SM)})\). Now we can directly apply Theorem 4.3 to finish the proof.\(\square \)
Proof of Theorem 1.4
Combining Proposition 4.4 and Proposition 4.5, we have the unique determination of \(\sigma \) and N from the boundary data.\(\square \)
Finally we end this section by proving Theorem 1.2, where the transport equation \(\partial _t f+v\cdot \nabla _x f+\sigma f+N(x,v,f)=K(f)\) in \(\Omega \) with N defined as \(N(x,v,f)=q(x,v)N_0(f)\), which is different from the setting in Section 3.
Proof of Theorem 1.2
By utilizing the techniques here and also in Section 3 (in particular, Theorem 3.3), we can conclude the following two results: (1) If \(\mu \) is given, then \(\sigma \) and N are uniquely determined by the boundary data \(\mathcal {A}_{\sigma ,\mu ,N}\).
(2) On the other hand, if \(\sigma \) is given, then \(\mu \) and N are uniquely determined by the boundary data \(\mathcal {A}_{\sigma ,\mu ,N}\).\(\square \)
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Assylbekov, Y.M., Yang, Y.: An inverse radiative transfer in refractive media equipped with a magnetic field. J. Geom. Anal. 25, 2148–2184, 2015
Bal, G.: Inverse transport theory and applications. Inverse Probl. 25, 053001, 2009
Bal, G., Jollivet, A.: Stability estimates in stationary inverse transport. Inverse Probl. Imaging 2, 427–454, 2008
Bal, G., Jollivet, A.: Stability estimates for time-dependent inverse transport. SIAM J. Math. Anal. 42(2), 679–700, 2010
Bal, G., Jollivet, A.: Generalized stability estimates in inverse transport theory. Inverse Probl. Imaging 12(1), 59–90, 2018
Bal, G., Monard, F.: Inverse transport with isotropic time-harmonic sources. SIAM J. Math. Anal. 44(1), 134–161, 2012
Balehowsky, T., Kujanpää, A., Lassas, M., Liimatainen, T.: An inverse problem for the relativistic Boltzmann equation. Commun. Math. Phys. 396, 983–1049, 2022
Bugheim, A., Klibanov, M.: Global uniqueness of class of multidimensional inverse problems. Soviet Math. Doklady 24, 244–247, 1981
Carleman, T.: Sur un problème d’unicité pour les systèmes d’équations aux derivées partielles à deux variables independentes. Ark. Mat. Astr. Fys. 2B, 1–9, 1939
Chen, I.-K., Kawagoe, D.: Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography. Inverse Probl. Imaging 13(2), 337–351, 2019
Chen, X., Lassas, M., Oksanen, L., Paternain, G.: Detection of Hermitian connections in wave equations with cubic non-linearity. J. Eur. Math. Soc. 24(7), 2191–2232, 2021
Choulli, M., Stefanov, P.: Scattering inverse pour l’équation du transport et relations entre les opérateurs de scattering et d’albédo. C. R. Acad. Sci. Paris 320, 947–952, 1995
Choulli, M., Stefanov, P.: Inverse scattering and inverse boundary value problems for the linear Boltzmann equation. Commun. Partial Differ. Equ. 21, 763–785, 1996
Choulli, M., Stefanov, P.: Reconstruction of the coefficients of the stationary transport equation from boundary measurements. Inverse Probl. 12, L19–L23, 1996
Choulli, M., Stefanov, P.: An inverse boundary value problem for the stationary transport equation. Osaka J. Math. 36, 87–104, 1998
Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 6. Springer, 2000
Feizmohammadi, A., Oksanen, L.: An inverse problem for a semi-linear elliptic equation in Riemannian geometries. J. Differ. Equ. 269(6), 4683–4719, 2020
Gaitan, P., Ouzzane, H.: Inverse problem for a free transport equation using Carleman estimates. Appl. Anal. 93, 1073–1086, 2014
Gölgeleyen, F., Yamamoto, M.: Stability for some inverse problems for transport equations. SIAM J. Math. Anal. 48(4), 2319–2344, 2016
Kang, H., Nakamura, G.: Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map. Inverse Probl. 18, 1079–1088, 2002
Klibanov, M., Pamyatnykh, S.: Lipschitz stability of a non-standard problem for the nonstationary transport equation via a Carleman estimate. Inverse Probl. 22, 881–890, 2006
Klibanov, M., Pamyatnykh, S.: Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate. J. Math. Anal. Appl. 343, 352–365, 2008
Krupchyk, K., Uhlmann, G.: Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities. Math. Res. Lett. 27(6), 1801–1824, 2020
Krupchyk, K., Uhlmann, G.: A remark on partial data inverse problems for semilinear elliptic equations. Proc. AMS 148(2), 681–685, 2020
Kurylev, Y., Lassas, M., Uhlmann, G.: Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations. Invent. Math. 212(3), 781–857, 2018
Lai, R.-Y., Li, Q.: Parameter reconstruction for general transport equation. SIAM J. Math. Anal. 52(3), 2734–2758, 2020
Lai, R.-Y., Lin, Y.-H.: Inverse problems for fractional semilinear elliptic equations. Nonlinear Anal. 216, 112699, 2022
Lai, R.-Y., Ohm, L.: Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations. Inverse Probl. Imaging 16(2), 305–323, 2022
Lai, R.-Y., Ren, K., Zhou, T.: Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption. SIAM J. Appl. Math., 2022. https://doi.org/10.1137/21M1436178
Lai, R.-Y., Uhlmann, G., Yang, Y.: Reconstruction of the collision kernel in the nonlinear Boltzmann equation. SIAM J. Math. Anal. 53(1), 1049–1069, 2021
Lai, R.-Y., Zhou, T.: Partial data inverse problems for nonlinear magnetic Schrödinger equations. Accepted in Mathematical Research Letters, arXiv:2007.02475, 2020.
Lai, R.-Y., Zhou, T.: An inverse problem for non-linear fractional magnetic Schrödinger equation. J. Differ. Equ. 343(15), 64–89, 2023
Lassas, M., Liimatainen, T., Lin, Y.-H., Salo, M.: Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations. Revista Matemática Iberoamericana, 2020. https://doi.org/10.4171/rmi/1242
Lassas, M., Liimatainen, T., Lin, Y.-H., Salo, M.: Inverse problems for elliptic equations with power type nonlinearities. Journal de Mathématiques Pures et Appliqués 145, 44–82, 2021
Lassas, M., Uhlmann, G., Wang, Y.: Inverse problems for semilinear wave equations on Lorentzian manifolds. Commun. Math. Phys. 360(2), 555–609, 2018
Li, L., Ouyang, Z.: Determining the collision kernel in the Boltzmann equation near the equilibrium. Proc. Am. Math. Soc. 151, 4855–4865, 2023
Liimatainen, T., Lin, Y.-H., Salo, M., Tyni, T.:Inverse problems for elliptic equations with fractional power type nonlinearities. arXiv:2012.04944, 2020.
Machida, M., Yamamoto, M.: Global Lipschitz stability in determining coefficients of the radiative transport equation. Inverse Probl. 30, 035010, 2014
McDowall, S., Stefanov, P., Tamasan, A.: Gauge equivalence in stationary radiative transport through media with varying index of refraction. Inverse Probl. Imaging 4, 151–167, 2010
McDowall, S., Stefanov, P., Tamasan, A.: Stability of the gauge equivalent in stationary inverse transport. Inverse Probl. 26, 025006, 2010
McDowall, S., Stefanov, P., Tamasan, A.: Stability of the gauge equivalent classes in inverse stationary transport in refractive media. Contemp. Math. 559, 85–100, 2011
McDowall, S.R.: An inverse problem for the transport equation in the presence of a Riemannian metric. Pacific J. Math. 216, 303–326, 2004
Ren, K., Zhang, R.: Nonlinear quantitative photoacoustic tomography with two-photon absorption. SIAM J. Appl. Math. 78(1), 479–503, 2018
Ren, K., Zhong, Y.: Unique determination of absorption coefficients in a semilinear transport equation. SIAM J. Math. Anal. 53(5), 1, 2021
Stefanov, P.: Inverse problems in transport theory, vol. 47. Inverse Problems; MSRI Publications, edited by G. Uhlmann, Inside Out, 2003
Stefanov, P., Uhlmann, G.: Optical tomography in two dimensions. Methods Appl. Anal. 10, 1–9, 2003
Stefanov, P., Zhong, Y.: Inverse boundary problem for the two photon absorption transport equation. SIAM J. Math. Anal. 54(3), 1, 2022
Wang, J.-N.: Stability estimates of an inverse problem for the stationary transport equation. Annales de l’I.H.P., section A 70(5), 473–495, 1999
Zhao, H., Zhong, Y.: Instability of an inverse problem for the stationary radiative transport near the diffusion limit. SIAM J. Math. Anal. 51(5), 3750–3768, 2019
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Ru-Yu Lai is partially supported by the National Science Foundation through grant DMS-2006731 and DMS-2306221. Gunther Uhlmann is partly supported by NSF, a Simons Fellowship, a Walker Family Professorship at UW, and a Si Yuan Professorship at IAS, HKUST. Hanming Zhou is partly supported by the NSF grant DMS-2109116.
Additional information
Communicated by T.-P. Liu.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lai, RY., Uhlmann, G. & Zhou, H. Recovery of Coefficients in Semilinear Transport Equations. Arch Rational Mech Anal 248, 62 (2024). https://doi.org/10.1007/s00205-024-02007-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00205-024-02007-6