In the present paper we study the nonlinear system *u*_{t} + [*ϕ*(*u*)]_{x} + *v* = 0, *v*_{t} + *ψ*(*u*)*v*_{x} = 0 as a model for the one-dimensional dynamics of dark matter. We prove that under certain conditions this system, such as the Gurevich-Zybin system, can also explain why the observed rotation speed (relative to the galactic center) of stars near galactic halos do not coincide with what it is expected in classical mechanics. The solutions are obtained in fully explicit formulas, in a convenient space of distributions, without using any result within the classical framework. For such purpose we use the *α*-solution concept which is defined within a product of distributions. Such a concept generalizes the classical solution concept and for evolution equations may also be seen as an extension of the weak solution concept to the nonlinear setting.

## I. INTRODUCTION AND CONTENTS

*t*∈ [0, +∞[ is the time variable,

*u*=

*u*(

*x*,

*t*) and

*v*=

*v*(

*x*,

*t*) represent the state variables and $\varphi ,\psi :C\u2192C$ are entire functions that take real values on the real axis. The variable

*u*will be considered as an hydrodynamic speed and the variable

*v*will be seen in such a way that

*v*

_{x}(

*x*,

*t*) =

*ρ*(

*x*,

*t*), being

*ρ*a density of matter.

*ϕ*,

*ψ*that contains the nonlinear Gurevich-Zybin system

^{1}and was derived from a hydrodynamic system in a book by Jeans

^{2}whose first edition goes back to 1928. See also, for example, Refs. 3–6, to account for several aspects under which this system was studied. A large number of papers and books have been published regarding the dark matter and the problem of galactic rotation. As an introduction we refer Refs. 7 and 8.

*u*,

*v*of the system (1) and (2) to the conditions

*m*> 0 and

*H*,

*δ*stand respectively for the Heaviside function and the Dirac delta measure supported at the origin. The solution (

*u*,

*v*) of problem (1), (2), (5), and (6) will be considered in a convenient space

*W*of pairs of functions (

*u*,

*v*) defined by

*C*

^{1}-functions. In a sense that will be explained we will prove that, if

*ψ*′(

*u*

_{1}) ≠ 0, problem (1), (2), (5), and (6) has an unique solution in

*W*, given by

*m*> 0, that moves according to the law (7) in a one-dimensional universe. From a physical viewpoint this will be interpreted in Sec. VIII in the setting of the problem of galactic rotation.

Usually, the solutions of problems like this one are given by weak limits of sequences of continuous functions. Frequently, such limits cannot be substituted into the equations mainly owing to the difficulties of multiplying distributions and, sometimes, these processes do not yield to mathematically consistent solutions (see Ref. 9, Sec. II). These difficulties can be overcome, as we will explain.

In our framework, the product of two distributions is a distribution that depends on the choice of a function *α* encoding the indeterminacy inherent to such products. This indeterminacy generally is not avoidable and in many cases it has a physical meaning; concerning this point see Refs. 10–13. As a consequence, the solutions of differential equations with such products may depend, or not, of *α*. We call such solutions *α*-solutions and, in each case, they must be defined in such a way that any classical solution is an *α*-solution for any *α*. Thus, the *α*-solutions can be seen as generalized classical solutions and, for evolution equations, they can also be considered as extensions of the concept of weak solution in the nonlinear setting (see Ref. 14, p. 868).

When the solutions depend on *α* the future behavior of the model cannot be fully predicted. This fact might be due to physical features omitted in the formulation of the model with the goal of simplifying it. See Refs. 15–17 concerning this point. In spite of the possible *α*-dependence, it is worthwhile to stress that for many problems (like our one) the *α*-solutions do not depend on *α*. Certainly this is one of the reasons for which several authors begun to adopt this method (see Refs. 18–30). Another reason regards to important simplifications: the Rankine-Hugoniot shock conditions and their multiple generalizations are not necessary and, as much as we know, the final result is the same (see Refs. 21, 31, and 32). Thus, even in the cases where the classical results can be applied, it is certainly more easy to use *α*-solutions if the operations with distributions involved are perfectly defined.

Interestingly, in a recent paper,^{23} Pang *et al.* compared the exact *α*-solutions of a totally degenerate system of conservation laws, with the numerical solutions obtained by using the Nessyahu-Tadmor scheme: the *α*-solutions coincide with the corresponding numerical solutions. For more problems where the concept of *α*-solution was also applied the reader may see Refs. 33–36 and other works of the author.

## II. THE MULTIPLICATION OF DISTRIBUTIONS

The present section concerns some formulas from our theory of distributional products that are of interest in the sequel. To get a general view of our distributional products see Ref. 37, Secs. 2 and 3 and also Ref. 38, Secs. 2–4. Details are given in Ref. 39.

Let $D$ be the space of indefinitely differentiable complex-valued functions defined on $R$, with compact support, and let $D\u2032$ be the space of Schwartz distributions. In our theory, each function $\alpha \u2208D$ with $\u222b\u2212\u221e+\u221e\alpha =1$ affords a general product $T\alpha \u0307S\u2208D\u2032$ of $T,S\u2208D\u2032$.

*τ*

_{a}denotes the usual translation operator in distributional sense. In general, associativity and commutativity do not hold. Recall that in the setting of the classical products of distributions, the commutative property is a convention inherent to the definition of such products and the associative property does not hold in general (see the monograph of Schwartz

^{40}, pp. 117, 118, 121, where these products are defined). The usual differential rules are satisfied, including the Leibniz formula, which must be written in the form

*D*is the derivative operator in distributional sense. This general product is not consistent with the Schwartz products of distributions with functions but it is possible to define certain

*α*-products in order to recover that consistency. This happens with the

*α*-product

*p*∈ {0, 1, 2, …, ∞}, $D\u2032p$ is the space of distributions of order $\u2264p$ in the sense of Schwartz ($D\u2032\u221e$ means $D\u2032$), $D\mu \u2032$ is the space of distributions whose support has Lebesgue measure zero,

*Tβ*is the usual Schwartz product of a $D\u2032p$-distribution with a

*C*

^{p}-function and (

*T*∗

*α*)

*f*is the usual product of a

*C*

^{∞}-function with a distribution. For instance, if

*β*is a continuous function, we have for any

*α*,

*α*-product (10) is consistent with all Schwartz products of $D\u2032p$-distributions with

*C*

^{p}-functions if the

*C*

^{p}-functions are placed at the right-hand side. It also keeps the bilinearity and satisfies (8) and (9), this one clearly under certain natural conditions; for $T\u2208D\u2032p$ we must suppose $S\u2208Cp+1\u2295D\mu \u2032$.

*α*-products. The following formula was constructed this way (for details see Ref. 41, Sec. 2),

*Tw*is the usual pointwise product of $T\u2208D\u2032\u22121$ with $w\u2208Lloc1$. Thus, locally,

*T*can be read as a function of bounded variation and $D\u2032\u22121$ as the space of locally bounded variation functions. For instance, since $H\u2208D\u2032\u22121$ and $H=H+0\u2208Lloc1\u2295D\mu \u2032$, we have

*α*. More generally, if $T\u2208D\u2032\u22121$ and $S\u2208Lloc1$ then $T\alpha \u0307S=TS$, for any

*α*, because by (12) we can write

*α*-product that is computed by the formula

*DY*=

*T*. The products $Y\alpha \u0307S$ and $Y\alpha \u0307(DS)$ are supposed to be computed by (10) or (12). The value of $T\alpha \u0307S$ given by (14) is independent of the choice of

*Y*such that

*DY*=

*T*(see Ref. 41, p. 1004 and Ref. 42, Sec. 2, for details). For instance, using (11), (13), and (14), we have for any

*α*,

*α*.

The compatibility of (10), (12) and (14) is effective, that is, if the same *α*-product can be computed by two different formulas, they give the same result.

Also in general, $supp(T\alpha \u0307S)\u2282suppS$ as it happens for the usual product of functions, but it may happen that $supp(T\alpha \u0307S)\u2284suppT$.

## III. THE *α*-SOLUTIONS FOR THE SYSTEM (1) AND (2)

Let *I* be an interval of $R$ with more that one point, and let $F(I)$ be the space of continuously differentiable maps $u\u0303:I\u2192D\u2032$ in the sense of the usual topology of $D\u2032$. For *t* ∈ *I*, the notation $[u\u0303(t)](x)$ is sometimes used to emphasize that the distribution $u\u0303(t)$ acts on functions $\xi \u2208D$ depending on *x*.

Let Σ(*I*) be the space of functions $u:R\xd7I\u2192R$ such that:

for each

*t*∈*I*, $u(x,t)\u2208Lloc1(R)$;$u\u0303:I\u2192D\u2032$, defined by $[u\u0303(t)](x)=u(x,t)$ belongs to $F(I)$.

*I*) into $F(I)$ identifies any function in Σ(

*I*) with a certain map in $F(I)$. Since $C1(R\xd7I)\u2282\Sigma (I)$, we can write the inclusions

*u*with $u\u0303$ and

*v*with $v\u0303$, the system (1) and (2) can be read as follows:

*Given* *α**, the pair* $(u\u0303,v\u0303)\u2208F(I)\xd7F(I)$ *will be called an* *α**-solution of the system (16) and (17) on* *I**, if* $\varphi \u25e6u\u0303(t)$*,* $\psi \u25e6u\u0303(t)$ *and the* *α**-product* $[\psi \u25e6u\u0303(t)]\alpha \u0307[Dv\u0303(t)]$ *are well defined distributions and this system is satisfied for all* *t* ∈ *I*.

We have the following results:

Note that, by a classical solution of (1) and (2) on $R\xd7I$, we mean a pair of *C*^{1}-functions (*u*, *v*) that satisfies (1) and (2) on $R\xd7I$.

For the proofs, it is enough to observe that any *C*^{1}-function *u* can be read as a continuously differentiable function $u\u0303\u2208F(I)$ defined by $[u\u0303(t)](x)=u(x,t)$ and to use the consistency of the *α*-products with the classical Schwartz products of distributions with functions.

*α*-products are not, in general, commutative. However, all we said for the systems (1), (2), (16), (17) is also valid for the systems (1), (2), (16), and (18) including Definition 1. Thus, and taking advantage of this situation we introduce the following definition that further extends the concept of

*α*-solution.

As a consequence an *α*-solution $(u\u0303,v\u0303)\u2208F(I)\xd7F(I)$ in this sense read, for each *t*, as a pair of distributions (*u*, *v*), affords a consistent extension of the concept of a classical solution for the system (1) and (2). Thus, and for short, we also call the pair (*u*, *v*) an *α*-solution of the system (1) and (2) on $R\xd7I$.

## IV. THE PROBLEM (1), (2), (5), AND (6) IN *W*

*I*= [0, +∞[ and (

*u*,

*v*) ∈

*W*. Remember that we assume

*ϕ*,

*ψ*, entire functions that take real values on the real axis and $u1,u2,m\u2208R$ with

*m*> 0. When we read this problem in $F(I)$, the identification $u\u2192u\u0303$ allows us to replace the system (1) and (2) by the system (16) and (17) and conditions (5) and (6) respectively by the following ones

*α*-solutions of the problem (16), (17), (19), and (20) in the space $W\u0303\u2282F(I)\xd7F(I)$ defined as follows: $(u\u0303,v\u0303)\u2208W\u0303$, if and only if,

*C*

^{1}-functions $a,b,g,\gamma :I\u2192R$ and all

*t*∈

*I*. Clearly, $W\u0303$ is the set corresponding to

*W*which was defined in the introduction.

*Given* *α**, the problem (16), (17), (19), and (20) with* *ψ*′(*u*_{1}) ≠ 0 *has an* *α**-solution* $(u\u0303,v\u0303)\u2208W\u0303$ *if and only if the following two conditions are satisfied:*

$\u222b\u2212\u221e0\alpha =\varphi \u2032\u2032(u1)2\psi \u2032(u1)$

*,*$\varphi (z)\u2212\varphi (u1)=(z\u2212u1)\psi (u1)+\varphi \u2032\u2032(u1)2\psi \u2032(u1)[\psi (z)\u2212\psi (u1)]$

*, for all**z*.

*In this case, the*

*α*

*-solution is given by*

*where*

*γ*

*is given by*

*This*

*α*

*-solution is unique in*$W\u0303$

*, does not depend on*

*α*

*and clearly*

*g*(0)

*τ*

_{γ(0)}

*δ*=

*mδ*, which means that

*γ*(0) = 0 and

*g*(0) =

*m*. Then, from (26) we get

*a*(0) +

*b*(0)

*H*=

*u*

_{1}+ (

*u*

_{2}−

*u*

_{1})

*H*and

*a*(0) =

*u*

_{1}and

*b*(0) =

*u*

_{2}−

*u*

_{1}follows.

*t*, $u\u0303(t)$ is the function defined by

*t*,

*α*-solution of (16) and (17) if and only if, for all

*t*, the following five conditions hold:

*a*′(*t*) = 0,*b*′(*t*) +*g*(*t*) = 0,−

*b*(*t*)*γ*′(*t*) +*ϕ*[*a*(*t*) +*b*(*t*)] −*ϕ*[*a*(*t*)] = 0,*g*′(*t*) = 0,$\u2212g(t)\gamma \u2032(t)+\psi [a(t)]g(t)+{\psi [a(t)+b(t)]\u2212\psi [a(t)]}g(t)\u222b\u2212\u221e0\alpha =0$.

*a*(

*t*) =

*u*

_{1},

*g*(

*t*) =

*m*and

*b*(

*t*) = −

*mt*+

*u*

_{2}−

*u*

_{1}. From (3′) and (5′) we get respectively

*η*is an entire function and

*η*(−

*mt*+

*u*

_{2}) = 0 for all

*t*. Since the zeros of an entire function

*η*≠ 0 are always isolated points we conclude that

*η*= 0, that is,

*z*, and taking the second derivative of both sides we get

*z*by

*u*

_{1}we conclude that

*t*. Clearly, this

*α*-solution is unique in $W\u0303$ and does not depend on

*α*. The value of $\rho \u0303(t)$ follows from (22). The statement is proved.■

Taking in Theorem 5 the first derivative of both sides of (b) and replacing *z* by *u*_{1}, we get *ϕ*′(*u*_{1}) = *ψ*(*u*_{1}) and a simple and practical necessary condition for the existence of an *α*-solution follows:

*α*-products are not, in general, commutative we may analyze what happens to Theorem 5 if in (17) we replace $[\psi \u25e6u\u0303(t)]\alpha \u0307[Dv\u0303(t)]$ by $[Dv\u0303(t)]\alpha \u0307[\psi \u25e6u\u0303(t)]$. Thus, and using (15), we must write instead of (28),

*Given* *α**, the problem (1), (2), (5), and (6), with* *ψ*′(*u*_{1}) ≠ 0*, has an* *α**-solution* (*u*, *v*) ∈ *W* *if and only if the following two conditions are satisfied:*

$\u222b\u2212\u221e0\alpha =\varphi \u2032\u2032(u1)2\psi \u2032(u1)$

*or*$\u222b0+\u221e\alpha =\varphi \u2032\u2032(u1)2\psi \u2032(u1)$*,*$\varphi (z)\u2212\varphi (u1)=(z\u2212u1)\psi (u1)+\varphi \u2032\u2032(u1)2\psi \u2032(u1)[\psi (z)\u2212\psi (u1)]$

*, for all**z*.

*In this case, the*

*α*

*-solution is given by*

*where*

*γ*

*is given by*

*This*

*α*

*-solution is unique in*

*W*

*, does not depend on*

*α*

*and clearly*

Therefore, as we said in the introduction, the density of matter *ρ*(*x*, *t*) shows a particle of mass *m* > 0 that moves according to the law (31).

*Let us consider the Gurevich-Zybin system (3) and (4) with the unknowns* (*u*, *v*) *subjected to conditions (5) and (6)*. *Clearly, we have* $\varphi (z)=z22$ *and* *ψ*(*z*) = *z**. For this particular case, we have* *ψ*′(*u*_{1}) = 1 ≠ 0 *and since* $\varphi \u2032\u2032(u1)2\psi \u2032(u1)=12$*, (a) turns out to be simply* $\u222b\u2212\u221e0\alpha =12$*, because* $\u222b0+\u221e\alpha =1\u2212\u222b\u2212\u221e0\alpha =12$*. Also it is easy to see that (b) is satisfied for any* *z**, and* $\gamma (t)=u1+u22t\u221214mt2$*. As a consequence, we conclude:*

*Given*

*α*

*, the Gurevich-Zybin system (3) and (4) with the unknowns subjected to conditions (5) and (6) has an*

*α*

*-solution*(

*u*,

*v*) ∈

*W*

*if and only if*$\u222b\u2212\u221e0\alpha =12$

*. In this case, the*

*α*

*-solution is unique, it is independent of*

*α*

*and it is given by*

In Ref. 36, Theorem 6, p. 7, the expressions for *u* and *v* are not exactly the same. Unfortunately, in this paper, the author committed a mistake; in the Proof of Theorem 5, p. 5, line −3, instead of $u\u0303(t)\alpha \u0307u\u0303(t)=a2(t)+[b2(t)\u2212a2(t)]\tau \gamma (t)H$, it must be $u\u0303(t)\alpha \u0307u\u0303(t)=a2(t)+[2a(t)b(t)+b2(t)]\tau \gamma (t)H$. Thus, after the correction of some expressions, we get the corrected Theorem 5 of Ref. 36 where *u* and *v* are given exactly by (32) and (33) and the density of matter by (34). Fortunately, this mistake does not modify the essence of the physical interpretation we have done in Ref. 36, as we will see in Sec. VI, (I).

## V. THE PROBLEM OF GALACTIC ROTATION

The speed of rotation of stars or gas within galaxies should decrease as a function of the distance of that matter to the center of such galaxies, as it is expected in classical mechanics. Instead, from astronomical observations regarding the “curves of galactic rotation” such speed of rotation remains approximately constant when going out to large distances from the galactic center. In 1937, Fritz Zwicky,^{8} applying the Virial Theorem of classical mechanics, was the first to discover that galaxies may contain much more mass then that which can be seen. Thus, to explain data, it was postulated that enormous halos exist involving galaxies made of unknown “dark matter” that does not interact with the usual baryonic matter or with the electromagnetic field (we cannot see the dark matter), but interact only with the gravitational field.

Astronomers were skeptical about dark matter until the appearing of the enormous work developed by Vera Rubin, that forced a large part of the astronomical community to take it seriously. For the interested reader we refer as an introduction.^{7,43} In what follows we will interpret our results.

## VI. PHYSICAL INTERPRETATION

*P*near the halo of a galaxy so that the hydrodynamic speed at

*P*in the instant

*t*= 0 be

*u*(

*x*, 0) = 0. This means that

*u*

_{1}=

*u*

_{2}= 0 in (5). For the

*x*-axis of our model we will chose a straight line belonging to the galactic plane, passing by

*P*and being ortogonal to the position vector of

*P*relative to the galactic center. This way,

*P*will be the origin of our

*x*-axis. Then, from (32) and (33) the hydrodynamic speed

*u*and the density of mass

*ρ*=

*v*

_{x}are given, by

*ρ*can be interpreted as the motion of a “dark particle” of mass

*m*> 0 that moves on the

*x*-axis according to the law $\gamma (t)=\u221214mt2$, that is, with constant acceleration $\gamma \u2032\u2032(t)=\u2212m2$. This acceleration may be seen, as the result of a constant “dark force”

*F*, of gravitational nature, that acts on the baryonic matter in negative sense, that is, in the same sense of the hydrodynamic speed

*u*(0,

*t*), for each

*t*> 0. As a consequence, the gravitational pull

*F*tends to increase (in absolute value) the speed of the matter at

*P*, that is, near the galactic halo, as observed in the curves of galactic rotation!

From Theorem 7 our general problem (1), (2), (5), and (6) has an *α*-solution (*u*, *v*) ∈ *W*, with *u*_{1} = *u*_{2} = 0, if and only if the following two conditions are satisfied:

$\u222b\u2212\u221e0\alpha =\varphi \u2032\u2032(0)2\psi \u2032(0)$ or $\u222b0+\u221e\alpha =\varphi \u2032\u2032(0)2\psi \u2032(0)$;

$\varphi (z)\u2212\varphi (0)=z\psi (0)+\varphi \u2032\u2032(0)2\psi \u2032(0)[\psi (z)\u2212\psi (0)]$ for all

*z*.

*u*and the density

*ρ*=

*v*

_{x}in this case are given by

*γ*(

*t*) is given by

*F*(

*t*) spanned by this acceleration acts to increase (in absolute value) the speed of the matter at

*P*if there exists an interval

*J*⊂ ]0, +∞[, with more than one point, such that for all

*t*∈

*J*,

*F*(

*t*) has the same sense of the hydrodynamic speed

*u*(0,

*t*), which means that

*u*(0,

*t*)

*γ*″(

*t*) > 0 for all

*t*∈

*J*, that is, if

*t*∈

*J*, where

*γ*(

*t*) is given by (35). Summing up, we have the following result:

*The problem (1), (2), (5), and (6) with* *u*_{1} = *u*_{2} = 0 *and* *ψ*′(0) ≠ 0 *has an* *α**-solution* (*u*, *v*) ∈ *W* *if and only if conditions (a′) and (b′) are satisfied. In such case, if there exists an interval of time* *J**, with more than one point, such that for all* *t* ∈ *J* *(36) is satisfied, the gravitational pull* *F*(*t*) *spanned by the acceleration* *γ*″(*t*)*, has, the same sense of the hydrodynamic speed* *u*(0, *t*)*, and the* “*dark force*” *F*(*t*) *acts to increase the speed of the matter at* *P* *during the interval of time* *J*.

*Let us consider the problem*

*with the unknowns*

*u*,

*v*

*subjected to the conditions*

*with*

*m*> 0

*. Regarding the general problem, we have*

*ϕ*(

*z*) =

*z*

^{4}−

*z*

^{2}

*,*

*ψ*(

*z*) =

*z*−

*z*

^{3}

*,*

*u*

_{1}=

*u*

_{2}= 0

*, and*

*ψ*′(0) = 1 ≠ 0

*. Hence, this problem has an*

*α*

*-solution*(

*u*,

*v*) ∈

*W*

*if and only if conditions (a′) and (b′) of Theorem 11 are satisfied. Since (b′) is trivially satisfied, we conclude that for all*

*α*

*satisfying (a′), that is, for all*

*α*

*such that*

*the problem (37)–(40) has an*

*α*

*-solution*(

*u*,

*v*) ∈

*W*

*given by*

*where*

*γ*(

*t*)

*is given by (35), that is,*

*Hence, assuming that the system (37) and (38) rules the dynamic of dark matter (which from a theoretical viewpoint can be true only approximately), the density of matter*

*ρ*(

*x*,

*t*) =

*v*

_{x}(

*x*,

*t*) =

*mδ*[

*x*−

*γ*(

*t*)]

*shows a star of mass*

*m*

*that moves according to the law (41). Thus, (36) turns out to be*

*and in the interval of time*$J=]2m,+\u221e[$

*, or any subinterval with more than one point, (36) is satisfied. Therefore, from the instant*$t=2m$

*on, the gravitational pull*

*F*(

*t*) =

*mγ*″(

*t*)

*acts in order to increase the speed of the star.*

*J*= ]0, +∞[ and the dark force

*F*(

*t*) spanned by the acceleration $\gamma \u2032\u2032(t)=\u2212m2$ is constant. Therefore, the possibility that the curves of galactic rotation do not adjust to this model during all this interval of time is real. Possibly, in the general model (1) and (2), a convenient choice of the functions

*ϕ*,

*ψ*can adjust this model to the curves of galactic rotation and better predict future events.

## ACKNOWLEDGMENTS.

The author is grateful to the referee for the suggestions which clearly improved the text.

The present research was supported by FCT, UID/MAT/04561/2019.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**C. O. R. Sarrico**: Conceptualization (equal); Investigation (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## REFERENCES

*Astronomy and Cosmology*

*δ*-shock waves in isentropic fluids

*δ*-shock waves in the

*p*-system: A distributional product approach

*δ*-waves in the nonlinear Kline-Gordon equation

*δ*′ wave solution to a totally degenerate system of conservation laws

*δ*′ waves in a model of three partial differential equations

*δ*′-waves in the zero pressure gas dynamic system